Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

Dai, J. G.; Miyazawa, Masakiyo

doi:10.1214/10-ssy022

Cited by 27 publications

(86 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted that these results still leave the LDP for d = 2 unresolved for some parameter cases (see [10] for a summary). More recently, Dai and Miyazawa [5] obtained exact asymptotics for SRBM in two dimensions using moment generating functions and techniques from complex analysis. However, the results are limited to asymptotic behavior along a ray of the quadrant.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

Liang

Hasenbein

2013

Stochastic Systems

View full text Add to dashboard Cite

We study the variational problem that arises from consideration of large deviations for semimartingale reflected Brownian motion (SRBM) in R 3 + . Due to the difficulty of the general problem, we consider the case in which the SRBM has rotationally symmetric parameters. In this case, we are able to obtain conditions under which the optimal solutions to the variational problem are paths that are gradual (moving through faces of strictly increasing dimension) or that spiral around the boundary of the octant. Furthermore, these results allow us to provide an example for which it can be verified that a spiral path is optimal. For rotationally symmetric SRBM's, our results facilitate the simplification of computational methods for determining optimal solutions to variational problems and give insight into large deviations behavior of these processes.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

Liang

Hasenbein

2013

Stochastic Systems

View full text Add to dashboard Cite

show abstract

“…Let A be the arc of the ellipse with endpoints s + 1 , s + 2 not passing through the origin, see Figure 3. For a given angle α ∈ [0, π/2] let us define the point θ(α) on the arc A as (6) θ(α) = argmax θ∈A θ | e α where e α = (cos α, sin α).…”

Section: Notation We Write the Asymptotic Expansionmentioning

confidence: 99%

“…The following theorem provides the main asymptotic term of π(r cos α, r sin α). (6). Let {θ(α 0 ), s 0 } (resp.…”

Section: Notation We Write the Asymptotic Expansionmentioning

confidence: 99%

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

Franceschi

Kourkova

2017

Stochastic Systems

View full text Add to dashboard Cite

Brownian motion in R 2 + with covariance matrix Σ and drift μ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found and its main term is identified depending on parameters (Σ, μ, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on R 2 + with reflections on the axes.

show abstract

“…One is the stationary equation in terms of a certain type of moment generating functions which is obtained from the Dynkin's formula. Because the background state is involved, this stationary equation is not the same as used for a two dimensional SRBM in [9], which is called a basic adjoint relation, BAR for short, but they are equivalent concerning finiteness. We derive its general dimensional version.…”

Section: Introductionmentioning

confidence: 99%

Tail asymptotics of the stationary distribution for a two-node generalized Jackson network

Miyazawa

2014

SIGMETRICS Perform. Eval. Rev.

Self Cite

View full text Add to dashboard Cite

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotic behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on Dynkin's formula of a Markov process, where a key ingredient is a suitable choice of test functions. Those results show how multidimensional tail asymptotic can be studied for the more than two dimensional case, which is known as a hard problem.

show abstract

Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

Cited by 27 publications

References 22 publications

Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

Tail asymptotics of the stationary distribution for a two-node generalized Jackson network

Contact Info

Product

Resources

About