Weak Dirichlet processes with a stochastic control perspective.

Fausto Gozzi (1) , Francesco Russo (2)

1 Dipartimento di Scienze Economiche e Aziendali
2 LAGA - Laboratoire Analyse, Géométrie et Applications

The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case that the value function is assumed to be continuous in time and once differentiable in the space variable ($C^{0,1}$) instead of once differentiable in time and twice in space ($C^{1,2}$), like in the classical results. For this purpose, the replacement tool of the It\^{o} formula will be the Fukushima-Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the sum of a local martingale $M$ plus an adapted process $A$ which is orthogonal, in the sense of covariation, to any continuous local martingale. The mentioned decomposition states that a $C^{0,1}$ function of a weak Dirichlet process with finite quadratic variation is again a weak Dirichlet process. That result is established in this paper and it is applied to the strong solution of a Cauchy problem with final condition. Applications to the proof of verification theorems will be addressed in a companion paper.

Mots clés: Stochastic calculus via regularization; weak Dirichlet processes; stochastic optimal control; Cauchy problem for parabolic partial differential equations.; Cauchy problem for parabolic partial differential equations

Domaines: Probabilités [math.PR]

Fausto Gozzi, Francesco Russo. Weak Dirichlet processes with a stochastic control perspective.. 2006. ⟨hal-00022839⟩