The following is the last in a series of posts on optimal stopping. In the previous post, we showed that the value function satisfied a particular partial differential equation (PDE) in the viscosity sense, thereby positing the existence of a solution to that PDE. In this post, we derive a comparison principle for the optimal stopping problem, which in turn guarantees uniqueness of solutions to the PDE. It follows that, roughly speaking, the value function and the solution to the PDE are one and the same.

We now look to transcribe the PDE of the previous post along with the relevant Cauchy data (i.e., terminal condition). Let $\rho > 0$ be arbitrary and $$ F(w,r,q,A)=-\operatorname{trace}(\sigma(w)\sigma(w)^{\top}A)-b(w)\cdot q+\rho r\text{ where }w=(t,x). $$ We can write the Cauchy problem as \begin{align} \min\left\{ -\partial_{t}v+F(\cdot,v(\cdot),Dv(\cdot),D_{x}^{2}v(\cdot)),(v-g)(\cdot)\right\} & =0 & \text{on }[0,T)\times\mathbb{R}^{d};\nonumber \\ (v-g)(T,\cdot) & =0 & \text{on }\mathbb{R}^{d}.\tag{1}\label{eq:pde_boundary} \end{align}

For completeness, we extend (in the obvious manner) our notion of viscosity solution from the previous post to take into account the boundary:

Uniqueness for elliptic (resp. parabolic) equations in the classical setting is often established via a maximum principle. Such a maximum principle often states that if two solutions $u$ and $v$ satisfy $u\leq v$ on the boundary $\partial\Omega$ of the bounded domain $\Omega$ on which the PDE is defined, then $u\leq v$ on the closure of the domain $\overline{\Omega}$ (i.e., everywhere). However, maximum principles are often derived by considering the case of $u$ and $v$ smooth, so that the first and second derivatives of $u-v$ satisfy the usual conditions for maxima (i.e., $D(u-v)=0$ and $D^{2}(u-v)\preceq0$).

However, in the context of viscosity solutions, no smoothness is assumed. The main tool to circumvent this apparent problem is the celebrated Crandall-Ishii Lemma [1]. We use the notation $|A|=\sup\left\{ A\xi\cdot\xi\colon\left|\xi\right|\leq1\right\} $ for $A\in\mathscr{S}(d)$, along with the parabolic semijets $\overline{\mathscr{P}}^{2,\pm}$ as defined in [1]. The semijets give an alternative characterization of viscosity solutions which we will not discuss here. We mention that we are unable to use the "parabolic" Crandall-Ishii Lemma [1 Theorem 8.3] directly due to an issue with the boundedness of the derivatives. We rely instead on the "elliptic" version [1 Theorem 3.2] and a variable-doubling argument.

We consider here the case of bounded solutions (e.g., $g$ is bounded in the first post of the series). We leave it to the reader to derive conditions for more interesting cases (e.g., solutions of sublinear growth).

It follows from the above that a viscosity solution $u$ (i.e., sub and super) of \eqref{eq:pde_boundary} satisfies $u^{*}\leq g\leq u_{*}$ on $\{T\}\times\mathbb{R}^{d}$ and hence $u^{*}\leq u_{*}$ everywhere. Moreover, the inequality $u_{*}\leq u^{*}$ is trivial from the definition of semicontinuous envelopes. Therefore, $u_{*}=u^{*}$, so that the function $u$ is continuous. Moreover, since for any two viscosity solutions $u$ and $v$ we have $u\leq v$ and $v\leq u$, it follows that $u=v$.

We can now apply the Crandall-Ishii Lemma (with $u_{1}=u$ and $u_{2}=-v$) to find $A_{n},B_{n}\in\mathscr{S}(d)$ and $a_{n}\in\mathbb{R}$ such that$$ \left(a_{n},D_x\varphi(x_{n},y_{n}),A_{n}+\epsilon I_d \right)\in\mathscr{\overline{P}}_{\mathcal{U}}^{2,+}u(t_{n},x_{n})\text{ and }\left(a_{n},-D_y\varphi(x_{n},y_{n}),B_{n}-\epsilon I_d \right)\in\mathscr{\overline{P}}_{\mathcal{U}}^{2,-}v(t_{n},x_{n}) $$ and $$-3\alpha_{n}I_{2d}\preceq\left(\begin{array}{cc} A_{n}\\ & -B_{n} \end{array}\right)\preceq3\alpha_{n}\left(\begin{array}{cc} I_{d} & -I_{d}\\ -I_{d} & I_{d} \end{array}\right).$$

Since $u$ is a subsolution and $v$ is a supersolution, it follows that \begin{align*} \min\left\{ -a_{n}+F(t_{n},x_{n},u(t_{n},x_{n}),\alpha(x_{n}-y_{n})+\epsilon x_{n},A_{n}+\epsilon I_d),(u-g)(t_{n},x_{n})\right\} & \leq0;\\ \min\left\{ -a_{n}+F(s_{n},y_{n},v(s_{n},y_{n}),\alpha(x_{n}-y_{n})-\epsilon y_{n},B_{n}-\epsilon I_d),(v-g)(t_{n},y_{n})\right\} & \geq0. \end{align*} With an abuse of notation, suppose $(u-g)(t_{n},x_{n})\leq0$ along some subsequence $(t_{n},x_{n},s_n,y_{n})_{n}$. Then, since $(v-g)(s_{n},y_{n})\geq0$ by the supersolution property, we have that $$\delta-\nu-\epsilon|x^\nu|^{2}+\varphi_n \leq u(t_{n},x_{n})-v(s_{n},y_{n})-\left(g(t_n,x_{n})-g(s_n,y_{n})\right)\leq0.$$ If we take $n$ large enough and $\epsilon$ small enough, the left-hand side of the above becomes strictly positive, yielding a contradiction. Therefore, with yet another abuse of notation, we can pick a subsequence $(t_{n},x_{n},s_n,y_{n})_n$ on which $(u-g)(t_{n},x_{n})>0$ for all $n$. On this subsequence, \begin{align*} -a_{n}+F(t_{n},x_{n},u(t_{n},x_{n}),\alpha(x_{n}-y_{n})+\epsilon x_{n},A_{n}+\epsilon I_d) & \leq0;\\ -a_{n}+F(s_{n},y_{n},v(s_{n},y_{n}),\alpha(x_{n}-y_{n})-\epsilon y_{n},B_{n}-\epsilon I_d) & \geq0. \end{align*} We now claim that if $$\left(\begin{array}{cc} A\\ & -B \end{array}\right)\preceq \operatorname{const.} \alpha\left(\begin{array}{cc} I_{d} & -I_{d}\\ -I_{d} & I_{d} \end{array}\right), $$ then \begin{multline*} F(s,y,r^{\prime},\alpha(x-y)-\epsilon y,B-\epsilon I_d)-F(t,x,r,\alpha(x-y)+\epsilon x,A+\epsilon I_d)\\ \leq\rho\left(r^{\prime}-r\right)+\operatorname{const.}(\alpha\,|x-y|^{2}+\epsilon\,(1+|x|^{2}+|y|^{2})). \end{multline*} $\operatorname{const.}$ denotes some nonnegative constant. We leave this as an exercise to the reader (use the Lipschitz continuity and linear growth of $b$ and $\sigma$). Using this claim, \begin{align*} 0 & \leq F(s_{n},y_{n},v(s_{n},y_{n}),\alpha(x_{n}-y_{n})-\epsilon y_{n},B_{n}-\epsilon I_{d})\\ & \qquad-F(t_{n},x_{n},u(t_{n},x_{n}),\alpha(x_{n}-y_{n})+\epsilon x_{n},A_{n}+\epsilon I_{d})\\ & \leq\rho\,(v(s_{n},y_{n})-u(t_{n},x_{n}))+\operatorname{const.}\,(\varphi(x_{n},y_{n})+\epsilon)\\ & \leq\rho\,(-\delta+\nu+\epsilon|x^\nu|^{2})+\operatorname{const.}\,(\varphi_n+\epsilon) \end{align*} Taking the limit superior of both sides as $n\rightarrow\infty$ and moving some terms around, we get $$\delta\leq\operatorname{const.}\,(\nu+\epsilon+\epsilon|x^\nu|^{2})$$ where $\operatorname{const.}$ is not necessarily the same as it was above. Now, simply take $\epsilon$ small enough to arrive at a contradiction.

## Bibliography

- Crandall, Michael G., Hitoshi Ishii, and Pierre-Louis Lions. "Userâ€™s guide to viscosity solutions of second order partial differential equations." Bulletin of the American Mathematical Society 27.1 (1992): 1-67.