Componentwise accurate Brownian motion computations using Cyclic Reduction

Abstract

Markov-modulated Brownian motion is a popular tool to model continuous-time phenomena in a stochastic context. The main quantity of interest is the invariant density, which satisfies a differential equation associated with the quadratic matrix polynomial P (z) = V z 2 −Dz +Q, where the matrices V and D are diagonal and Q is the transition matrix of a discrete-time Markov chain. Its solution is typically constructed by computing an invariant pair of P (z) associated with its eigenvalues in the left half-plane, or by solving the matrix equation X 2 V − XD + Q = 0. We show that these tasks can be solved using a componentwise accurate algorithm based on Cyclic Reduction, generalizing the recently appeared algorithms for the linear case (V = 0). We give a proof of the numerical stability of our algorithm in the componentwise sense; the same proof applies to Cyclic Reduction in a more general M-matrix setting which appears in other applications such as the modelling of QBD processes.

Matrix equations in Markov modulated Brownian motion: theoretical properties and numerical solution

“…Numerical examples show that the algorithm applying ADDA to the shifted NARE is superior to the other doubling algorithms in comparison. This also holds when compared to Nguyen and Poloni's quadratically convergent algorithm [20] that is based on the quadratic matrix equation obtained by Asmussen.…”

“…In this paper we have shown, by using linear algebra tools, an explicit algebraic relation between the QME and the NARE that characterize MMBM. We have compared the performances of several existing algorithms for computation of first-passage probabilities of the MMBM, among which the component-wise stable and quadratically convergent algorithm by Nguyen and Poloni [20] for solving the QME. In the null recurrent case, SDA and ADDA applied to the original NARE are not satisfactory, since they don't provide an accurate approximation of the solution.…”

“…The following theorem shows the relation between the solutions of our NARE and the quadratic matrix equation in Nguyen and Poloni [20].…”

“…In this section, by using properties of matrix pencils and matrix polynomials, we explicitly relate the solutions of the quadratic matrix equations given in Nguyen and Poloni [20] and the solution X(s) of the NARE (11).…”

“…Several numerical methods have been suggested in the literature, which are based on Cyclic Reduction [19], the eigendecomposition of a linearization [17], or more generally a block diagonal decomposition [1]. More recently, Nguyen and Poloni [20] proposed an algorithm of our special attention. It is an extension of the algorithm developed by Nguyen and Latouche [19] that is based on the cyclic reduction method and designed for the MMBM with σ > 0.…”