We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. In both cases, we develop new theoretical tools to provide explicit sufficient conditions under which our probabilistic representations hold. As an application, we consider several examples including multi-dimensional semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo method.
In this note, we propose two different approaches to rigorously justify a pseudoMarkov property for controlled diffusion processes which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. The first approach develops a sketch of proof proposed by Fleming and Souganidis [9]. The second approach is based on an enlargement of the original state space and a controlled martingale problem. We clarify some measurability and topological issues raised by these two approaches.
In this paper, we aim to develop the theory of optimal stochastic control for branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing a criterion that is expressed as the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio [27] to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to recover a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.
In this paper, we provide some results on Skorokhod embedding with local time and its applications to the robust hedging problem in finance.First we investigate the robust hedging of options depending on the local time by using the recently introduced stochastic control approach, in order Julien Claisse, Corresponding authoŕ Ecole Polytechnique
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