# Stochastics and Finance Seminar

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

## 2021

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

Speaker: Yu-Jui Huang (University of Colorado)

Title: tba

## Abstract:

tba

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** 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.

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** 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.

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** 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

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** 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.

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** 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

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.

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** 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.