Xiaolu Tan

Optimal transportation under controlled stochastic dynamics

We consider an extension of the Monge–Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale. The optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions. Our first main result is an extension of the Kantorovitch duality to this context. We also suggest a finite-difference scheme combined with the gradient projection algorithm to approximate the dual value. We prove the convergence of the scheme, and we derive a rate of convergence. We finally provide an application in the context of financial mathematics, which originally motivated our extension of the Monge–Kantorovitch problem. Namely, we implement our scheme to approximate no-arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity.

A numerical algorithm for a class of BSDEs via the branching process

We generalize the algorithm for semi-linear parabolic PDEs in Henry-Labordere to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren, Keller, Touzi and Zhang and extended in Ekren, Touzi and Zhang.

Optimal Skorokhod embedding under finitely-many marginal constraints

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod embedding problem to the case of finitely-many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results which are formulated by means of probability measures on an enlarged space. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.

A general Doob-Meyer-Mertens decomposition for g-supermartingale systems

We provide a general Doob-Meyer decomposition for

Discrete-time probabilistic approximation of path-dependent stochastic control problems

g

A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations

-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Pengs (1999) version for right-continuous

On the Monotonicity Principle of Optimal Skorokhod Embedding Problem

g

Weak approximation of second-order BSDEs.

-supermartingales. As examples of application, we prove an optional decomposition theorem for

Numerical approximation of BSDEs using local polynomial drivers and branching processes

g

Exact Simulation of Multi-Dimensional Stochastic Differential Equations

-supermartingale systems, and also obtain a general version of the well-known dual formation for BSDEs with constraint on the gains-process, using very simple arguments.