# Stochastics and Finance Seminar

Seminars are held in person and on zoom. Please direct enquiries about this seminar to Anna Aksamit

## 2024

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** Friday, January 12, 2:00pm, ** Carslaw AGR

Speaker: Bin Zou (University of Connecticut)

Title: Strategic Loss Reporting

## Abstract:

We study a strategic loss reporting problem for an insured who seeks an optimal strategy to dictate what losses she should report to the insurer, with the purpose to maximize her expected wealth (risk-neutral case) or expected utility of wealth (risk-averse case). To explore the potential advantage of underreporting her insurable losses, the insured follows a barrier strategy and only reports losses above the barrier to the insurer. Under a tractable 2-class bonus-malus model and given that the insured has purchased full insurance, we obtain a unique equilibrium reporting strategy in (semi)closed form. We find that the two equilibrium barriers are the same and strictly greater than zero, offering a theoretical explanation for the underreporting of insurable losses. If time allows, we will discuss the generalization to the case where the insured has deductible insurance and may even take into account strategic reporting in the decision of contract deductibles.

This talk is based on joint work with Jingyi Cao, Dongchen Li, and Jenny Young.

## 2023

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** Tuesday, October 24, 2:00pm, ** Carslaw AGR and zoom

Speaker: Noufel Frikha (Panthéon-Sorbonne University)

Title: Well-posedness of McKean-Vlasov SDEs, related PDEs on the Wasserstein space and some new quantitative estimates for propagation of chaos

## Abstract:

In this presentation, I will discuss recent findings concerning the well-posedness of some non-linear stochastic differential equations (considering the McKean-Vlasov framework) in both weak and strong senses. These results go beyond what can be derived from standard Cauchy-Lipschitz theory, as outlined in works such as Sznitman's monograph. Then, I will show how the underlying noise regularizes the equation and allows to prove that the transition density of the dynamics exists and is smooth, especially in the measure direction, under the uniform ellipticity assumption. Such smoothing effects then in turn allow to establish the existence and uniqueness for the Cauchy problem associated to the Kolmogorov PDE stated on the Wasserstein space with irregular terminal condition and source term. This PDE on an infinite dimensional space plays a key role in order to derive new quantitative estimates of propagation of chaos for the mean-field approximation by systems of interacting particles.

This presentation is based on two recent works in collaboration with P.-E. Chaudru de Raynal (Université de Nantes):

- P.-E. Chaudru de Raynal, N. Frikha, Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space, Journal de mathématiques pures et appliquées, 159:1-167, 2022.

- P.-E. Chaudru de Raynal, N. Frikha, From the backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDEs, Journal de mathématiques pures et appliquées, 156:1-124, 2021.

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** Tuesday, October 17, 2:00pm, ** Carslaw AGR and zoom

Speaker: Huansang Xu (University of Sydney)

Title: Equity Protection Swaps: A New Type of Risk Management Products for Pension Funds

## Abstract:

An equity protection swap (EPS) is a proposed new type of insurance product for pension accounts, which is largely based on an equity swap structure. It is a financial derivative, which is reminiscent of a total return swap but has also some features of the annuity insurance product known as the registered index-linked annuity (RILA). The buyer of an EPS agrees to share portfolio’s gains with its provider if the realized return on the reference portfolio is above a predetermined positive level and obtain a partial protection if a loss on that portfolio occurs. Formally, a generic EPS consists of the protection and fee legs with different participation rates, which are negotiated by the provider of an EPS and its buyer. We first derive a general pricing formula for a standard EPS by obtaining the static hedging strategy based on traded European options. Next, we argue that it order to make the contract appealing to a typical holder of pension account, the provider should select an appropriate participation rate for the fee leg in relation to the protection level required by the holder so that fair premium for the contract at its inception date is null. We present numerical examples based on the market data to illustrate our theoretical findings and demonstrate the benefits of an EPS as an efficient portfolio’s insurance tool for its holder.

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** Tuesday, September 19, 2:00pm, ** Carslaw AGR and zoom

Speaker: Szymon Peszat (Jagiellonian University)

Title: Gradient and Hessian formulae for transition semigroups of generalized Ornstein-Uhlenbeck processes

## Abstract:

Let $(P_t)$ be the transition semigroup on the space $B_b(E)$ of bounded measurable function on a Banach space $E$, of the process defined by the linear equation with additive noise $$ {\rm d} X= AX {\rm d} t + B{\rm d} W, \qquad X(0)=x\in E. $$ Our goal is to establish gradient and hessian formulae \begin{align*} \nabla P_t\phi (x)[y] &=\mathbb{E}\,\phi(X(t))Y(t,y), \\ \nabla ^2 P_t\phi (x)[y_1,y_2]&=\mathbb{E}\,\phi(X(t))Y(t,y_1,y_2), \end{align*} as well as estimates $$ \|\nabla P_t\phi(x)\|\le \rho(t)\|\phi\|_{B_b(E)},\qquad \|\nabla ^2P_t\phi(x)\|\le \tilde \rho(t)\|\phi\|_{B_b(E)}, $$ with properly chosen rates $\rho$ and $\tilde \rho$.

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** Tuesday, June 20, 2:00pm, ** Carslaw AGR and zoom

Speaker: Jörn Wichmann (Monash University)

Title: Temporal regularity of power-law fluids under stochastic perturbations

## Abstract:

Humans have always been interested in fluids. However, many interesting aspects have not been understood yet. In this talk we focus on regularity results for power-law fluids in the regime of laminar flow. These are modelled by the $p$-Stokes system. Already in the absence of turbulence, additional difficulties arise through the non-linear structure of the system as well as the low regularity of the stochastic perturbation. We will elaborate on the difficulties and try to build up an intuition on what regularity is natural. Finally, we present the temporal regularity results and give an application of the natural regularity within the development of numerical algorithms.

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** Tuesday, May 30, 2:00pm, ** zoom

Speaker: Jiahao Wu (Monash University)

Title: Simultaneous upper and lower bounds of American option prices with hedging via neural networks

## Abstract:

We introduce two methods to solve the American-style option pricing problem and its dual form at the same time using neural networks. Without applying nested Monte Carlo, the first method uses a series of neural networks to simultaneously compute both the lower and upper bounds of the option price, and the second one accomplishes the same goal with one global network. The avoidance of extra simulations and the use of neural networks significantly reduce the computational complexity and allow us to price Bermudan options with frequent exercise opportunities in high dimensions, as illustrated by the provided numerical experiments. As a by-product, these methods also derive a hedging strategy for the option, which can also be used as a control variate for variance reduction.

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** Tuesday, May 16, 8:00pm, ** zoom

Speaker: Dörte Kreher (Humboldt University of Berlin)

Title: A cross-border market model with limited transmission capacities

## Abstract:

We study a cross-border market model between two countries F and G, in which the transmission capacities that enable transactions between market participants of different countries are limited. Starting from two reduced-form representations of national limit order book dynamics, we allow incoming market orders to be matched with standing volumes of the foreign market, resulting in cross-border trades. Since the transmission capacities in our model are limited, our model alternates between regimes in which cross-border trades are possible and regimes in which incoming market orders can only be matched against limit orders of the same origin. We start with a complex microscopic description of the market dynamics and derive its high frequency behavior, when the size of an individual order converges to zero while the order arrival rate tends to infinity. In the scaling limit the market dynamics behaves as follows: if transmission capacities are available, the volume dynamics at the best available price is described by a four-dimensional linear Brownian motion in the positive orthant with oblique reflection at the axes and reinitilization whenever two components hit zero simultaneously. This approximation provides the necessary analytical tractability to compute key quantities of interest like the duration until the next price change or the duration of importing resp. exporting from country F to G. The talk is based on joint work with Cassandra Milbradt.

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** Tuesday, May 9,** 2:00pm, Eastern Avenue Lecture Theatre 315

Speaker: Harini Desiraju (SMS, University of Sydney)

Title: Gaussian multiplicative chaos measures, Painlevé equations, and conformal blocks

## Abstract:

Conformal blocks appear in several areas of mathematical physics from random geometry to black hole physics. A probabilistic notion of conformal blocks using gaussian multiplicative chaos measures was recently formulated by Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun (arxiv:2003.03802). In this talk, I will show that the semiclassical limit of the probabilistic conformal blocks recovers a special case of the elliptic form of Painlevé VI equation, thereby proving a conjecture by Zamolodchikov. This talk is based on an upcoming paper with Promit Ghosal and Andrei Prokhorov.

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** Tuesday, May 2,** 2:00pm, Eastern Avenue Lecture Theatre 315

Speaker: Ruyi Liu (SMS, University of Sydney)

Title: Two Equivalent Families of Linear Fully Coupled Forward-Backward Stochastic Differential Equations

## Abstract:

Forward-backward stochastic differential equations (FBSDEs) connect to many problems in Finance in the light of the stochastic maximum principle. While the well-posedness are usually hard to get , especially for the 1 dimension cases, because the forward equations and backward equations coupled with each other. In this talk, we first present some families of coupled FBSDEs which have varying coefficients but share the same well-posedness. Additionally, we also examine another equivalent family of FBSDEs using the linear transformation method, which enables us to decouple FBSDEs into partial coupling while maintaining the existence and uniqueness of solutions.

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** Tuesday, April 4,** 2:00pm, Eastern Avenue Lecture Theatre 315

Speaker: Kevin Qu (School of Physics, University of Sydney)

Title: Dynamical and computational properties of heavy-tailed deep neural networks

## Abstract:

In deep neural networks (DNNs), the assumption of Gaussian statistics in theoretical studies has been pervasive. However, empirical data has shown that heavy- tailed connectivity is prevalent in commonly used pretrained DNNs. To elucidate the fundamental impact of such heavy-tailed connectivity on the dynamical and computational properties of DNNs, we develop a novel mean field theory that integrates theories of heavy-tailed random matrices and non-equilibrium statistical physics. Our theoretical framework demonstrates that heavy-tailed weights enable the emergence of the extended criticality (edge-of-chaos) in a broad parameter region, leading to improved and faster learning of real-world tasks without the need to fine-tune the weight statistics, compared to networks with Gaussian weights. Notably, we find that heavy-tailed DNNs exhibit multifractal eigenvectors and internal neural representations of input data, which are associated with profound computational advantages such as enhanced robustness and increased signal-to-noise ratio in the context of few-shot learning. Overall, our findings challenge the commonly held assumption of Gaussian statistics in DNNs and offer new insights into the underlying mechanisms of deep learning.

This is a joint project with Asem Wardak and Pulin Gong under the Complex Systems, School of Physics.

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** Tuesday, March 28,** 2:00pm, zoom

Speaker: Max Menzies (Tsinghua University) and Nick James (University of Melbourne)

Title: Diversification benefits of random portfolios of different spread and size

## Abstract:

There has been a wealth of recent literature that has analysed financial market structure, particularly correlation, as it changes over time. Researchers have frequently noticed both market and sector effects, and have even filtered the principal components of the correlation matrix to find stock groups. First, we study several time-varying quantities that investigate collective behaviour and structure, with a focus on market sectors. Then, motivated by several practical considerations, we investigate the diversification benefits of randomly sampled portfolios with a focus on total size and spread across different sectors.

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** Tuesday, March 21,** 2:00pm, Carslaw 535 and zoom

Speaker: Monique Jeanblanc (University of Evry-val d'Essonne)

Title: Shrinkage of semimartingales

## Abstract:

In this talk we will study projections of semimartingales on various filtrations. The goal is to obtain the characteristics of the projection in terms of the original form and to give explicitly, under adequate hypotheses, the various predictable representation theorems.

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** Tuesday, March 7,** 2:00pm, Carslaw 535 and zoom

Speaker: Kostiantyn Ralchenko (Taras Shevchenko National University of Kyiv)

Title: Drift parameters estimation in the fractional Vasicek model

## Abstract:

We investigate the fractional Vasicek model described by the stochastic Langevin equation driven by fractional Brownian motion. We study the maximum likelihood estimators for unknown drift parameters based on the continuous-time observations of the solution. We derive their joint asymptotic distribution and prove their asymptotic independence. Both ergodic and non-ergodic cases are considered.

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** Wednesday, February 8,** 2:00pm, Carslaw AGR and zoom

Speaker: Erika Hausenblas (University of Leoben)

Title: A stochastic version of the Schauder-Tychonoff Theorem and systems of nonlinear SPDEs arising in biochemical systems

## Abstract:

The topic of the talk are nonlinear partial differential equations disturbed by noise. We will present recent results about the existence of martingale solutions using a stochastic version of the Schauder - Tychonoff type theorem.

Nonlinear partial differential equations appear naturally in many biological or chemical systems. E.g., activator-inhibitor systems play a role in morphogenesis and may generate different patterns. Another example is chemotaxis, which describes a fundamental biological process in which cells migrate following the direction of spatial cues.

Random fluctuations are ubiquitous in the real world. The randomness leads to various new phenomena and may have a non-trivial impact on the behaviour of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behaviour, which helps to understand the real processes and is also often more realistic. Due to the interplay of noise and nonlinearity, noise-induced transitions, stochastic resonance, metastability, or noise-induced chaos may appear. Noise in stochastic Turing patterns expands the range of parameters in which Turing patterns appears.
However, in many cases, the nonlinear systems are not monotone, nor do they satisfy a maximum principle. So, the existence of a solution can only be shown using compactness arguments.

In the talk, we first introduce these types of systems. Secondly, we will introduce the notion of martingale solutions and present our main result. Finally, we will outline the proof of our main result, i.e., the proof of the existence of martingale solutions.

## 2022

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** Wednesday, November 30,** 1:00pm, Carslaw AGR and zoom

Speaker: Max Nendel (Bielefeld University)

Title: A parametric approach to the estimation of risk functionals based on Wasserstein distance

## Abstract:

In mathematical finance and actuarial science, the assessment of risk is closely related to matching the distribution of underlying risk factors in an accurate way; a major issue in this process is so-called model uncertainty or epistemic uncertainty. The latter refer to the impossibility of perfectly capturing the randomness of future states in a single stochastic framework. We explore a static setting, where we allow for uncertainty around a baseline distribution measured via Wasserstein distance. We investigate to which extent this form of uncertainty can be parametrized for asymptotically small amounts of uncertainty and different scalings. This leads to a risk measure which incorporates a safety margin with respect to nonparametric uncertainty and still can be represented through a class of parametrized models using a first order approximation. In a second step, we discuss the effect of additional constraints. A mean constraint leads to simple correction term, whereas, in the case of a martingale constraint, an additional layer of radomization enters the parametrization. In a last step, we present numerical methods that build on our theoretical findings, and discuss applications in an insurance and finance context. The talk is based on joint work with Alessandro Sgarabottolo.

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** Tuesday, November 22,** 1:00pm, Carslaw AGR and zoom

Speaker: Emanuela Gussetti (Bielefeld University)

Title: Pathwise central limit theorem and moderate deviations via rough paths for a class of SPDEs with multiplicative small noise

## Abstract:

We employ the theory of rough paths to establish a path-wise central limit theorem (CLT) and a moderate deviation principle (MDP) for the deviations of a class of SDEs and SPDEs perturbed with a small multiplicative noise from the deterministic solution. The CLT can be interpreted as a path-wise derivative of the Ito-Lyons map. The result follows from the application of a path-wise Malliavin-like calculus for rough paths and compactness methods. From the exponential equivalence principle, we can derive easily a MDP. In particular, we do not apply the weak convergence approach usually employed in this framework. As an application, we derive a path-wise central limit theorem and a moderate deviation principle for the stochastic Landau-Lifschitz-Gilbert equation in 1D. We derive also path-wise convergence to the CLT limit for equations driven by linear Ito noise and a corresponding MDP. We discuss some applications to non-linear filtering problems.

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** Thursday, November 17,** 2:00pm, Carslaw AGR and zoom

Speaker: Michael Roeckner (Bielefeld University)

Title: Markov property of weak solutions to McKean-Vlasov SDEs

## Abstract:

In this talk
we shall present a general method to show that the laws of the weak solution to a
McKean-Vlasov SDE form a (nonlinear) Markov process. Examples include McKean-Vlasov
SDEs whose associated nonlinear Fokker-Planck equations is a porous media equation
perturbed by the divergence of a vector field depending nonlinearly on the solution,
where both the diffusivity function and the vector field are allowed to depend
explicitly on the spatial variable.

References:

Ren, Panpan; Roeckner, Michael; Wang, Feng-Yu. Linearization of nonlinear Fokker-Planck equations and applications. J. Differential Equations 322 (2022), 1-37.

Barbu, Viorel, Roeckner, Michael. Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov SDEs: The degenerate case. arXiv:2203.00122.

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** Wednesday, November 16,** 2:00pm, Carslaw AGR and zoom

Speaker: Michael Roeckner (Bielefeld University)

Title: Taming Uncertainty and Profiting from Randomness

## Abstract:

The title of this talk is a short form of the title of a major research
project (a so-called Collaborative Research Centre, funded by the German Research
Foundation, DFG), currently running at Bielefeld University. This project is based on
the claim that randomness, which is ubiquitous and cannot be avoided in mathematical
modelling, neither in the natural sciences, nor in the humanities, has in fact two
faces: One is a kind of a nuisance, which destroys observations or
measurements, is hard to quantify and causes uncertainty about the correctness of
ones final conclusions. So, one has to look for means for taming it and to
develop corresponding mathematical tools. On the other hand, in many cases randomness
can help to understand observed phenomena, which otherwise would remain a mystery. In
addition, its mathematical analysis may lead to a deeper understanding of the underlying
mechanisms, thus leading to resolving seemingly contradictory conclusions from the
theoretical and the experimental outcomes respectively. The latter is with no doubt a
valuable profit.

In the talk I shall present some examples for the above claim
and try to explain what, in particular, stochastic analysis, offers for taming
uncertainty as well as for profiting from randomness. One eminent instance for
both the latter will be the mathematical approach to understanding and analysing
dynamical processes under the influence of noise, which led to the development of
the field of stochastic partial differential equations. But despite this for
non-experts maybe not so attractive and too technically sounding term, I think these and
their usefulness can be understood quite well on an intuitive level from an appropriate
view point, as I shall try to explain in the talk.

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** Tuesday, October 25,** 1:00pm, Carslaw AGR and zoom

Speaker: Tal Morgenstern (University of Sydney)

Title: Incorporating Valuation Adjustments to Derivative Pricing

## Abstract:

The main purpose of the thesis is to introduce a set of derivative pricing models with the objective of breaking away from the assumptions commonly present in the classical setting. For this purpose we have developed multiple frameworks in order to mainly deal with counterparty credit risk and funding costs in a variety of situations. Each framework introduces the mechanisms governing the primary (risky) assets and consequently the derivative assets within the economy when releasing the assumptions present in traditional modelling.

One of the key objectives is to obtain Value Adjustments that will incorporate counterparty and own entity credit risk as well as funding costs. These adjustments have been intensively studied by both academics and practitioners in recent years, so we have therefore leveraged some of the body of work available to develop (under certain assumptions) specific frameworks for their computation. As of the present date, our research has shown there is still room for debating the most suitable approach which will likely change the accepted rules of risk-neutral pricing. We hope the work presented in this thesis could contribute to illustrate some of the consequences of operating in markets under counterparty credit risk and funding costs, while introducing an alternative to the existing approaches.

In this thesis, we are primarily focused on the incorporation to risk-neutral derivative pricing of the Valuation Adjustments, considered herewith as: counterparty credit risk value adjustment (Credit Value Adjustment), own entity credit risk value adjustment (Debit Value Adjustment), funding costs value adjustment (Funding Value Adjustment) and collateral costs value adjustment (Collateral Value Adjustment). The setup needed to comprehensively model these adjustments, requires changes to many aspects of the classical theory of risk-neutral valuation.

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** Tuesday, October 18,** 1:00 pm - 2pm, Carslaw AGR and zoom

Speaker: Guillaume Roger (Monash University)

Title: Dynamic contracting under imperfect observability

## Abstract:

This paper is concerned with a continuous time, dynamic contracting problem in which the principal observes performance at discrete intervals only. In spite of this, incentive compatibility can be enforced pathwise, but at additional costs to the principal. The trick is to find the correct martingale representation of the payoff process. The discrete nature of the information of the principal gives rise to a mixed problem with continuous and discrete controls. We characterise the contract and compute the probability of overpayment, as well as the expected cost of these over-payments. Finally we implement the contract in securities, which display real-life characteristics like special dividends.

## 2022

###
** Tuesday, October 11,** 8:00 pm - 9pm, zoom

Speaker: Yufei Zhang (London School of Economics)

Title: Exploration-exploitation trade-off for continuous-time reinforcement learning

## Abstract:

Recently, reinforcement learning (RL) has attracted substantial research interests. Much of the attention and success, however, has been for the discrete-time setting. Continuous-time RL, despite its natural analytical connection to stochastic controls, has been largely unexplored and with limited progress. In particular, characterising sample efficiency for continuous-time RL algorithms remains a challenging and open problem.

In this talk, we develop a framework to analyse model-based reinforcement learning in the episodic setting. We then apply it to optimise exploration-exploitation trade-off for linear-convex RL problems, and report sublinear (or even logarithmic) regret bounds for a class of learning algorithms inspired by filtering theory. The approach is probabilistic, involving analysing learning efficiency using concentration inequalities for correlated continuous-time observations, and applying stochastic control theory to quantify the performance gap between applying greedy policies derived from estimated and true models.

## 2022

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** Tuesdays, September 13 and September 20,** 1:00 pm - 2:30pm, Carslaw AGR and zoom

Speaker: David Lee (Sorbonne Université)

Title: An introduction to the theory of Rough Integration and Rough Paths

## Abstract:

In this mini-course we will introduce the main ideas and concepts behind Terry Lyons' theory of Rough paths.
This theory can be seen as an alternative integration theory to handle paths of low regularity. The mathematician, Martin Hairer, took inspiration from the theory of Rough Paths to develop the theory of Regularity Structures to tackle singular stochastic partial differential equations. It was this theory that led Martin Hairer to receive the Fields medal in 2014.

In the first lecture we will first remind the reader of certain classical integration theories, such as Riemann-Stieltjes or Ito, and their limitations when dealing with objects such as Fractional Brownian motion. We will also introduce the famed Sewing Lemma to help us prepare for the main tools in rough path theory.

In the second lecture we will define the notion of a rough path and controlled paths. From this we will define the rough integral and state some important consequences.

The course is aimed at the individual who has a background in measure theory and stochastic analysis.

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** Tuesday, May 24,** 2:00 pm, zoom talk

Speaker: Julian Sester (Nanyang Technological University)

Title: Robust statistical arbitrage strategies and their detection with neural networks

## Abstract:

In this talk, we discuss the notion of robust statistical arbitrage, which refers to profitable trading strategies that take into account ambiguity about the underlying time-discrete financial model. Our investigations rely on the mathematical characterization of (non-robust) statistical arbitrage, which was originally introduced by Bondarenko in 2003. In contrast to pure arbitrage strategies, statistical arbitrage strategies are not entirely risk-free, but the notion allows to identify strategies which are profitable on average, given the outcome of a specific sigma-algebra. In particular, such strategies may exist even in arbitrage-free markets. Besides a characterization of robust statistical arbitrage, we also provide a super-/sub-replication theorem for the construction of statistical arbitrage strategies based on path-dependent options.

Relying on these theoretical results, we then discuss an approach, based on deep neural networks, that allows identifying robust statistical arbitrage strategies in real-world financial markets. The presented novel methodology does not suffer from the curse of dimensionality nor does it depend on the identification of cointegrated pairs of assets and is therefore applicable even on high-dimensional financial markets or in markets where classical pairs trading approaches fail. Moreover, we provide a method to build an ambiguity set of admissible probability measures that can be derived from observed market data. Thus, the approach can be considered as being model-free and entirely data-driven. We showcase the applicability of our method by providing empirical investigations with highly profitable trading performances even in 50 dimensions, during financial crises, and when the cointegration relationship between asset pairs stops to persist.

(based on joint works with Eva Lütkebohmert, Ariel Neufeld and Daiying Yin)

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** Tuesday, May 10,** 2:00 pm, zoom talk

Speaker: Kazutoshi Yamazaki (University of Queensland)

Title: Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities

## Abstract:

We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Levy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players' strategies as well as the quantification of the value of available exercise opportunities. This talk is based on joint works with Neofytos Rodosthenous and Jose Luis Perez.

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** Wednesday, April 27,** 8:00 pm, zoom talk

Speaker: Max Nendel (Bielefeld University)

Title: Nonlinear transition semigroups and model uncertainty in finance

## Abstract:

When considering stochastic processes for the modeling of real world phenomena, e.g., stock prices, a major issue is so-called model uncertainty or epistemic uncertainty. The latter refer to the impossibility of perfectly capturing all probabilistic information about the future in a single stochastic framework. In a dynamic setting, this leads to the task of constructing consistent families of nonlinear transition semigroups. In this talk, we present two ways to incorporate model uncertainty into Markovian dynamics. One approach considers parameter uncertainty in the generator of a Markov process while the other considers perturbations of a reference model within a Wasserstein proximity. In both cases, we are able to compute the infinitesimal (nonlinear) generator of the semigroup, and show that, in typical situations, these two a priori different approaches de facto lead to the same nonlinear transition semigroup. Moreover, we discuss financial applications and the connection between upper envelopes of Markovian transition semigroups, dynamic risk measures, and abstract Hamilton-Jacobi-Bellman-type differential equations. The talk is based on joint works with Robert Denk, Sven Fuhrmann, Michael Kupper, and Michael Röckner.

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** Tuesday, April 12,** 2:00 pm, zoom talk

Speaker: Małgorzata O'Reilly (University of Tasmania)

Title: Stochastic fluid models and their application potential in finance

## Abstract:

Matrix-analytic methods (MAMs) is an area in applied probability focusing on constructing models and methods of analysis that can be applied efficiently, using fast algorithms and computers. Stochastic fluid model (SFM) is a fundamental class of Markov models in the theory of MAMs, which offers solution methods for a wide range of real-world problems. In this talk, I will discuss the results for various types of SFMs and their application potential in finance.

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** Tuesday, April 5,** 2:00 pm, zoom talk

Speaker: Shidan Liu (Monash University)

Title: Robust pricing-hedging duality for multi-action options

## Abstract:

We will be talking about the robust pricing-hedging duality of an exotic kind of options, where the buyer is allowed to choose some action from an action space at each time step, in a discrete-time setup. By space enlargement techniques, the superhedging problem for such exotic options can be reformulated as a problem for European options, which has been well studied. Then by constructing suitable analytic sets, a measurable selection theorem can be applied to obtain the duality result. Since the action space can be uncountable, an immediate application could be superhedging multiple American options or swing options on oil or natural gas.

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** Wednesday, March 30,** 8:00 pm, zoom talk

Speaker: Claudio Fontana (University of Padova)

Title: Term structure modeling with overnight rates beyond stochastic continuity

## Abstract:

In the current reform of interest rate benchmarks, a central role is played by risk-free rates (RFRs), such as SOFR (secured overnight financing rate) in the US. A key feature of RFRs is the presence of jumps and spikes at periodic time intervals as a result of regulatory and liquidity constraints. This corresponds to stochastic discontinuities (i.e., jumps occurring at predetermined dates) in the dynamics of RFRs. In this work, we propose a general modelling framework where RFRs and term rates can have stochastic discontinuities and characterize absence of arbitrage in an extended HJM setup. When the term rate is generated by the RFR itself, we show that it solves a BSDE, whose driver is determined by the HJM drift restrictions. We develop a tractable specification driven by affine semimartingales, also extending the classical short rate approach to the case of stochastic discontinuities. In this context, we show that a simple specification allows to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Finally, we study hedging in the sense of local risk-minimization when the underlying term structures have stochastic discontinuities. Based on joint work with Zorana Grbac and Thorsten Schmidt.

## 2021

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** Wednesday, November 10,** 8:00 pm, zoom talk

Speaker: Miryana Grigorova (University of Leeds)

Title: Superhedging of options in a non-linear incomplete financial market model

## Abstract:

We will study the superhedging price (and superhedging strategies) of European and American options in a non-linear incomplete market model with default, with a particular focus on the American options case which is more involved. We will provide a dual representation of the seller’s (superhedging) price for the American option in terms of a mixed stochastic control/stopping problem with non-linear expectations/ evaluations, and in terms of non-linear Reflected BSDEs with constraints. If time permits, we will also present a duality result for the buyer’s price in terms of a stochastic game of control and stopping with non-linear expectations/ evaluations. The talk is based on joint works with Marie-Claire Quenez and Agnès Sulem.

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** Wednesday, November 3,** 3:00 pm, zoom talk

Speaker: Edward Kim (University of Sydney)

Title: General Nonlinear Dynkin Games with Unorderd Payoffs

## Abstract:

Two-person zero-sum stopping games where payoffs are evaluated by taking conditional expectations are classical and were first analysed by Dynkin (1969). More recently, some authors have studied nonlinear variants of the Dynkin game by evaluating payoffs under a nonlinear evaluation induced by a solution of a backward stochastic differential equation (BSDE) and have shown that the theory naturally applies to stopping policies for hedging game-type options. Under suitable conditions, a Nash equilibrium of the game can be characterised using appropriately defined first hitting times of a doubly reflected BSDE. In turn, this Nash equilibrium can be interpreted as a rational exercise / break-even time pair for a given party of the game contract. A constant assumption in the literature is that the stochastic payoffs of the nonlinear Dynkin game are ordered. However, this is somewhat artificial and is not always a natural assumption to make in applications. In this talk, we will show that this ordering assumption is not needed. Namely, we will discuss extensions of some results for the classical Dynkin game with unordered payoffs from Guo (2014) to the case of a nonlinear Dynkin game. In particular, we will show necessary and sufficient conditions for a Dynkin game with unordered RCLL payoffs to have a value. An important consequence of this result is that DRBSDEs (with appropriate adjustments) can still serve as useful tools for characterising solutions even in the case of fully unordered nonlinear Dynkin games.

###
** Wednesday, October 20,** 2:00 pm, zoom talk

Speaker: Arturo Kohatsu-Higa (Ritsumeikan University)

Title: On the differential behavior of stopped diffusions

## Abstract:

In this presentation, we will discuss a number of issues arising from the study of stopped diffusions. In particular, together with N. Frikha (Paris-Diderot) and L. Li (UNSW, Australia), we obtained an unbiased formula for the integration by parts of a one dimensional stopped process. This result based on Markov chain structures has a number of interesting geometrical aspects which in current work with Dan Crisan (Imperial), we are clarifying. I will present some of the results that we are trying to obtain and the reasoning which supports our insights.

###
** Wednesday, October 13,** 2:00 pm, zoom talk

Speaker: Daniel Hauer (University of Sydney)

Title: Functional Calculus via the extension technique: a first hitting time approach

## Abstract:

In this talk, I will present a solution to the problem:

* "Which type of linear operators can be realized by the
Dirichlet-to-Neumann operator associated with the operator
$-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension
problem?"*,

which was raised in the pioneering work [Comm. Par.Diff. Equ. 32
(2007)] by Caffarelli and Silvestre. But I even intend to go a step
further by replacing the negative Laplace operator $-\Delta$ on
$\mathbb{R}^{d}$ by an $m$-accretive operator $A$ on a general Banach space
$X$ and the Dirichlet-to-Neumann operator by the
Dirichlet-to-Wentzell operator. I show how to prove uniqueness of
solutions to the extension problem in the general Banach spaces
framework, which seems to be new in the literature and of
independent interest. I outline a type of functional calculus using
probabilistic tools from excursion theory. With this new method, I
am able to characterize all linear operators $\psi(A)$, where $\psi$
is a complete Bernstein function
($\mathcal{C}\mathcal{B}\mathcal{F}$), resulting in a new
characterization of the famous \emph{Phillips' subordination
theorem} within this class $\mathcal{C}\mathcal{B}\mathcal{F}$.

This talks is based on the research results provided in the recent
Arxiv submission.

###
** Wednesday, October 6,** 2:00 pm, zoom talk

Speaker: Nicolas Langrene (CSIRO)

Title: Portfolio optimization with a prescribed terminal wealth distribution

## Abstract:

This paper studies a portfolio allocation problem, where the goal is to reach a prescribed wealth distribution at a final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which is solved with a gradient descent algorithm. This involves solving an associated Hamilton–Jacobi–Bellman and Fokker–Planck equations with a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We then consider adding consumption during the investment process, to take into account distributions that are either not attainable, or sub-optimal.

This talk is based on a paper accepted for publication in Quantitative Finance .

###
** Wednesday, September 22,** 2:00 pm, zoom talk

Speaker: Zhou Zhou (University of Sydney)

Title: Optimal relative performance criteria in mean field contribution games

## Abstract:

We consider mean-field contribution games, where players in a team choose some effort levels at each time period, and the aggregate reward for the team depends on the aggregate cumulative performance of all the players. Each player aims to maximize the expected reward of her own share subject to her cost of effort. To reduce free-rider issue, we propose some relative performance criteria (RPC), based on which the reward is redistributed to each player. We are interested in those RPCs which implement the optimal solution for the corresponding centralized problem, and we call such RPC an optimal one. That is, the expected payoff of each player under the equilibrium associated with an optimal RPC is as large as the value induced by the corresponding problem where players completely cooperate. We first analyze a one-period model with homogeneous players, and obtain natural RPCs of different forms. Then we generalize these results to a multi-period model in discrete time. Next, we investigate a two-layer mean-field game: The top-layer is an inter-team game (team-wise competition) in which the reward of a team is impacted by the relative achievement of the team with respect to other teams; the bottom layer is an intra-team contribution game where an RPC is implemented for reward redistribution among team members. We establish the existence of equilibria for the two-layer game and characterize the intra-team optimal RPC. Finally, we extend the (one-layer) results of optimal RPCs to the continuous-time setup as well as to the case with heterogenous players.

###
** Wednesday, September 15,** 2:00 pm, zoom talk

Speaker: Libo Li (UNSW)

Title: Random Times and GBSDEs

## Abstract:

In this talk, we will discuss two related topics. The first is the construction of random time, where we extend using, multiplicative systems to construct random time with a given survival process. Secondly, motivated by the arbitrage-free pricing of European and American style contracts with the counterparty credit risk, we investigate the well-posedness of BSDE and RBSDE in the progressive enlargement of a reference filtration with a random time through the method of reduction.

###
** Wednesday, June 16,** 2:00 pm, zoom talk

Speaker: Ashwaq Zarban (UNSW)

Title: Pricing European Exchange Options under a Double Regime-Switching Jump-Diffusion model

## Abstract:

In this talk, we propose a pricing formula for European exchange options, where the dynamics of the underlying assets are driven by a double regime-switching jump-diffusion. We assume both the model parameters and the price level of the risky share depend on a continuous-time, finite-state, observable Markov chain. Our result is an extension of Cheang and Chiarella (2011), who have priced European exchange options under the jump-diffusion setting, and Shen, Fan and Siu (2014), who have priced European call options under a double regime-switching model. An analytical option pricing formula is obtained by using the inverse Fourier transform.

###
** Thursday, May 27,** 1:00 pm, zoom talk

Speaker: Yu-Jui Huang (University of Colorado)

Title: Mortality and Healthcare: a Stochastic Control Analysis under Epstein-Zin Preferences

## Abstract:

This paper studies optimal consumption, investment, and healthcare spending under Epstein-Zin preferences. Given consumption and healthcare spending plans, Epstein-Zin utilities are defined over an agent's random lifetime, partially controllable by the agent as healthcare reduces mortality growth. To the best of our knowledge, this is the first time Epstein-Zin utilities are formulated on a controllable random horizon, via an infinite-horizon backward stochastic differential equation with superlinear growth. A new comparison result is established for the uniqueness of associated utility value processes. In a Black-Scholes market, the stochastic control problem is solved through the related Hamilton-Jacobi-Bellman (HJB) equation. The verification argument features a delicate containment of the growth of the controlled morality process, which is unique to our framework, relying on a combination of probabilistic arguments and analysis of the HJB equation. In contrast to prior work under time-separable utilities, Epstein-Zin preferences largely facilitate calibration. In four countries we examined, the model-generated mortality closely approximates actual mortality data; moreover, the calibrated efficacy of healthcare is in broad agreement with empirical studies on healthcare across countries. (This is joint work with Joshua Aurand).

###
** Wednesday, April 28,** 2:00 pm, zoom talk

Speaker: Xuedong He (Chinese University of Hong Kong)

Title: Portfolio selection under median and quantile maximization

## Abstract:

Although maximizing median and quantiles is intuitively appealing and has an axiomatic foundation, it is difficult to study the optimal portfolio strategy due to the discontinuity and time inconsistency in the objective function. We use the intra-personal equilibrium approach to study the problem. Interestingly, we find that the only viable outcome is from the median maximization, because for other quantiles either the equilibrium does not exist or there is no investment in the risky assets. The median maximization strategy gives a simple explanation to why wealthier people invest more percentage of their wealth in risky assets. This is a joint work with Zhaoli Jiang and Steven Kou.

###
** Wednesday, April 14,** 2:00 pm, zoom talk

Speaker: Xiang Yu (Hong Kong Polytechnic University)

Title: Optimal consumption with reference to past spending maximum: exponential utility and S-shaped utility cases

## Abstract:

In this talk, we present two recent studies on the optimal consumption with the non-negativity constraint and the path-dependent reference to the past consumption peak under the exponential utility and the S-shaped utility. In both problems, the relative performance is measured by the distance between the current consumption rate and a fraction of the historical consumption maximum. The reference process is non-addictive in the sense that the investor is allowed to strategically consume below the reference level. By applying the dynamic programming argument and identifying the value function depending on the wealth variable and the reference variable, we can express the associated HJB equation in the piecewise manner across difference regions together with some free boundary conditions. In both problems, the thresholds for the wealth level can be characterized by nonlinear functions of the reference variable such that the optimal consumption in each region can be obtained in the feedback form that satisfies: (i) zero consumption; (ii) consumption below the reference; (iii) consumption above the reference but below the historical maximum; (iv) consumption sitting at the previous historical maximum; (v) consumption creating a new global maximum. Distinct optimal consumption behavior and financial implications will be concluded under two types of utility functions, and some comparison results with respect to loss aversion on the relative consumption will be provided.

###
** Wednesday, March 24,** 8:00 pm, zoom talk

Speaker: Tiziano De Angelis (University of Turin)

Title: Dynkin games with partial and asymmetric information

## Abstract:

I will review some recent results obtained in collaboration with Ekström, Glover, Merkulov and Palczewski concerning existence of equilibria for Dynkin games with partial and asymmetric information. I will illustrate explicit solutions to two specific problems: a zero-sum game with asymmetric information on the drift of a geometric Brownian motion and a non-zero sum game with uncertain competition. I will then present a general result for the existence of a saddle point in zero-sum non-Markovian Dynkin games. The construction of all equilibria relies upon the use of randomised stopping times. The talk is based on https://arxiv.org/abs/1810.07674 (Math. Oper. Res. 2020, to appear) https://arxiv.org/abs/1905.06564 (Stoch. Process. Appl. 130 (2020), pp. 6133-6156) https://arxiv.org/abs/2007.10643

###
** Wednesday, March 10,** 8:00 pm, zoom talk

Speaker: Matteo Burzoni (University of Milan)

Title: Mean Field Games with absorption and a model of bank run

## Abstract:

We consider a MFG problem obtained as the limit of N-particles systems with an absorbing region. Once a particle hits such a region, it leaves the game and the rest of the system continues to play with N-1 particles. We study existence of equilibria for the limiting problem in a framework with common noise and establish the existence of epsilon Nash equilibria for the N-particles problems. These results are applied to a novel model of bank run. This is a joint work with L. Campi.

###
** Wednesday, February 24,** 10:00 am, zoom talk

Speaker: Marcel Nutz (Columbia University)

Title: Entropic Optimal Transport

## Abstract:

Applied optimal transport is flourishing after computational advances have enabled its use in real-world problems with large data sets. Entropic regularization is a key method to approximate optimal transport in high dimensions while retaining feasible computational complexity. In this talk we discuss the convergence of entropic optimal transport to the unregularized counterpart as the regularization parameter vanishes, with a focus on the local behavior. Based on joint works with Espen Bernton (Columbia), Promit Ghosal (MIT), Johannes Wiesel (Columbia).

## 2020

###
** Tuesday, October 20,** 3:00 pm, zoom talk

Speaker: Dr Chao Zhou (National University of Singapore)

Title: Relative wealth concerns with partial information and heterogeneous priors

## Abstract:

We establish a Nash equilibrium in a market with N agents with CARA utility and the relative performance criteria when the market return is unobservable. Each investor has a Gaussian prior belief on the return rate of the risky asset. The investors can be heterogeneous in both the mean and variance of the normal random variable. By a separation result and a martingale argument, we show that the optimal investment strategy under a stochastic return rate model can be characterized by a fully-coupled FBSDE with linear coefficients. Two sets of deep neural networks are used for the numerical computation to first find each investor’s estimate of the mean return rate and then solve the FBSDEs. We are the first to establish the uniqueness result for the class of FBSDEs with stochastic coefficients. The deep learning scheme for solving the game under partial information is also novel. We demonstrate the efficiency and accuracy by comparing with the numerical solution from PDE for the linear filter case and apply the algorithm to the general case of nonlinear hidden variable process. Simulations of investment strategies demonstrate a herd effect that investors trade more aggressively under relative performance, in most cases for our specified market parameters. Statistical properties of the investment strategies and the portfolio performance, including the Sharpe ratios and VRRs are examed. This is a joint work with Chao DENG and Xizhi SU.

###
** Tuesday, October 6,** 3:00 pm, zoom talk

Speaker: Dr Guiyuan Ma (Xi'an Jiaotong University)

Title: Optimal portfolio execution problem with stochastic price impact

## Abstract:

In this paper, we provide a closed-form solution to an optimal portfolio execution problem with stochastic price impact and stochastic net demand pressure. Specifically, each trade of an investor has temporary and permanent price impacts, both of which are driven by a continuous-time Markov chain; whereas the net demand pressure from other inventors is modelled by an Ornstein-Uhlenbeck process. The investor optimally liquidates his portfolio to maximize his expected revenue netting his cumulative inventory cost over a finite time. Such a problem is first reformulated as an optimal stochastic control problem for a Markov jump linear system. Then, we derive the value function and the optimal feedback execution strategy in terms of the solutions to coupled differential Riccati equations. Under some mild conditions, we prove that the coupled system is well-posed, and establish a verification theorem. Financially, our closed-form solution shows that the investor optimally liquidates his portfolio towards a dynamic benchmark. Moreover, the investor trades aggressively (conservatively) in the state of low (high) price impact.

###
** Tuesday, September 15,** 3:00 pm, zoom talk

Speaker: Dr Ruyi Liu (Shandong University)

Title: Pairs-trading under geometric Brownian motions: An optimal strategy with cutting losses

## Abstract:

Pairs trading is about simultaneously trading a pair of stocks. A pairs trade is triggered when their prices diverge and consists of a short position of the strong stock and a long position of the weak one. Pairs trading bets on the reversal of their price strengths. In this paper, we study the optimal pairs trading problem under general geometric Brownian motions and focus on trading with cutting losses. The objective is to trade the pairs over time to maximize an overall return with a fixed transaction cost. Our optimal policy is characterized by threshold curves obtained by solving the associated HJB equations. We provide sufficient conditions that guarantee the optimality of our trading rules. We also present numerical examples to illustrate.

###
** Tuesday, September 1,** 3:00 pm, zoom talk

Speaker: Prof Xiaolu Tan (The Chinese University of Hong Kong)

Title: A $C^{0,1}$-functional Ito formula and its applications in finance

## Abstract:

We obtain a functional (path-dependent) extension of the Ito formula for $C^{0,1}$ functions in Bandini and Russo (2017). We then provide some original applications in finance of this new formula, by considering an option replication problem and a super-replication problem. Joint work with Bruno Bouchard.

###
** Tuesday, July 28,** 2:00 pm, zoom talk

Speaker: Ms Kris Wu (UNSW)

Title: Valuation of American VIX Call Options under the Generalized Mixture Model

## Abstract:

In this paper, we study the pricing of American VIX call option under generalized mixture of 3/2 and 1/2 (Heston) models. According to Detemple and Kitapbayev (2018), there are two optimal stopping boundaries under this mixture model. By taking the Laplace-Carson transform of the free-boundary problem, we were able to obtain numerically the optimal stopping boundaries and the price of American VIX call option. In addition, we also derive a closed form formula for the price of a perpetual American VIX call option.

###
** Tuesday, July 21,** 2:00 pm, zoom talk

Speaker: Prof Samuel Drapeau (Shanghai Jiao Tong University)

Title: Robust Uncertainty Analysis

## Abstract:

In this talk, we will showcase how methods from optimal transport and distributionally robust optimisation allow to capture and quantify sensitivity to model uncertainty for a large class of problems. We consider a generic stochastic optimisation problem. This could be a mean-variance or a utility maximisation portfolio allocation problem, a risk measure computation, a standard regression or a deep learning problem. At the heart of the optimisation is a probability measure, or a model, which describes the system. It could come from data, simulations or a modelling effort for which there is always exists a degree of uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated measure. Our main results provide explicit formulae for the first order correction to both the value function and the optimiser. We further extend our results to optimisation under linear constraints. Our sensitivity analysis of the distributionally robust optimisation problems finds applications in statistics, machine learning, mathematical finance and uncertainty quantification. In the talk, we will discuss several financial examples anchored in a one-step financial model and compute their sensitivity to model uncertainty. These include: option pricing, mean-variance portfolio selection, optimised certainty equivalent and similar risk assessments. We will also address briefly some other applications, such as explicit formulae for first-order approximations of square-root LASSO and square-root Ridge optimisers and measures of NN architecture robustness wrt to adversarial data. This talk is based on joint works with Daniel Bartl, Jan Obloj and Johannes Wiesel.

###
** Tuesday, July 14,** 2:00 pm, zoom talk

Speaker: Prof Georg Gottwald (University of Sydney)

Title: Simulation of non-Lipschitz stochastic differential equations driven by α-stable noise: a method based on deterministic homogenisation

## Abstract:

The talk introduces an explicit method to integrate α-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the Lévy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct α-stable laws. This is joint work with Ian Melbourne.

###
** Tuesday, July 7,** 2:00 pm, zoom talk

Speaker: Dr Jie Fan (Monash University)

Title: From mimicking to local Brownian motions

## Abstract:

Motivated by questions from finance, we are interested in constructing new processes from existing ones while preserving certain desired properties. For instance, constructing martingales with given marginal distributions allows us to have alternative (and hopefully better) models for asset price while retaining the (European) option prices. In this talk, I will review some results on constructing martingales with given marginal distributions, and introduce local Brownian motions, which are processes that behave locally like a Brownian motion.

###
** Tuesday, June 30,** 2:00 pm, zoom talk

Speaker: Prof Ben Goldys (U Sydney)

Title: G-expectation via Nisio semigroup -- Part 2

## Abstract:

The concept of G-expectation introduced by Peng takes as its starting point a non-trivial result from the theory of fully nonlinear partial differential equations. I will present another approach to G-expectation based on an old idea of M. Nisio, who introduced a non-linear semigroup corresponding to an optimal control problem. However, the Nisio approach had some some serious limitations. I will present an improvement of her theory, that will allow us to recover some basic properties of G-expectation. The existence of a unique viscosity solution to the corresponding fully nonlinear partial differential equation will follow naturally, in the same way as for the Kolmogorov equations associated to Markov processes. The talk is based on a recent paper by Max Nendel and Michael Roeckner (2019) and on our joint work in preparation.

###
** Tuesday, June 23,** 2:00 pm, zoom talk

Speaker: Prof Ben Goldys (U Sydney)

Title: G-expectation via Nisio semigroup -- Part 1

## Abstract:

The concept of G-expectation introduced by Peng takes as its starting point a non-trivial result from the theory of fully nonlinear partial differential equations. I will present another approach to G-expectation based on an old idea of M. Nisio, who introduced a non-linear semigroup corresponding to an optimal control problem. However, the Nisio approach had some some serious limitations. I will present an improvement of her theory, that will allow us to recover some basic properties of G-expectation. The existence of a unique viscosity solution to the corresponding fully nonlinear partial differential equation will follow naturally, in the same way as for the Kolmogorov equations associated to Markov processes. The talk is based on a recent paper by Max Nendel and Michael Roeckner (2019) and on our joint work in preparation.

###
** Tuesday, June 16,** 2:00 pm, zoom talk

Speaker: Dr Libo Li (UNSW)

Title: Strong approximation of the alpha-CEV and alpha-CIR process

## Abstract:

We propose a positivity-preserving implicit numerical scheme for jump-extended Cox-Ingersoll-Ross (CIR) process and Constant-Elasticity-of-Variance (CEV) process, where the jumps are governed by a compensated spectrally positive alpha-stable Levy process for alpha in (1, 2). This class of models have first been studied in the context of continuous branching processes with interaction and/or immigration, and in this class a model has been introduced to mathematical finance for modelling sovereign interest rates and the energy market. Numerical schemes for jump-extended CIR and CEV processes, to the best of our knowledge, have all focused on the case of finite activity jumps.

###
** Tuesday, March 10,** 2:00 pm, AGR

Speaker: Dr Yuuki Ida (Ritsumeikan University, Kyoto)

Title: Hyperbolic symmetrization of Heston type diffusion

## Abstract:

The symmetrization of diffusion processes was originally introduced by Imamura, Ishigaki and Okumura, and was applied to pricing of barrier options. We introduced a hyperbolic version of the symmetrization of a diffusion by symmetrizing drift coefficient. In view of applications under a SABR model, which is transformed to a hyperbolic Brownian motion with drift. In the talk, in order to apply the hyperbolic symmetrization technique to Heston model, we introduce an extension where diffusion coefficient is also symmetrized. Some numerical results are also presented.

###
** Tuesday, March 3,** 2:00 pm, AGR

Speaker: Dr Kihun Nam (Monash University)

Title: Global well-posedness of non-Markovian multidimensional superquadratic BSDEs

## Abstract:

Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay between the local well-posedness of fully coupled path-dependent forward backward stochastic differential equations (FBSDEs) and backward iterations of the superquadratic BSDE. The superquadratic BSDE studied in this article includes quadratic BSDEs appearing in stochastic differential game and price impact model. We also study the well-posedness of superquadratic FBSDE using the corresponding BSDE results. Our result also provides the well-posedness of a system of path-dependent quasilinear partial differential equations where the nonlinearity has superquadratic growth in the gradient of the solution.

###
** Thursday, January 16,** 2:00 pm, AGR

Speaker: Dr Ivan Guo (Monash University)

Title: Path-dependent optimal transport with applications

## Abstract:

We introduce a generalisation of the classical martingale optimal transport problem that relaxes the usual marginal distribution constraints to arbitrary convex constraints on the space of probability measures. Duality is established, which leads to a path-dependent Hamilton-Jacobi-Bellman equation, in which the solution localises to the state variables of the constraints, while bypassing the usual dynamic programming principle. Our result has a variety of applications, including: model calibration on path-dependent as well as VIX derivatives; admissibility of option prices (analogous to the first fundamental theorem of asset pricing); portfolio selection problems with target wealth distributions; and robust hedging in continuous time.

## 2019 Semester 2

### Tuesday, December 10, 2:00 pm, AGR

Speaker: Prof Bohdan Maslowski (Charles University, Prague)

Title: Linear Stochastic PDEs Driven by Volterra Processes

## Abstract:

In the first part, the basic setting for infinite-dimensional linear stochastic equations with memory-dependent noise will be recalled and some results on the existence, uniqueness, regularity and large time behaviour of solutions will be presented. The general results will be illustrated by examples of the most most popular Volterra processes, such as fractional Brownian motion and Rosenblatt process. In the second part, some optimal control problems for such systems will be discussed for the case of quadratic cost functionals. We will also consider the Kalman-Bucy type filter and the corresponding integral equations for the optimal estimate and covariance of the error will be derived. All results will be compared to the standard case of Gauss-Markov driving processes. The talk is based on joint papers with P. Coupek and V. Kubelka.

### Tuesday, November 26, 2:00 pm, AGR

Speaker: Ms Chunxi Jiao (University of Sydney)

Title: Computable Primal and Dual Bounds for Some Stochastic Control Problems

## Abstract:

We investigate the linear programming framework for stochastic control with a view towards the numerical implementation (Lasserre’s hierarchy) for obtaining pointwise bounds and global bounding functions for the value function. The primal minimisation corresponds to the well-studied moment problem based upon a set of necessary equality constraints on the occupation and boundary measures, whereas the dual maximisation is built on a set of sufficient inequality constraints on the test polynomial function with a flexible choice of optimality criteria. Under suitable technical conditions, optimised bounds are convergent to the value function as the polynomial degree tends to infinity. The dual maximisation is particularly effective as its single implementation yields a remarkably tight global bound in the form of polynomial function over the whole problem domain, and with a suitable objective function, one may improve the global bound on regions of interest.

### Tuesday, November 19, 3:00 pm, AGR

Speaker: Dr Jacek Krawczyk (University of Sydney)

Title: The Role of Payoff Distribution in Dynamic Portfolio Management. The Case of Lump-sum Pensions. Part II – Robustness of cautious-relaxed investment policies to target contingency and selfish manager preferences.

## Abstract:

A cautious-relaxed investment policy is one which optimises a target-based kinked utility measure. A cautious-relaxed investment policy can generate a left (negatively) skewed payoff distribution that helps people form strong expectations of a satisfactory final payoff. This means that applying a cautious-relaxed investment policy will help avoid frequently obtaining low returns - so, losses - and at the same time promises higher payoffs with greater certainty. The question then arises as to what extent are such strategies realistic in the presence of e.g., target variation and inflation, fund manager selfishness or transaction costs. In my presentation, I will use a computational method (“SOCSol”) to find approximately-optimal decision rules and the corresponding payoff distributions for several such cases. Therefore the reported results will be parameter specific. The effect of varying the payoff target on the payoff distribution is that increasing the target causes the distribution to become less left skewed, causing higher probabilities of loss; even if the fund manager's explicit preferences differ from the investor’s, the latter’s payoff should not suffer; in case the target is contingent on an exogenous stochastic process the payoff distribution depends on a correlation between inflation and the risky asset price.

### Tuesday, November 12, 2:00 pm, AGR

Speaker: Mr Yihan Zou (University of Glasgow)

Title: American Real Option Pricing with Stochastic Volatility and Multiple Priors

## Abstract:

In this article we study stochastic volatility models in a multiple prior setting and investigate prices of American options from the perspective of an ambiguity averse agent. Using the theory of reflected backward stochastic differential equations (RBSDEs), we formalize the problem and solve it numerically by a simulation scheme for RBSDEs. We also propose an alternative to obtain the American option value without using the theory of RBSDEs. We analyze the accuracy of the numerical scheme with single prior models, of which American options could also be efficiently evaluated by the least squares Monte Carlo (LSM) approach. By comparing to the single prior case, we highlight the importance of the dynamic structure of the agent’s worst case belief. At last we explore the applicability of numerical schemes in a setting with multidimensional real option and ambiguity.

### Tuesday, November 5, 2:00 pm, AGR

Speaker: Prof Jan Obloj (University of Oxford)

Title: Robust Finance. Part II – Fundamental Theorems

## Abstract:

We pursue robust approach to pricing and hedging in mathematical finance. We develop a general discrete time setting in which some underlying assets and options are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. We include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows to quantify the impact of making assumptions. We establish suitable FTAP and Pricing-Hedging duality results which include as special cases previous results of Acciaio et al. (2013), Bouchard and Nutz (2015), Burzoni et al. (2016) as well the Dalang-Morton-Willinger theorem. Finally, we explain how to treat further problems, such as insider trading (information quantification) or American options pricing. The talk will cover a body of results developed in collaboration with A. Aksamit, M. Burzoni, S. Deng, M. Frittelli, Z. Hou, M. Maggis, X. Tan and J. Wiesel.

### Tuesday, October 29, 2:00 pm, AGR

Speaker: Dr Jacek Krawczyk (University of Sydney)

Title: The Role of Payoff Distribution in Dynamic Portfolio Management. The Case of Lump-sum Pensions. Part I – Merton model asset allocation versus target-based kinked utility optimisation

## Abstract:

The Merton asset allocation model advocates a strategy that results in an optimal terminal payoff distribution that is right (positively) skewed. A right-skewed distribution ascribes a high probability to low payoffs, and a low probability to large payoffs. Such distribution and a model which generate them, will be unsuitable for investors who are cautious to obtain an acceptable payoff with high probability. This implies that such investors will have a payoff target in mind. They will then seek policies to optimise a payoff measure which is non-symmetric with respect to the target i.e., a measure which will penalised losses. I will introduce several target-based kinked utility measures whose optimisation can generate left (negatively) skewed payoff distributions i.e., such that ascribe a high probability to (relatively) large payoffs and low probability to low payoffs. Measures like that respond to the ideas of Thaler, Kahneman, Tversky and Simon. These Nobel Prize recipients have proposed that human rationality not always can be understood as choosing to act as expected utility maximisers. These researchers tried to bring decision theory closer to real life. In particular, it is Kahneman and Tverski’s prospect theory that tries to mathematically model real-life choices. Their theory describes the way people choose between probabilistic alternatives that involve risk, where the probabilities of outcomes are known. The theory states that people make decisions based on the potential value of losses and gains rather than on the final payoff value. This implies that people have a payoff target in mind. People then seek policies to optimise a payoff measure which is non-symmetric with respect to a target. I will show examples of target-based kinked utility measures and comment on the strategies they generate as well as on the resulting payoff distributions.

### Tuesday, September 24, 2:00 pm, AGR

Speaker: Prof Jan Obloj (University of Oxford)

Title: Robust finance. Part I -- Proof-of-concept Applications using Historical Time Series and Market Option Prices

## Abstract:

In this talk I introduce briefly the robust paradigm which strives to interpolate the modelling spectrum: from agnostic model-free to classical model-specific. It offers methods to walk the spectrum and quantify the impact of making assumptions and/or using market data. I explain briefly how classical fundamental notions and theorems in quantitative finance extend to the robust setting – I plan to cover this in more detail in “Part II – fundamental theorems”. In this Part I, I mostly focus on simple concrete examples. I use vanilla option prices, together with agent-prescribed bounds on key market characteristics, to drive the interval of no-arbitrage prices and the associated hedging strategies. The setting can be seen as a constrained variant of the classical optimal transportation problem and comes with a natural pricing-hedging duality. I discuss numerical methods based on discretisation and LP implementation and on a deep NN optimisation. I look at ways to coherently combine option prices data with past time series data, leading to a dynamic robust risk estimation. I explain how such non-parametric statistical estimators of key quantities (e.g., superhedging prices, 10-days V@R) superimposed with option prices can be treated as information signals. Based on joint works with Stephan Eckstein, Gaoyue Guo, Tongseok Lim and Johannes Wiesel.

### Tuesday, September 17, 2:00 pm, AGR

Speaker: Mr Guanting Liu (UNSW)

Title: On a Positivity Preserving Numerical Scheme for Jump-extended CEV Process: The alpha-stable Case

## Abstract:

We propose a positivity preserving implicit Euler–Maruyama scheme for a jump-extended constant elasticity of variance (CEV) process where the jumps are governed by a compensated spectrally positive alpha-stable process for alpha in (1,2). Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV process, the model considered here has inﬁnite activity jumps. For this speciﬁc model, we calculate the strong rate of convergence. Jump-extended models of this type were initially studied in the context of branching processes and was recently introduced to the ﬁnancial mathematics literature to model sovereign interest rates, power and energy markets.

### Tuesday, August 20, 2:00 pm, AGR

Speaker: Dr Kristoffer Glover (UTS)

Title: Optimally Stopping a Brownian Bridge with an Unknown Pinning Time: A Bayesian Approach

## Abstract:

We consider the problem of optimally stopping a Brownian bridge with an unknown pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we allow the stopper to update their belief about the value of the pinning time through sequential observations of the process. Uncertainty in the pinning time influences both the conditional dynamics of the process and the expected (random) horizon of the optimal stopping problem. Structural properties of the optimal stopping region are shown to be qualitatively different under different prior distributions, however we provide a sufficient condition for the existence of a one-sided stopping region. Certain gamma and beta distributed priors are shown to satisfy this condition and these cases are subsequently considered in detail. A two-point prior distribution is also considered in which a richer structure emerges (with multiple optimal stopping boundaries).

## 2019 Semester 1

### Tuesday, July 30, 2:00 pm, AGR

Speaker: Prof Antoine Ayache (Université de Lille)

Title: Almost Sure Approximations in Hölder Norms of a General Stochastic Process Defined by a Young Integral

## Abstract:

We focus on a stochastic process Y defined by a pathwise Young integral of a general form. Thanks to the Haar basis, we connect the classical method of approximation of Y through Euler scheme and Riemann-Stieltjes sums with a new approach consisting in the use of an appropriate series representation of Y. This representation is obtained through a general compactly supported orthonormal wavelet basis. An advantage offered by the new approach with respect to the classical one is that a better almost sure rate of convergence in Hölder norms can be derived, under a general chaos condition. Also, this improved rate turns out to be optimal in some situations; typically, when the integrand and integrator associated to Y are independent fractional Brownian motions with appropriate Hurst parameters. Joint work with Céline Esser and Qidi Peng.

### Thursday, July 25, 2:00 pm, AGR

Speaker: Dr Alexandru Hening (Tufts University)

Title: The Competitive Exclusion Principle in Stochastic Environments

## Abstract:

The competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. Even though this is a fundamental principle in ecology, it has been observed empirically that in some settings it will fail. One example is Hutchinson's `paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that in some instances (deterministic) temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times. In this talk I will look at how environmental noise interacts with competitive exclusion. I will show that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.

### Tuesday, July 23, 2:00 pm, AGR

Speaker: Dr Alexandru Hening (Tufts University)

Title: Harvesting of Populations in Stochastic Environments

## Abstract:

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold \(x^*>0\) such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is \(C^2\) and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

### Tuesday, May 28, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Mr Edward Kim (University of Sydney, School of Mathematics and Statistics)

Title: BSDEs Driven by Discontinuous Martingales

## Abstract:

We prove some new results on BSDEs, reflected BSDEs, and doubly reflected BSDEs driven by a multi-dimensional discontinuous martingale. We generalize the setups studied by Peng and Xu (2009) and Quenez, Grigorova, Sulem and Dumitrescu (2018) who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. Our results are not covered by existing results on BSDEs driven by a Brownian motion and a Poisson random measure.

### Tuesday, May 21, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Ben Goldys (University of Sydney, School of Mathematics and Statistics)

Title: On the Existence of Stochastic Flows

## Abstract:

An important question in the theory of stochastic (ordinary and partial) differential equations is about the existence and regularity of the stochastic flow associated to solutions of the equation. It has been studied by many authors, including Elworthy, Flandoli, Malliavin, Bismut, Ikeda and Watanabe, and Kunita. It is still a major open problem in the case of stochastic PDEs and even for ordinary stochastic differential equations many questions remain unanswered. In this talk I will discuss the difficulties arising in the case of stochastic PDEs and present some partial answers. This is a joint work with Szymon Peszat.

### Tuesday, May 14, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Mr James Yang (University of Sydney, School of Mathematics and Statistics)

Title: Singular Perturbation of Zero-sum Linear-Quadratic Stochastic Differential Games

## Abstract:

We develop a framework for zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by multiscale state equations. The multiscale nature of the problem can be leveraged to reformulate the associated generalised Riccati equation in terms of a deterministic singular perturbation problem. In doing so, we can show that, for small enough \(\epsilon\), the existence of solution to the associated generalised Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. Furthermore, we construct two asymptotic estimates to the closed-loop value of the game. The first is by constructing an approximate closed-loop strategy and the second is by observing the limiting value as \(\epsilon\) goes towards zero. This is a joint work with Ben Goldys and Zhou Zhou.

### Tuesday, April 30, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Anna Aksamit (University of Sydney, School of Mathematics and Statistics)

Title: The Robust Pricing–Hedging Duality for American Options in Discrete Time Financial Markets (Part II)

## Abstract:

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry-Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing-hedging duality for American options. This is joint work with Shuoqing Deng, Jan Obłój and Xiaolu Tan.

### Tuesday, April 23, 3:10 pm, Your favourite beach or park

Easter break

## Abstract:

Partly cloudy. Medium (50%) chance of showers in the afternoon and evening. Light winds.### Tuesday, April 16, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Marek Rutkowski (University of Sydney, School of Mathematics and Statistics)

Title: Valuation under Credit Risk, Margins and Funding Costs (Part II)

## Abstract:

We present a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. We show that this unifying theoretical framework clarifies the relationship between the two valuation approaches: the so-called adjusted cash flows approach and the classic replication approach. In particular, results of this work cover most previous papers where the authors studied specific replication-based models. This is joint work with Damiano Brigo, Cristin Buescu, Marco Francischello and Andrea Pallavicini.

### Tuesday, April 9, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Marek Rutkowski (University of Sydney, School of Mathematics and Statistics)

Title: Valuation under Credit Risk, Margins and Funding Costs (Part I)

## Abstract:

We develop a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication-based approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. We first discuss practical issues motivating our research and we stress that the classic approach is not suitable when dealing with over-the-counter collateralized contracts under differential funding costs. This is joint work with Damiano Brigo, Cristin Buescu, Marco Francischello and Andrea Pallavicini.

### Tuesday, April 2, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Anna Aksamit (University of Sydney, School of Mathematics and Statistics)

Title: The Robust Pricing–Hedging Duality for American Options in Discrete Time Financial Markets (Part I)

## Abstract:

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality. We show that it is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. This is joint work with Shuoqing Deng, Jan Obłój and Xiaolu Tan.

### Tuesday, March 26, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Zhou Zhou (University of Sydney, School of Mathematics and Statistics)

Title: Transport Plans with Domain Constraints (Part II)

## Abstract:

This talk is the second part of my presentation commenced on March 12. I will first recall the main results presented in the first part. Subsequently, I will focus on the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle, which gives us a geometric way of identifying the optimizer.

### Tuesday, March 19, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Prof Marek Musiela (University of Oxford and Oxford-Man Institute of Quantitative Finance)

Title: Multivariate Fractional Brownian Motion and Generalizations of SABR Model

## Abstract:

SABR is a generalization of the Black and Scholes model which was adopted as the market standard for quoting cap and swaption volatilities. At the time it fitted the market relatively well. Its success was driven by the approximate formula for implied volatility developed by Pat Hagan. More recently, SABR was applied to price equity and FX options. This generated new challenges for the SABR framework and suggested a new class of models inspired by SABR, where the process defining the noise is a bivariate fractional Brownian motion with parameter \((1/2, H)\). In the classical SABR, the correlation parameter between the two Brownian motions determines many of its mathematical properties. In the modified SABR, a Brownian motion defining the dynamics of stochastic volatility is replaced by a fractional Brownian motion. This leads to the question how one should define the dependence structure between the Brownian motion driving the asset dynamics and the fractional Brownian motion used to define the volatility. In this talk, I propose dependence structure implicit to the definition of multivariate fractional Brownian motion with multivariate self-similarity parameter \(H\). I also consider the consequences and suggest further modifications.

### Tuesday, March 12, 3:10 pm, Wilkinson Seminar Room 231

Speaker: Dr Zhou Zhou (University of Sydney, School of Mathematics and Statistics)

Title: Transport Plans with Domain Constraints (Part I)

## Abstract:

Let \(\Omega\) be one of \(\mathbb{X}^{N+1},C[0,1],D[0,1]\): product of Polish spaces, space of continuous functions from \([0,1]\) to \(\mathbb{R}^d\), and space of RCLL (right-continuous with left limits) functions from \([0,1]\) to \(\mathbb{R}^d\), respectively. We first consider the existence of a probability measure \(P\) on \(\Omega\) such that \(P\) has the given marginals \(\alpha\) and \(\beta\) and its disintegration \(P_x\) must be in some fixed \(\Gamma(x)\subset \mathfrak{P}(\Omega)\), where \(\mathfrak{P}(\Omega)\) is the set of probability measures on \(\Omega\). The main application we have in mind is the martingale optimal transport problem in mathematical finance when the martingales are assumed to have bounded volatility/quadratic variation. We show that such probability measure exists if and only if the \(\alpha\) average of the so-called \(G\)-expectation of bounded continuous functions with respect to the measures in \(\Gamma\) is less than their \(\beta\) average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. This is joint work with Erhan Bayraktar and Xin Zhang.