I was thinking on the following problem about ordinary differential equations applied in economics: Compute the expected life time $u(S)$ of an investment on an asset at current price $S_0$ if it will sold whenever the price reaches $S_1<S_0$ or $S_2>S_0$ assuming that the price $S$ follows a random motion in one dimension with equal probability of jumping to the right and to the left.
To solve this problem I was thinking on the Poisson equation in one dimension, this is $$u''(S)=-1/k,$$ where $k=\lim_{\Delta t \rightarrow 0}\frac{\Delta S^2}{2 \Delta t},$ which is an ODE with boundary conditions $$u(S_1)=u(S_2)=0.$$
Is that right? In this case the expected life time searched is the solution of the previous equation with the two boundary conditions.
Thank you in advance!
Yes, that sounds like a good plan. And it is not at all difficult to solve that differential equation.