E

where the inner expectation on the right hand side of Eq. (7.93) is over the sample paths starting at X(t + dt) = x + dX. Of course, the inner expectation is also u(x + dX, t + dt) so that we conclude u(x, t) = r(x, t)e-&dt + o(dt) + EdX{u(x + dX, t + dt)g (7.94)

Finish the calculation to show that u(x, t) satisfies the differential equation

Next, assume that a, b and r are functions of x but not functions of time. Set u(x,t) = v(x)e- At. What equation does v(x) satisfy?

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