Ergodicity and First Passage Probability of Regime-Switching Geometric Brownian Motions

A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain. In this work, the author gives a complete characterization of the recurrent property of this process. The long time behavior of this process such as its p-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.     

Upper estimate of martingale dimension for self-similar fractals

We study upper estimates of the martingale dimension d _m of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that d _m  = 1 for natural diffusions on post-critically finite self-similar sets and that d _m is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.     

Fractional Brownian motion with complex variance via random walk in the complex plane and applications

A model of complex-valued fractional Brownian motion has been built up recently as the limit of a random walk in the complex plane, but this model involves radial steps only. It is shown that, by using non-radial steps, this model can be easily extended to define a fractional Brownian motion with complex-valued variance. The relations between complex-valued Brownian motion and the heat equation of order n is clarified and mainly one obtains the general expression of the probability density functions for these processes. One shows that the maximum entropy principle (MPE) provides the probability density of the complex-valued fractional Brownian motion, exactly like for the standard Brownian motion. And lastly, one shows that the heat equation of order 2n (which is the Fokker–Planck equation (FPE) of the complex-valued Brownian motion) has a solution which is similar to that of the FPE of fractional order introduced before by the author, therefore, to some extent, an identification between the complex-valued model via random walk in the complex plane and the model involving a derivative of fractional order.     

Attractiveness of Brownian queues in tandem

Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.

     

Torsional Rigidity for Regions with a Brownian Boundary

Let ?? ^m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? ^m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? ^m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??²\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??³\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W _r(t)[0, t] of radius r(t) = o(t ^-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?³ and W ₁[0, t] in ? ^m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? ^m , which has received a lot of attention in the literature in past years.     

On generalized m-th root finsler metrics

In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them reduce to Riemannian metrics.     

An Extension of Vervaat's Transformation and Its Consequences

Vervaat⁽¹⁸⁾ proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s (0, 1), then we extend this result by determining a random time m _s such that when we exchange the pre-m _s-part and the post-m _s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m _s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.     

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian H _n of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process {H _[nt]}_0≤t≤1 behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.     

A Local Refinement Strategy for Constructive Quantization of Scalar SDEs

We present a fully constructive method for quantization of the solution X of a scalar SDE in the path space L _p [0,1] or C[0,1]. The construction relies on a refinement strategy which takes into account the local regularity of X and uses Brownian motion (bridge) quantization as a building block. Our algorithm is easy to implement, its computational cost is close to the size of the quantization, and it achieves strong asymptotic optimality provided this property holds for the Brownian motion (bridge) quantization.     

Smoothing noisy data with spline functions

It is shown how to choose the smoothing parameter when a smoothing periodic spline of degree 2m?1 is used to reconstruct a smooth periodic curve from noisy ordinate data. The noise is assumed “white”, and the true curve is assumed to be in the Sobolev spaceW ₂ ^(2m) of periodic functions with absolutely continuousv-th derivative,v=0, 1, ..., 2m?1 and square integrable 2m-th derivative. The criteria is minimum expected square error, averaged over the data points. The dependency of the optimum smoothing parameter on the sample size, the noise variance, and the smoothness of the true curve is found explicitly.