Abstract
We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.
Highlights
In many scenarios, one is interested in the dynamics of an entity that is constrained to a subset of its possible state space by some mechanism
For three and higher dimensional semimartingale reflecting Brownian motions (SRBMs), the relative bias still grows subexponentially, and the relative variance grows with the exponential rate inf x∈B I(x) − T (0) that is strictly smaller than inf x∈B I(x): our estimator is theoretically superior to the standard Monte Carlo estimator whose relative variance grows with exponential rate inf x∈B I(x)
We develop particle-based simulation algorithms for estimating rare event probabilities related to SRBMs in a nonnegative orthant
Summary
One is interested in the dynamics of an entity that is constrained to a subset of its possible state space by some mechanism. We develop splitting algorithms to estimate specific tail probabilities for SRBMs. We focus on the particular rare event that a positive recurrent SRBM exits the L1 cube of radius n before returning to the cube of radius. We are able to rescale and shift our local subsolution to ensure that it satisfies the necessary boundary conditions and to derive a subsolution to the VP (Theorem 2) Given that it is not, in general, possible to exactly simulate an SRBM, we use the Euler method to simulate a discrete time approximation process. Using our results on the behavior of the discretized process, our splitting algorithms can be used for estimating the rare event probability for either SRBMs or the approximation processes with identical logarithmic asymptotics. If f (t)=g(t) → 0 as t → ∞, f (t) O(g(t)) if f (t) ≤ Cg(t) for all t, f (t) Θ(g(t)) if cg(t) ≤ | f (t)| ≤ Cg(t) for all t, where C and c are positive constants
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