stochastic solution and classical solution

Let ${Q = O \times (0,T)}$ , where ${O}$ is an open domain in ${\mathbb{R}^n}$ . Consider equation

$\displaystyle u_t + Lu = 0 \ \ \ \ \ (1)$

where

$\displaystyle L = \frac 1 2 a^{ij} D_{ij} + b^i D_{i} + c \ \ \ \ \ (2)$

We are going to identify sufficient condition, under which continuous stochastic solution is indeed a classical solution. The results are from ([JT06]).

First, we define stochastic solution of (1). Consider ${\mathbb{R}^n}$ -valued stochastic process ${X_t}$ solving

$\displaystyle X_t^i = x_0^i + \sum_{j=1}^n \int_{t_0}^t \sigma^{ij}(X_s, s) d W_s^j + \int_{t_0}^t b^i(X_s, s) ds, \ \ \ \ \ (3)$

where ${(a^{ij}) = (\sigma^{ij}) (\sigma^{ij})'}$ . Note that, (3) has unique strong solution if

Assumption 1 (Local regularity)

${a^{ij}, b^i \in C(Q)}$ ,
${b^i(\cdot,t) \in C^{0,1}_{loc}(O)}$ ,
${a^{ij}(\cdot, t) \in C^{0,1}_{loc}(O)}$ , or ${a(\cdot, t) \in C^{0,1/2}_{loc}(O)}$ if ${n= 1}$ .

We should note that by Lemma 6.1.1 of [Friedman 1976] ([Fri76]), under Assumption 4, ${\sigma^{ij}(\cdot, t) \in C^{0,1}_{loc}(O)}$ , (or ${\sigma(\cdot, t) \in C^{0,1/2}_{loc}(O)}$ ) exists iff ${a^{ij}(\cdot, t) \in C^{0,1}_{loc}(O)}$ (or ${a(\cdot, t) \in C^{0,1/2}_{loc}(O)}$ ). To avoid the explosion of ${X}$ from (3), we assume

Assumption 2 (Uniform linear growth) ${|\sigma(x,t)| + |b(x,t)| \le C (1 + |x|).}$

Definition 1 (Stochastic solution) A stochastic solution to (1) is a function ${u}$ defined on ${Q}$ satisfying,

$\displaystyle u(x_0,t_0) = \mathbb{E} \Big[ e^{\int_{t_0}^\tau c (X_s, s) ds} u(X_\tau, \tau) \Big], \ \forall (x_0, t_0) \in Q. \ \ \ \ \ (4)$

where ${\tau}$ is the first exit time from ${Q}$ , i.e.

$\displaystyle \tau = \inf\{ t>t_0: X_s \notin Q \}.$

We need one more assumption to avoid explosion of stochastic solution ${u}$ from (4).

Assumption 3 ${|c| \le C}$ .

Next, we present sufficient condition, under which a classical solution is a stochastic solution.

Theorem 2 Suppose ${F}$ is a classical solution, and Assumptions 1 and 3 hold. In addition, if either of two followings are ture,

${Q}$ is bounded;
${Q = \mathbb{R}_+^n \times (0,T)}$ , Assumption 2 hold, and ${F}$ satisfies
$\displaystyle \hbox{\rm (Polynomial growth)} \quad |F(x,t)| \le C (1 + |x|)^m \ \hbox{ for some } C, m>0,$

then ${F}$ is a stochastic solution.

Proof: Assumption 1 is needed to have strong solution ${X}$ of (3). Ito’s formula shows that ${ e^{\int_{t_0}^{t\wedge \tau} c (X_s, s) ds} F(X_{t\wedge \tau}, t\wedge \tau) }$ is a local martingale. Also, one can show ${\mathbb{E} [ \sup_t |F(X_t, t)| ] \le C \mathbb{E} [\sup_t (1 + |X_t|)^m] < C}$ , hence

$\displaystyle e^{\int_{t_0}^{t\wedge \tau} c (X_s, s) ds} F(X_{t\wedge \tau}, t\wedge \tau)$

is a uniformly integrable martingale in ${[t_0,T]}$ , and so is a martingale. $\Box$

Next, we consider converse of Theorem 2: when do we have stochatics solution being a classical solution.

Lemma 3 Consider a bounded cylinder ${Q_1 = O_1 \times (t_1, t_2)}$ being ${\partial Q_1 \in C^{2+\delta}}$ for some ${\delta \in (0,1)}$ . Suppose ${L}$ is nondegenerate with constant of ellipticity ${\kappa>0}$ . Also, assume ${|a,b,c|_{\delta, \frac \delta 2} \le K}$ . Let ${\varphi}$ be a continuous function on the parabolic boundary ${\partial^* Q_1}$ . Then, there is a unique classical solution to (1) in ${Q_1}$ with boundary data ${\varphi}$ .

If boundary data ${\varphi}$ satisfies smooth and consistency conditions of (10.4.2-3) of [Kry96], then it is the result of Theorem 10.4.1 of [Kry96]. However, ${\varphi}$ is just assumed to be continuous without any regularity. In this case, Shauder’s interior estimate will be useful. A proof of similar result of Lemma 3 is provided in [JT06]. (The proof will be completed later)

Assumption 4 (Positive definite) ${a^{ij} \xi^i \xi^j>0}$ .

The next is Theorem 2.7 of [JT06].

Theorem 4 Suppose Assumptions 1, 2, 3, and 4 hold. If ${F}$ is a continuous stochastic solution, then ${F}$ is a classical solution.

Proof: Consider arbitrary ${Q_1 = B_1 \times (t_1, t_2)}$ with ${\bar Q_1 \subset Q}$ with ${\partial B_1 \in C^\infty}$ . By strong Markov property, for any ${(x_0, t_0)\in Q_1}$ , we have

$\displaystyle F(x_0, t_0) = \mathbb{E} \Big[ \mathbb{E}\Big[ e^{\int_{t_0}^\tau c (X_s, s) ds} F(X_\tau, \tau) | \mathcal{F}_{\tau_1}\Big] \Big] = \mathbb{E} \Big[ e^{\int_{t_0}^{\tau_1} c (X_s, s) ds} F(X_{\tau_1}, \tau_1) \Big],$

where ${\tau_1}$ is the first hitting time of ${(X_t, t)}$ to ${Q_1}$ . This implies, ${F}$ is stochastic solution in ${Q_1}$ with continuous boundary data ${F|_{\partial^* Q_1}}$ . Thus, ${F\in C^{2,1}(Q_1)}$ . Therefore, ${F\in C^{2,1}_{loc}(Q)}$ . $\Box$

Omega

Deterministic Randomness

stochastic solution and classical solution

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