Almost sure optimality and optimality in probability for stochastic control problems over an infinite time horizon
type: Item
instance of: scholarly article
title: Almost sure optimality and optimality in probability for stochastic control problems over an infinite time horizon
zbMATH Open document ID: 0951.93076
DOI: 10.1023/A:1018974112132
published in: Annals of Operations Research
publication date: 1999-12-02T00:00:00Z
review text: A pathwise optimality criterion for stochastic control problems is proposed, which is aimed at reducing the risk connected with the fluctuation of cost: Consider \[ dx^u_t= f(x^u_t,u_t)d \tau+\sigma (x^u_t,u_t) dw_t, \] with cost function \[ J_T(u)= \int^T_0 c(x_t^u,u_t) dt, \] and look for a control \(u^*\) optimal in the sense that \[ \lim_{T\to \infty}g_T \bigl[J_T (u^*)-J_T(u) \bigr]^+ =0\text{ a.s. for any }u\in {\mathcal U_\infty}, \] with \(g\) positive, non-increasing, typically \(o(1/t)\). This may be of interest, e.g., in economical applications, when risky situations are particularly dangerous. The control problem is solved under a number of technical conditions, and two examples are discussed, in particular the linear Gaussian model with quadratic cost function.
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 49K45
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 1373565
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: pathwise optimality criterion
zbMATH Keywords: stochastic control problems
zbMATH Keywords: cost function
author: Paolo Dai Pra
author: Giovanni Di Masi
author: Barbara Trivellato
reviewed by: Heinrich Hering
MaRDI profile type: Publication
full work available at URL: a:1018974112132
OpenAlex ID: W2122910631
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description: scientific article; zbMATH DE number 1373565