## Optimizing a variable-rate diffusion to hit an infinitesimal target at a set time.(English)Zbl 1327.60153

Summary: 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 the 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.

### MSC:

 60J60 Diffusion processes 93E20 Optimal stochastic control
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