I'm looking at a trio of processes which arises in a stochastic control situation. I have a process $(V_t)$ which I may control, and $(V_t)$ influences a diffusive stock price process $(S_t)$. The evolution of my wealth function $(W_t)$ depends on both $(V_t)$ and $(S_t)$. Without giving the explicit SDEs, are there any techniques one can evaluate $E[W_t]$ numerically?
Many thanks.