×

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 \[ dX(t)= f(X(t))\,dt+ (v(X(t))| u(t)|)^{{1\over 2}} dW(t), \] with initial \(X(0)= x\in[d_1, d_2]\), 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 \[ J(x)= \int^{\tau(x)}_0 (\textstyle{{1\over 2}} q(X(t)) u^2(t)+ \lambda)\,dt+ K(X(\tau(x))), \] where \(\tau(x)\) 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
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Karlin, S. and Taylor, H. (1981). A Second Course in Stochastic Processes . Academic Press, New York. · Zbl 0469.60001
[2] Kuhn, J. (1985). The risk-sensitive homing problem. J. Appl. Probab. 22 167–172. · Zbl 0607.93065 · doi:10.2307/3213947
[3] Lefebvre, M. (1989). An extension of a stochastic control theorem due to Whittle. IEEE Trans. Automat. Control 34 567–568. · doi:10.1109/9.24219
[4] Lefebvre, M. (1997). Reducing a nonlinear dynamic programming equation to a Kolmogorov equation. Optimization 42 125–137. · Zbl 0887.93072 · doi:10.1080/02331939708844355
[5] Lefebvre, M. (2001). A different class of homing problems. Systems Control Lett. 42 347–352. · Zbl 0974.93071 · doi:10.1016/S0167-6911(00)00107-9
[6] Whittle, P. (1982). Optimization Over Time 1 . Wiley, Chichester. · Zbl 0557.93001
[7] Whittle, P. (1990). Risk-Sensitive Optimal Control . Wiley, Chichester. · Zbl 0718.93068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.