Romuald Elie

Stochastic Target Problems with Controlled Loss

We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e., finding the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike in the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics and are shown to recover the explicit solution of Follmer and Leukert [Finance Stoch., 3 (1999), pp. 251-273].

Optimal Control under Stochastic Target Constraints

We study a class of Markovian optimal stochastic control problems in which the controlled process

BSDE representations for optimal switching problems with controlled volatility

Z^{\nu}

Discrete-time Approximation of Multidimensional BSDEs with oblique reflections

is constrained to satisfy an almost sure constraint

BSDEs with mean reflection

Z^{\nu}(T)\in G\subset\mathbb{R}^{d+1}

Approximate hedging for nonlinear transaction costs on the volume of traded assets

\mathbb{P}

Discrete time approximation of decoupled Forward-Backward SDE with jumps

-a.s. at some final time

A simple constructive approach to quadratic BSDEs with or without delay

T>0

Discrete-Time Approximation of BSDEs and Probabilistic Schemes for Fully Nonlinear PDEs

. When the set is of the form