Smoothness of expectation

Question

Suppose that $X_t$ is a strong solution to the SDE, $$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by the Brownian motion.

Let $T > 0$ be a constant. Suppose $X_t = x \in (-1,1)$. Define, $\tau := \inf \{s \ge t: X_s \notin (-1,1)\} \wedge T$ and $$J(t,x) := \mathbb E \left[\int_t^\tau f(X_s, C_s) \, ds + h(\tau, X_\tau)\right].$$

I am interested in knowing what can be said about the smoothness of $J(\cdot, \cdot)$ in $t$ and $x$. In particular, if $f(x,c)$ is continuous in $x$ and u.s.c in $c$, and $h(t,x)$ is continuous in both the arguments, is $J(t,x)$ continuous in $t$ and usc in $x$? How about continuous in $x$?

Answer 1

First we apply time change to write $X_{t}=x+B_{A_{t}}$ with $A_{t}:=\int_{0}^{t}C_{s}^{2}ds$ with inverse $g(s)=A^{-1}(s)$. So we have

$$\tau_{X,t}=t+\inf\{s\geq 0: B_{A_{s+t}}\in (-1-x,1-x)\}=t+\inf\{g(s)\geq t: B_{s}\in (-1-x,1-x)\},$$

and thus $\tau_{X,t}=t+g(T_{I,t})$ where $I:= (-1-x,1-x)$ and $T\geq A_{t}$ is the exit time from $I$ after time $A_{t}$.

So to get continuity in t,x, we use the continuity of the above processes eg. exit time $T_{I,t}$ is continuous using a reflection principle argument as here https://math.stackexchange.com/questions/3202704/derivation-of-joint-and-conditional-density-of-a-brownian-motion-and-its-maximum/3209052#3209052

For general dependence on parameters see here: Differentiable dependence on the initial condition of the solution of a SDE