As part of my research, I have come across the following problem and I am trying to tackle it.
Let $(X_t)_{0 \leq t \leq T}$ be a mean controlled Brownian Motion with the following dynamics
\begin{equation}
X_t = B_t + \int_0^t \theta_s ds
\end{equation}
and we have the following cost function
\begin{equation}
C(t,x;\theta) = \mathbb{E}_{t,x} \left[ \int_t^T \left( \frac{1}{2} X_s^2 + |\theta_s|\right) ds \right]
%\[\frac{1}{2} \]
\end{equation}
I would like to establish that Value function $V(t,x):= \min_{\theta} C(t,x;\theta)$ to be the unique viscosity solution of the following Free Boundary PDE. Although intuitively it's clear, I am not able to prove it formally.
\begin{align} \min \left\lbrace \partial_tV + \frac{1}{2} \partial^2_{xx}V + \frac{1}{2} x^2, 1-|\partial_xV| \right\rbrace &=0 \\ V(T,x) &= 0 \end{align}
Also, I am interested in several aspects of the viscosity solution $V(t,x)$ to the above PDE.
- Are there any sufficient conditions under which we can expect $V$ to be classical solution i.e. $V \in C^{1,2}$?
- From the structure of the PDE, we can see that $V$ must be locally Lipschitz. Moreover, $V(t,\cdot)$ should be linear outside of $(-L(t),L(t))$. Can we prove this rigorously? Also, can we write any ODE for $L(t)$?
Thank you very much!