2
$\begingroup$

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\tau$: $$V_t=\sup_{\tau,(A_s)_{s\geq t}}E\Big{[}\int^{\tau}_te^{-\rho (s-t)}f(s,X_s,A_s)ds+e^{-\rho (\tau-t)}g(\tau, X_{\tau})|\mathcal{F}_t\Big{]}$$ such that $$dX_t=\mu(t,X_t,A_t)dt+\sigma(t,X_t,A_t)dB_t, \quad X_t =x$$ and $\mathcal{F}_t$ is the filtration generated by the brownian motions up to time t. It is $$0 = \sup_a \Big{[} g(t,x)-v(t,x),-\rho v(t,x) + f(t,x,a) + (\partial_tv)(t,x)+\mu(t,x,a) (\partial_x v)(t,x) + \frac{\sigma^2(t,x,a)}{2}(\partial_{xx}(t,x) \Big{]}$$.

In an application I stumble upon following the control problem where the optimal stopping time changes the drift and(/or) volatility of the process: $$V_t=\sup_{\tau,(A_s)_{s\geq t}}E\Big{[}\int^{\tau}_te^{-\rho (s-t)}f(s,X_s,A_s)ds+e^{-\rho (\tau-t)}g(\tau, X_{\tau},A_{\tau})|\mathcal{F}_t\Big{]}$$ such that $$dX_t=\mu_1(t,X_t,A_t)dt+\sigma_1(t,X_t,A_t)dB_t, \quad X_t =x, \quad if \quad t \leq \tau$$ and $$dX_t=\mu_2(t,X_t,A_{\tau})dt+\sigma_2(t,X_t,A_{\tau})dB_t, \quad X_t =x, \quad if \quad t > \tau$$.

The differences are that the optimal stopping time $\tau $ change the law of motion of $X_t$ and the the control at that time $A_{\tau}$ affect the drift and volatility of the process and the terminal payoff at stopping.

I couldn't find the HJB equation (assuming everything is well defined). Any hints on how I could proceed to find it?

$\endgroup$
1
$\begingroup$

This is not an answer. But I think you can find the answer in T. Björk, Finite dimensional optimal filters for a class of Ito processes with jumping parameters, Stochastics, 4 (1980), 167–183.

or the textbook here.

Your SDE is a special case of the regime switching class.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.