Differential Systems Associated to Brownian Motion

Abstract

For a given T > 0 let F: ℝ ↦ ℝ, f : [0, T] × ℝ ↦ ℝ₊ and g : [0, T] × ℝ ↦ ℝ be continuous functions. Assume also that F and g are bounded. If, moreover, g and f are locally uniformly (in t) Hölder continuous, the Feynman-Kac formula (cf. No. II.24) says that

$$v\left( {t,x} \right): = E\left( {F\left( {{{W}_{T}}} \right)\exp \left( { - \int_{t}^{T} {f\left( {s,{{W}_{s}}} \right)ds} } \right) + \int_{t}^{T} {g\left( {s,{{W}_{s}}} \right)\exp \left( { - \int_{t}^{s} {f\left( {v,{{W}_{v}}} \right)} dv} \right)ds|{{W}_{t}} = x} } \right)$$

is the unique solution of the problem

$$ - {{u'}_{t}}(t,x) = \frac{1}{2}{{u''}_{{xx}}}(t,x) - f(t,x)u(t,x) + g(t,x),\quad (t,x) \in (0,t) \times R,\quad u(T,x) = F(x),\quad x \in R$$

.