Let the exit time density q + (t; x 0 ,t 0 ) have dependence on the

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Let the exit time density q+(t; x0,t0) have dependence on the initial state X(t0) = x0. We write τ = t − t0 so that q+(t; x0,t0) = q+(x0,τ). Show that the partial differential equation formulation is given by

q+  =  aq+ 02 02q+ xo 2  + l < xo < u, T > 0,

with auxiliary conditions:

q(u, t) = 0, q (l, t) = 0 and q+(xo, 0) = 8(xo).Solve for q+(x0,τ) using the partial differential equation approach and compare the solution with that given in (4.1.53b). Also, show that

fat (s 10 -19 (5 q(s; xo, to) ds + + S to q+(s; xo, to) ds + P(x, t; xo, to) dx = 1,

where P(x,t; x0,t0) is the transition density function defined in Problem 4.9.

Problem 4.9.

Let P(x,t; x0,t0) denote the transition density function of the restricted Brownian process Wμt = μt + σZt with two absorbing barriers at x = 0 and x = ℓ. Using the method of separation of variables (Kevorkian, 1990), show that the solution to P(x,t; x0,t0) admits the following eigen-function expansion [which differs drastically in analytic form from that in (4.1.48)]

P(x, f; xo, to) (-1 -10) l e-k(-10) sin k=1  - sin k l

where the eigenvalues are given by

2 A4 = 2 *** ) ( + 22P(x,t; x0,t0) satisfies the forward Fokker–Planck equation with auxiliary conditions: P(0,t) = P(ℓ,t) = 0 and P(x,t+0 ; x0,t0) = δ(x−x0). 

Pelsser (2000) derived the above solution by performing the Laplace inversion using Bromwich contour integration. 

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