In this work, we prove error expansions for the approximation of BSDEs when using the cubature method. To profit fully from these expansions, e.g. to design high order approximation methods, we need however to control the complexity growth of the cubature method. In our work, this is achieved by using interpolation methods. We present several numerical results that confirm the efficiency of our method. This is a joint work with C. Garcia (UCL)
We propose a general methodology to describe behaviors in optimal stopping problems without the time-consistent property (general discounting, rank dépendent utility). We ground our theory on the classical intra-personal game formulation but in contrast with optimal control problems, we do not rely on a variational formulation of Nash equilibrium. We discuss existence, uniqueness and Pareto optimality of proposed solutions. This is a joint work with Yu-Jui Huang (Boulder) and Xunyu Zhou (Columbia).
We model the behavior of three agent classes acting dynamically in a limit order book of a financial asset. Namely, we consider market makers (MM), high-frequency trading (HFT) firms, and institutional brokers (IB). Given a prior dynamic of the order book, similar to the one considered in the Queue-Reactive models [14, 20, 21], the MM and the HFT define their trading strategy by optimizing the expected utility of terminal wealth, while the IB has a prescheduled task to sell or buy many shares of the considered asset. We derive the variational partial differential equations that characterize the value functions of the MM and HFT and explain how almost optimal control can be deduced from them. We then provide a first illustration of the interactions that can take place between these different market participants by simulating the dynamic of an order book in which each of them plays his own (optimal) strategy.
In this talk, we are going to introduce the partial hedging problem in mathematical finance. First, we present the stochastic optimal control problem and the PDE that the value function solves (in the viscosity sense), together with the comparison theorem, which ensures uniqueness for the solution of the PDE. We then introduce a numerical scheme to approximate the solution. This numerical scheme is based on Piecewise Constant Policy Timestepping (PCPT). We prove its convergence and show some numerical examples. This is a joint work with Jean-François Chassagneux and Christoph Reisinger.