Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function: $$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$ such that $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dB_t, \quad X_t =x$$ and $\mathcal{F}_t$ is the filtration generated by the brownian motions up to time t. It is, using the martingale representation theorem: $$dV_t=rV_tdt-f(t,X_t)dt+\theta_tdB_t$$ For some square integrable process $\theta_t$.

However, in an application, I'm struggling to find the law of motion of the change in the value function when one perturbs the initial condition $x$ ie the law of motion of $W_t(x)=\dfrac{\partial V_t}{\partial x}$. Any hints or reference on how I could proceed to find it?

• What do you mean by "law of motion"? What I see is the Feynman-Kac formula. – zhoraster Sep 13 '15 at 11:05
• Thank you for asking, by law motion I mean how can I write it as an Ito process $dW_t=\mu dt+\sigma dB_t$. A differential equation satisfied by $W$ is also okay. Applying Feynman-Kac to $V_t$ is not exactly what I'm trying to do. Thanks. – skillfeedback Sep 14 '15 at 22:48
• then what are you trying to do? Writing the stochastic differential for derivative? Something else? – zhoraster Sep 15 '15 at 3:12
• Dear @zhoraster, I'm trying to write the stochastic differential for the derivative yes. – skillfeedback Sep 15 '15 at 14:05