This question arise from Lemma 4.14 of Kwon, H. D., & Palczewski, J. (2022). Exit game with private information. arXiv preprint arXiv:2210.01610. Let us consider the following optimal stopping problem $u(x)=\sup_\tau\mathbb E_x[\int^\tau_0e^{-rs}D(X_s)ds+e^{-r}\theta]-\theta$ where $X_t$ is an Ito diffusion following sde $dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$ with Lipschitz $\mu$ and $\sigma$ and $D(x)$ is bounded and continuous. We would like to apply Ito formula (or maybe by applying Ito-Tanaka formula in some way as claimed in the paper) to $e^{-rt}u(X_t)$ so that $$e^{-rt}u(X_t)=u(x)+\int^t_0e^{-rs}(\mathscr L-r)u(X_s)ds+\int^t_0e^{-rs}dM_s$$ where $\mathscr L$ is the infinitesimal generator of $X_t$. They claim that
- $u$ is $C^1$ and the second derivative lies in $L^\infty_{loc}$
- $M_t$ is a square integrable martingale.
My question is
- Is there any way to prove their first claim? The explicit formula of $u$ is given in the paper as $u(X_t)=(\frac{\theta-d(\alpha(\theta))}{\phi(\alpha(\theta))}\phi(x)+d(x))\mathbf 1_{x>\alpha(\theta)}+\theta\mathbf 1_{x\leq\alpha(\theta)}-\theta$ where $\phi(s)$ is the decreasing fundamental solution to the ODE $(\mathscr L -r)u=0$ and $d(x)=\mathbb E_x[\int^\infty_0e^{-rs}D(X_s)ds]$. But I see no claim on the regularity of $\phi(x)$ and $d(x)$. Is there any way to prove the regularity for those two objects?
- Is the regularity claimed sufficient for the Ito formula to work? Or do we need to use the Ito Tanaka formula in some way as claimed in the paper? How the Ito Tanaka formula is used to prove the result?
- Is the regularity sufficient to prove that $M_t=\int^t_0\sigma(X_s)u'(X_s)dW_s$ (per my own calculation using the Ito formula) is a square-integrable martingale?
- Is there any reference that I can check for results of the regularity of the value function of the optimal stopping problems?
A small note on notation: the $u$ function in this question is $\tilde u$ in the paper. There is a difference of $\theta)