Complete the proof of Proposition 23.6 (Kolmogorov's forward equation). Data From 23.6 Proposition 23.6 Proposition (forward equation.

Complete the proof of Proposition 23.6 (Kolmogorov's forward equation).

Data From 23.6 Proposition

     

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23.6 Proposition (forward equation. Kolmogorov 1931). Let (X+) to denote a diffusion Ex. 23.5 with generator (A, D(A)) such that Alego (R) is given by (23.1) with drift and diffusion coefficients b = e (Rd, Rd) and a = e(Rd, Rdxd). Assume that X+ has a probability density p(t,x,y) satisfying P. at , ((0,00) x Rdx Rd). Then Kolmogorov's second or forward equation holds: op(t, x,y) ot L*(y, Dy)p(t,x,y) for all t> 0, x, y Rd. (23.6) Proof. This proof is similar to the proof of Proposition 23.5, where we use Lemma 7.10 In the form Tu = TAU = TL(,D)u for u e eco (Rd), and then integration by parts in the resulting integral equality. The fact that we do not have to interchange the integral and the operator L(x, D) as in the proof of Proposition 23.5, allows us to relax the boundedness assumptions on p, b and a, but we pay a price in the form of differentiability assumptions for the drift and diffusion coefficients.