The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:

Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we control the intensity $\lambda$. Let $X$ be a process on $[0,T]$ with dynamics

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_t=x,$$

with $x,\mu,\sigma,\gamma\in\mathbb R$, $t\in [0,T]$, $u\in \mathcal A$.

Consider an optimization problem of the form

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_T) +\int_t^T f(X^u_s,u_s)ds\right).$$

Under certain assumptions the previous value function is a (viscosity) solution of the HJB equation:

$$v_t + \mu v_x + \frac{1}{2}\sigma^2v_{xx} + \sup_{u\in U} \left\{\lambda^u(t)(v(t,x+\gamma) - v(t,x)) + f(x,u)\right\}$$

$$v(T,.) = g(.) $$

My question is: how would the previous equation (and its terminal condition) change if we were to change the terminal time $T$ for a stopping time $\tau\leq T$ in the definition of the value function. i.e.,

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_{\tau}) +\int_t^{\tau} f(X^u_s,u_s)ds\right).$$

Can anything be said in this situation?

What if we added the hypothesis if $\tau$ being absorbent for $X$? i.e., $X$ is constant over $[\tau,T]$.

Any reference would be very much appreciated. Oksendal-Sulem's book entitled "Applied Stochastic Control of Jump Diffusions", for example, as comprehensive as it is, does not deal with value functions with time dependence.

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:

Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we control the intensity $\lambda$. Let $X$ be a process on $[0,T]$ with dynamics

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_t=x,$$

where $x,\mu,\sigma,\gamma\in\mathbb R$, $t\in [0,T]$, $u\in \mathcal A$.

Consider an optimization problem of the form

$$ V(t,x) = \sup_{u\in \mathcal A} E\left( g(X^u_T) +\int_t^T f(X^u_s,u_s)ds\right).$$

Under certain assumptions the previous value function is a (viscosity) solution of the HJB equation:

$$v_t + \mu v_x + \frac{1}{2}\sigma^2v_{xx} + \sup_{u\in U} \left\{\lambda^u(t)(v(t,x+\gamma) - v(t,x)) + f(x,u)\right\}=0$$

$$v(T,.) = g(.) $$

My question is: how would the previous equation (and its terminal condition) change if we were to change the terminal time $T$ for a stopping time $\tau\leq T$ in the definition of the value function. i.e.,

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_{\tau}) +\int_t^{\tau} f(X^u_s,u_s)ds\right).$$

Can anything be said in this situation?

What if we added the hypothesis of $\tau$ being absorbent for $X$? i.e., $X$ is constant over $[\tau,T]$.

Any reference would be very much appreciated. Oksendal-Sulem's book entitled "Applied Stochastic Control of Jump Diffusions", for example, as comprehensive as it is, does not deal with value functions with time dependence.

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:

Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we control the intensity of $\lambda$. Let $X$ be a process on $[0,T]$ with dynamics

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_t=x,$$

with $x,\mu,\sigma\in\mathbb R$, $t\in [0,T]$, $u\in \mathcal A$.

Consider an optimization problem over of the form

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_T) +\int_t^T f(X^u_s,u_s)ds\right).$$

Under certain assumptions the previous value function is a (viscosity) solution of the HJB equation:

$$v_t + \mu v_x + \frac{1}{2}\sigma^2v_{xx} + \sup_{u\in U} \left\{\lambda^2u(t) (v(t,x+\gamma) - v(t,x)) + f(x,u)\right\}$$

$$v(T,.) = g(.) $$

My question is: how would the previous equation change if we were to change the initial time $t$ for a stopping time $\tau_1\leq T$ in the SDE of $X$ and the terminal time $T$ for another stopping time $\tau_1\leq \tau_2\leq T$ in the definition of the value function. i.e.,

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_{\tau_1}=x,$$

and

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_{\tau_2}) +\int_{\tau_1}^{\tau_2} f(X^u_s,u_s)ds\right).$$

Any reference would be very much appreciated. Oksendal-Sulem's book entitled "Applied Stochastic Control of Jump Diffusions", for example, as comprehensive as it is, does not deal with value functions with time dependence.

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:

Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we control the intensity $\lambda$. Let $X$ be a process on $[0,T]$ with dynamics

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_t=x,$$

with $x,\mu,\sigma,\gamma\in\mathbb R$, $t\in [0,T]$, $u\in \mathcal A$.

Consider an optimization problem of the form

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_T) +\int_t^T f(X^u_s,u_s)ds\right).$$

Under certain assumptions the previous value function is a (viscosity) solution of the HJB equation:

$$v_t + \mu v_x + \frac{1}{2}\sigma^2v_{xx} + \sup_{u\in U} \left\{\lambda^u(t)(v(t,x+\gamma) - v(t,x)) + f(x,u)\right\}$$

$$v(T,.) = g(.) $$

My question is: how would the previous equation (and its terminal condition) change if we were to change the terminal time $T$ for a stopping time $\tau\leq T$ in the definition of the value function. i.e.,

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_{\tau}) +\int_t^{\tau} f(X^u_s,u_s)ds\right).$$

Can anything be said in this situation?

What if we added the hypothesis if $\tau$ being absorbent for $X$? i.e., $X$ is constant over $[\tau,T]$.

Any reference would be very much appreciated. Oksendal-Sulem's book entitled "Applied Stochastic Control of Jump Diffusions", for example, as comprehensive as it is, does not deal with value functions with time dependence.

The following is one version of the HJB equation:

Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we control the intensity of $\lambda$. Let $X$ be a process on $[0,T]$ with dynamics

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_t=x,$$

with $x,\mu,\sigma\in\mathbb R$, $t\in [0,T]$, $u\in \mathcal A$.

Consider an optimization problem over of the form

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_T) +\int_t^T f(X^u_s,u_s)ds\right).$$

Under certain assumptions the previous value function is a (viscosity) solution of the HJB equation:

$$v_t + \mu v_x + \frac{1}{2}\sigma^2v_{xx} + \sup_{u\in U}\left\{\lambda^u(t) (v(t,x+\gamma) - v(t,x)) + f(x,u)\right\}$$ $$v(T,.) = g(.) $$

My question is: how would the previous equation change if we were to change the initial time $t$ for a stopping time $\tau_1\leq T$ in the SDE of $X$ and the terminal time $T$ for another stopping time $\tau_1\leq \tau_2\leq T$ in the definition of the value function. i.e.,

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_{\tau_1}=x,$$

and

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_{\tau_2}) +\int_{\tau_1}^{\tau_2} f(X^u_s,u_s)ds\right).$$

Any reference would be very much appreciated. Oksendal-Sulem's book for example, as comprehensive as it is, does not deal with value functions with time dependence.

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:

Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we control the intensity of $\lambda$. Let $X$ be a process on $[0,T]$ with dynamics

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_t=x,$$

with $x,\mu,\sigma\in\mathbb R$, $t\in [0,T]$, $u\in \mathcal A$.

Consider an optimization problem over of the form

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_T) +\int_t^T f(X^u_s,u_s)ds\right).$$

Under certain assumptions the previous value function is a (viscosity) solution of the HJB equation:

$$v_t + \mu v_x + \frac{1}{2}\sigma^2v_{xx} + \sup_{u\in U} \left\{\lambda^2u(t) (v(t,x+\gamma) - v(t,x)) + f(x,u)\right\}$$

$$v(T,.) = g(.) $$

My question is: how would the previous equation change if we were to change the initial time $t$ for a stopping time $\tau_1\leq T$ in the SDE of $X$ and the terminal time $T$ for another stopping time $\tau_1\leq \tau_2\leq T$ in the definition of the value function. i.e.,

$$dX^u_s = \mu ds + \sigma dW_s + \gamma dN^u_s$$ $$X_{\tau_1}=x,$$

and

$$ V(t,x) = \sup_{u\in \mathcal A_t} E\left( g(X^u_{\tau_2}) +\int_{\tau_1}^{\tau_2} f(X^u_s,u_s)ds\right).$$

Any reference would be very much appreciated. Oksendal-Sulem's book entitled "Applied Stochastic Control of Jump Diffusions", for example, as comprehensive as it is, does not deal with value functions with time dependence.