Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary Distribution

By Jim G. Dai, Masakiyo Miyazawa. 2011

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We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bxκ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data. A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of th...

Subfields: Finance, Management Information Systems, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0272a601f499a2b7063e43a2ebabc6411667f7a9