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)

  1. {a^{ij}, b^i \in C(Q)},
  2. {b^i(\cdot,t) \in C^{0,1}_{loc}(O)},
  3. {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,

  1. {Q} is bounded;
  2. {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

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