2008/06/11 by Marco Fuhrman, Federica Masiero, Gianmario Tessitore · 1 citation
Mathematics · #math.PR
published as Siam Journal on Control and Optimization 48 Issue: 7 (2010) 4624-4651
arxiv created 2008/06/11 · arxiv updated 2013/04/10
We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions \mathbf C, with generator \mathcal L. We then consider a backward stochastic differential equation depending on X, with unknown processes (Y,Z), and we study properties of the resulting system, in particular we identify the process Z as a deterministic functional of X. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space \mathbf C driven by \mathcal L, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process X.