2006/04/14 by Fausto Gozzi, Gozzi, Fausto, Francesco Russo +1 · 2 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.math/0604326
openalex publication_date 2006/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (C0,1) instead of once differentiable in time and twice in space (C1,2), like in the classical results. For this purpose, the replacement tool of the Itô 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 C0,1 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.