A Stochastic Target Problem for Branching Diffusions

Idris Kharroubi (1) , Antonio Ocello (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. We then show that the value function can be reduced to a new function with a finite dimensional argument by a so called branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.

Mots clés: Stochastic control; fintech; cryptocurrencies options; super-replication branching diffusion process; dynamic programming principle; Hamilton-Jacobi-Bellman equation; viscosity solution

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Idris Kharroubi, Antonio Ocello. A Stochastic Target Problem for Branching Diffusions. 2022. ⟨hal-03695036v3⟩