Proabilistic representation of the smallest solution of linear Parabolic PDE

We will discuss probabilistic representation of the smallest solution of linear parabolic equation under the absence of uniqueness of the solution.
Consider a stochastic differential equation with filtered probability space {(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)} of the form

\displaystyle   d X(s) = b(X(s)) ds + \sigma(X(s)) dW_s, \ X(t) = x \ge 0. \ \ \ \ \ (1)

Assumption1: SDE (1) has a unique strong solution {X^{t,x}} satisfying

\displaystyle \mathbb{P}\{X^{t,x}(t) \ge 0\} = 1 \hbox{ for all } t\ge 0. \Box

Let {f: [0,\infty) \rightarrow [0,\infty)} be a given function, and {\hat V(x,t)} be

\displaystyle   \hat V(x,t) = \mathbb{E}[f(X^{t,x}(T))|\mathcal{F}_t]. \ \ \ \ \ (2)

Then, {\hat V} satisfies following PDE in some sense:

\displaystyle   u_t + \mathcal{L} u = 0, \ u(x,T) = f(x), \ \ \ \ \ (3)

where

\displaystyle \mathcal{L} u = \frac 1 2 \sigma^2(x) u_{xx} + b(x) u_x.

Problem: If Assumption 1 holds, and furhter assume {\hat V \in C^{2,1}}, then {\hat V} is the smallest lower-bounded classical solution of (3). \Box

Proof: For simplicity, we denote {X^{0,x}} by {X} in this below. Note that

\displaystyle \hat Y(t) \triangleq \hat V(X(t),t) = \mathbb{E}_t [f(X^{t,X(t)}(T))] = \mathbb{E}_t [f(X(T)]

is a martingale process. Suppose {V} is an aribtrary lower bounded solution of (3), then Ito’s formula applying to {Y(t) \triangleq V(X(t), t)} leads to

\displaystyle Y(t) = V(X(0),0) + \int_0^t V_x (X(s),s) \sigma(X(s)) dW(s),

and {Y(t)} is a lower bounded local martingale, hence is a super martingale.

Therefore, we have

\displaystyle Y(0) \ge \mathbb{E}[Y(T)] = \mathbb{E}[f(X(T))] = \hat Y(0)

and this implies

\displaystyle V(x,0) \ge \hat V(x,0).

\Box

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