Uncertainty and Probability PDE Approach: Modified Hamilton–Jacobi–Bellman Equation

Markov Models

the state variableSsolves aforward SDE, e.g.

dSt=μ(St)dt+σ(St)dWt, S 0 = 1.

Let

L=μ(x)∂

∂x

+σ^2 (x)∂

2

∂x^2

be the infinitesimal generator ofS.

By Itˆo’s formula,v(t,St) is a martingale if

vt(t,x) +Lv(t,x) = 0 (1)

similarly,v(t,St) is ag –martingaleif

vt(t,x) +Lv(t,x) +g(t,vx(t,x)σ(x)) = 0 (2)