To derive the backward Fokker–Planck equation, we consider 

where u is some intermediate time satisfying t₀ 

From the forward Fokker–Planck equation derived in Problem 3.8, we obtain.

Problem 3.8

Let the dynamics of the stochastic state variable S_t be governed by the Ito process

For a twice differentiable function f (S_t), the differential of f (S_t) is given by

Perform parts integration of the integral in (ii).

By performing parts integration of the last integral and taking the limit u → t₀, show that ψ(S_t,t; S₀,t₀) satisfies

Transcribed Image Text:

- Lov (St, t; So, to) = V (St, t; , u) (s, u; So, to) de,