Control of a Solution of a Stochastic Integral Equation

Let $\xi(t)$ be a Wiener process in $E_n$, $\alpha_n$ a non-anticipative vector function, $\delta=\{\alpha_t\}$, $x_t^{\delta,x}$ a solution of
$$ x_t=x+\int_0^t\sigma(x_s,\alpha_s)d\xi_s+\int_0^t b(x_s,\alpha_s)\,ds, $$
$\varphi=\varphi(x)$. In this paper, smouthness of functions
$$ v(x)=\sup_{\delta,\tau}\mathbf{M}\biggl[\int_0^\tau e^{-\lambda t}f(x_t^{\delta,x},\alpha_t)\,dt+e^{-\lambda\tau}\varphi(x_\tau^\delta,x)\biggr] $$
is investigated.
Under conditions of smouthness type on $\sigma,b,f,\varphi$ it is proved that $v\in W_{p,\textrm{loc}}^2$ (Sobolev space). If, in addition, $\sigma\sigma^*$ is strictly positive-definite, then
$$ \sup_\alpha (L^\alpha v+f^\alpha)\leq 0\ (\textrm{a.e.}), \quad \sup_\alpha (L^\alpha v+f^\alpha)=0\ (\textrm{a.e.}\ \{x: v(x)>\varphi(x)\}). $$

The structure of $\varepsilon$-optimal policies $\delta$ and $\varepsilon$-optimal stopping times $\tau$ is also studied.