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A homing problem for diffusion processes with control-dependent variance. (English) Zbl 1061.93099

This paper deals with optimal control problems of 1-dimensional diffusion processes defined by the stochastic differential equation

$dX\left(t\right)=f\left(X\left(t\right)\right)\phantom{\rule{0.166667em}{0ex}}dt+{\left(v\left(X\left(t\right)\right)|u\left(t\right)|\right)}^{\frac{1}{2}}dW\left(t\right),$

with initial $X\left(0\right)=x\in \left[{d}_{1},{d}_{2}\right]$, where $u$ is a control process and $W$ a standard Brownian motion. So, $X$ has a control-dependent infinitesimal variance. The aim is to minimize the cost criterion

$J\left(x\right)={\int }_{0}^{\tau \left(x\right)}\left(\frac{1}{2}q\left(X\left(t\right)\right){u}^{2}\left(t\right)+\lambda \right)\phantom{\rule{0.166667em}{0ex}}dt+K\left(X\left(\tau \left(x\right)\right)\right),$

where $\tau \left(x\right)$ is the hitting time to the boundary points, ${d}_{1}$, ${d}_{2}$, and $\lambda$ a real constant.

Using the dynamic programming equation, the author investigates the value function and optimal control. In particular, he obtains explicit expressions for the value function and the optimal control when the functions $f$, $v$ and $q$ are proportional to a power of $x$.

##### MSC:
 93E20 Optimal stochastic control (systems) 60J60 Diffusion processes