Weak Dirichlet processes with a stochastic control perspective

By Fausto Gozzi, Francesco Russo. 2006

63 citations recorded by the source corpus.

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 where 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 Ito 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 decomposition mentioned 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.

Subfields: Computational Theory and Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d38fbe9c12ff6e04ca4e1f2ae4ce7d02f5045c48

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Citations in the corpus

Elements of Stochastic Calculus via Regularization (2006, methodology): “The following result was established in [23].”