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King's College London

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The programme for this event can be found here.

This workshop overviews new directions of research in stochastic control, covering cutting-edge topics such as optimal transport, mean-field games, and control of McKean Vlasov (backward) stochastic differential equations etc., including recent advances on both theory and numerics.

This will take place at Imperial College London in the Clore Lecture theatre all day on Monday and until noon on Tuesday, and Huxley 144 on Tuesday afternoon.

  • We can offer 5 PhD students £100 GBP for travel support. Interested candidates should submit a CV to Enfale Farooq (e.farooq@lse.ac.uk)

Speakers:

Title: Itô-Dupire formula for $C^1$-functionals and approximate viscosity solutions of PPDE

Abstract: We review some recent results on Itô’s formula for path-dependent functionals that are either only $C^1$, or, concave in space and non-increasing in time. This leads to the study of the regularity of candidate solutions to path-dependent parabolic PDEs for which we introduce a notion of approximate viscosity solutions. Applications to perfect hedging in markets with price impact and to super-hedging under model uncertainty will be discussed.

 

Title: Correlated equilibria and mean field games

Abstract: Mean field games are limit models for symmetric N-player games, as N tends to infinity, where the prelimit models are solved in terms of Nash equilibria. A generalization of the notion of Nash equilibrium, due to Robert Aumann (1973, 1987), is that of a correlated equilibrium. Here, we discuss, in a simple discrete time setting, the mean field game limit for correlated equilibria. We give a definition of correlated mean field game solution, prove that it arises as limit of N-player correlated equilibria in restricted ("open-loop") Markov feedback strategies, and show how to construct approximate N-player equilibria starting from a correlated mean field game solution. We also show how to adapt the definition of correlated solution to the case where the players in the pre-limit game are allowed to deviate following semi-Markov strategies, i.e. depending on the other players’ states via the empirical measure. This talk is based on joint works with Ofelia Bonesini and Markus Fischer (Padova University).

 

Title: Mean field game of mutual holding and systemic risk

Abstract: We introduce a mean field model for optimal holding of a representative agent of her peers as a natural expected scaling limit from the corresponding $N-$agent model. The induced mean field dynamics appear naturally in a form which is not covered by standard McKean-Vlasov stochastic differential equations. We study the corresponding mean field game of mutual holding in the absence of common noise. Our first main result provides an explicit equilibrium of this mean field game, defined by a bang-bang control consisting in holding those competitors with positive drift coefficient of their dynamic value. As a second main result, we use this mean field game equilibrium to construct (approximate) Nash equilibria for the corresponding $N$-player game. All of these results extend to the defaultable agent setting. Our last main result is a characterization of the default probability.

 

Title: Reducing Obizhaeva-Wang type trade execution problems to LQ stochastic control problems

Abstract: In this talk we start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear quadratic (LQ) stochastic control problem. Using the well developed theory on LQ problems we describe the solution to the obtained LQ one and trace it back up to the solution to the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. The talk is based on joint work with Julia Ackermann and Mikhail Urusov.

 

Title: Optimal Transport perspective on robustness of a stochastic optimization problem to model uncertainty

Abstract: We consider the sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present several applications and extensions of the above results in decision theory, mathematical finance and beyond. We consider in particular robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and examine in detail optimal investment, utility indifference pricing and Davis’ option pricing in a simple one-period model.

The talk is based on joint works with Daniel Bartl, Samuel Drapeau and Johannes Wiesel.

 

Title: Controlled measure-valued martingales: a viscosity solution approach

Abstract: We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We establish the ‘classical’ results of stochastic control for these problems: specifically, we show that the value function for the problem can be characterised as the unique solution to a Hamilton-Jacobi-Bellman equation in the sense of viscosity solutions. In order to obtain this, we exploit structural properties of the MVM processes; in particular, our results include existence of controlled MVMs and an appropriate version of Itô’s formula for such processes. We also illustrate how problems of this type arise in a number of applications including model-independent derivatives pricing. The talk is based on joint work with Alex Cox, Martin Larsson and Sara Svaluto-Ferro.

 

Title: On pricing rules and optimal strategies in general Kyle-Back models

Abstract: The folk result in Kyle-Back models states that the value function of the insider remains unchanged when her admissible strategies are restricted to absolutely continuous ones. In this talk I will show that, for a large class of pricing rules used in current literature, the value function of the insider can be finite when her strategies are restricted to be absolutely continuous and infinite when this restriction is not imposed. This implies that the folk result doesn't hold for those pricing rules and that they are not consistent with equilibrium. I will derive the necessary conditions for a pricing rule to be consistent with equilibrium and prove that, when a pricing rule satisfies these necessary conditions, the insider's optimal strategy is absolutely continuous, thus obtaining the classical result in a more general setting.

This, furthermore, allows justifying the standard assumption of absolute continuity of insider's strategies since one can construct a pricing rule satisfying the derived necessary conditions that yield the same price process as the pricing rules employed in the modern literature when insider's strategies are absolutely continuous.

 

Title: On time-inconsistent stopping problems and mixed strategy stopping times

Abstract: Time inconsistent problems of stochastic control and in particular optimal stopping, have been intensively studied in recent years. The main focus has been on the game theoretic formulation and the search for subgame perfect Nash equilibria. Different equilibrium concepts have been proposed and for some classes of problems the existence of (pure) equilibria has been proved. In this talk we introduce the concept of mixed equilibria in this framework and show how this leads to the solvability of larger classes of problems. In addition, we discuss the usefulness of the emergent class of special randomized stopping times in other contexts.

The talk is based on joint work with Kristoffer Lindensjö (Stockholm University).

 

Title: Trading with the Crowd

Abstract: We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact,

along with taking into account a common general price predicting signal.

The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents. We prove that the equilibrium trading speed and the value function of an agent in the finite $N$-player game converges to the corresponding trading speed and value function in the mean field game at rate $O(N^{-2})$. In addition, we prove that the mean field optimal strategy provides an approximate Nash-equilibrium for the finite-player game.

 

Title: An equilibrium model of production and capacity expansion

Joint work with Junchao Jia and Michael Zervos.

 

Title: Optimal management of pumped hydroelectric production with state constrained optimal control (joint work with Tiziano Vargiolu)

Abstract: We present a novel technique to solve the problem of managing optimally a pumped hydroelectric storage system. This technique relies on representing the system as a stochastic optimal control problem with state constraints, these latter corresponding to the finite volume of the reservoirs. Following the recent level-set approach presented in O. Bokanowski, A. Picarelli, H. Zidani, State-constrained stochastic optimal control problems via reachability approach, SIAM J. Control and Optim. 54 (5) (2016), we transform the original constrained problem in an auxiliary unconstrained one in augmented state and control spaces, obtained by introducing an exact penalization of the original state constraints. The latter problem is fully treatable by classical dynamic programming arguments.

 

Title: Multi-dimensional reflected BSDEs and applications

Abstract: In this talk, I want to present some new existence and uniqueness results for classes of multi-dimensional reflected BSDEs. I will first motivate the study of multi-dimensional reflected BSDEs by the presentation of randomized switching problems. These are extensions of classical switching problems, which allow for uncertainty in the state reached by the system after the switching times. The value process and the optimal control associated to these new switching problems can be determined by using specific obliquely reflected BSDEs in convex domain. I will then introduce reflected BSDEs in non-convex domain and explain the main difficulty encountered in their study.

This talk is based on joint works with C. Bénézet, S. Nadtochiy and A. Richou.

 

Title: Regret in Trading Decisions: A Model and Empirical Study

Abstract: In this talk we will present a simple model of asset sales under dynamic regret. We will use this to motivate our empirical study of trading decisions using a large discount brokerage dataset. Our focus is to test how regret induced by not selling a stock at its maximum price shapes the propensity to sell.

We undertake a number of descriptive analyses and more formal analysis via proportional hazard modelling.

 

Title: Pathwise construction of revealed utilities, a forward inverse problem

Abstract: Access here

Organisers/contacts:

Please contact Dr Dumitrescu with any questions.

This event is sponsored by the London Mathematical Society, Imperial College London,  London School of Economics and King's College London.

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Event details


Imperial College London, South Kensington Campus
Imperial College London, South Kensington Campus, Exhibition Road, London, SW7 2AZ

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