Optimizing a variable-rate diffusion to hit an infinitesimal target at a set time

Abstract

I consider a stochastic optimization problem for a time-changed Bessel process whose diffusion rate is constrained to be between two positive values $r_1<r_2$. The problem is to find an optimal adapted strategy for the choice of diffusion rate in order to maximize the chance of hitting an infinitesimal region around the origin at a set time in the future. More precisely, the parameter associated with "the chance of hitting the origin" is the exponent for a singularity induced at the origin of the final time probability density. I show that the optimal exponent solves a transcendental equation depending on the ratio $\frac{r_2}{r_1}$ and the dimension of the Bessel process.

19 pages, I generalized the result from the previous version of the article and made small corrections

References in corpus (1)

• Nonlinear Eigenvalues and Bifurcation Problems for Pucci's Operator