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Stochastic vector difference equations with stationary coefficients

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Paul Glasserman  and
David D. Yao
Paul Glasserman*
Affiliation:
Columbia University
David D. Yao*
Affiliation:
Columbia University
*
∗Postal address: Graduate School of Business, and
∗∗IE/OR Department, Columbia University, New York, NY10027, USA.
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Abstract

We give a unified presentation of stability results for stochastic vector difference equations based on various choices of binary operations and , assuming that are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with . Under this and mild additional conditions, the process has a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.

Keywords

RANDOM MATRICESSTABILITYSUBADDITIVE ERGODIC THEORY

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 851 - 866
Copyright
Copyright © Applied Probability Trust 1995 

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