To derive the backward FokkerPlanck equation, we consider where u is some intermediate time satisfying t 0

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To derive the backward Fokker–Planck equation, we consider 

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

where u is some intermediate time satisfying t

0    - (St, t; , u)w (, u; So, to) d  + 1 (St, t;, u)- - (E, u; So, to) d.   -From the forward Fokker–Planck equation derived in Problem 3.8, we obtain.

you ave - (St, t; &, u) (, u; So, to) de   - Led- SE + -[(, u)(, u; So, to)] 2 (, u) [6-8, 4:50.40]] (5.

Problem 3.8

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

dSt = u(St, t) d t +o (St, t) dZt.

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

af (St, t) f df = [u(S. 1) 25 + 0 (5, 1) 8/17] a St 2 as? ]dt +0 af dt + o (St, t) dZt. a St

We let (St, t; So, to) denote the transition density function of S, at the future time t, conditional on the

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(St,t; S0,t0) satisfies

to  So  (, u; So, to)  8(  So) as u  to.  + u(So, to) - 2(So, to) 2/ 2  + = 0.

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