Hlder范数下关于Brown运动增量的泛函局部收敛速度

本文研究了Brown运动在H?lder范数与容度下的泛函极限问题.利用大偏差小偏差方法,获得了Brown运动增量局部泛函极限的收敛速度,推广了文[4]中的结果.     

Martingale representation theorem for set-valued martingales

The present paper contains a martingale representation theorem for set-valued martingales defined on a filtered probability space with a filtration generated by a Brownian motion. It is proved that such type martingales can be defined by some generalized set-valued stochastic integrals with respect to a given Brownian motion. The main result of the paper is preceded by short part devoted to the definition and some properties of generalized set-valued stochastic integrals.     

On the potential in non‐Gaussian chain polymer models

In this paper, we investigate the potential for a class of non‐Gaussian processes so‐called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M‐Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.     

A version of Pitman's 2M — X theorem for geometric Brownian motions

We show that geometric Brownian motion with parameter μ, i.e., the exponential of linear Brownian motion with drift μ, divided by its quadratic variation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion with drift of Pitman's representation theorem for the three-dimensional Bessel process. Time inversion and generalized inverse Gaussian distributions play crucial roles in our proofs.     

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.     

分式Brownian运动的多重相交局部时

本文研究了分式布朗运动的多重相交局部时的问题.利用白噪声分析的方法,获得了分式布朗运动的多重相交局部时的展开式.进行适当的截取,展开式在白噪声广义泛函意义下存在,并给出它们的核函数.推广了布朗运动的多重相交局部时.     

Non-independence of excursions of the Brownian sheet and of additive Brownian motion

A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.

     

The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.     

Hitting Time Problems of Sticky Brownian Motion and Their Applications in Optimal Stopping and Bond Pricing

This paper investigates the hitting time problems of sticky Brownian motion and their applications in optimal stopping and bond pricing. We study the Laplace transform of first hitting time over the constant and random jump boundary, respectively. The results about hitting the constant boundary serve for solving the optimal stopping problem of sticky Brownian motion. By introducing the sharpo ratio, we settle the bond pricing problem under sticky Brownian motion as well. An interesting result shows that the sticky point is in the continuation region and all the results we get are in closed form.

     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

A note on first-passage times of continuously time-changed Brownian motion

The probability of a Brownian motion with drift to remain between two constant barriers (for some period of time) is known explicitly. In mathematical finance, this and related results are required, for example, for the pricing of single-barrier and double-barrier options in a Black-Scholes framework. One popular possibility to generalize the Black-Scholes model is to introduce a stochastic time scale. This equips the modelled returns with desirable stylized facts such as volatility clusters and jumps. For continuous time transformations, independent of the Brownian motion, we show that analytical results for the double-barrier problem can be obtained via the Laplace transform of the time change. The result is a very efficient power series representation for the resulting exit probabilities. We discuss possible specifications of the time change based on integrated intensities of shot-noise type and of basic affine process type.     

Applications of the continuous-time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit λ>0 at time t is not affected by conditioning the Brownian motion to stay below a line segment from (0,c) to (t,λ). We extend a result of Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.     

Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes

We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.     

Explicit Solution of a Non-Linear Filtering Problem for Lévy Processes with Application to Finance

In this paper we explicitly solve a non-linear filtering problem with mixed observations, modelled by a Brownian motion and a generalized Cox process, whose jump intensity is given in terms of a Lévy measure. Motivated by empirical observations of R. Cont and P. Tankov we propose a model for financial assets, which captures the phenomenon of time inhomogeneity of the jump size density. We apply the explicit formula to obtain the optimal filter for the corresponding filtering problem.     

Conditioned stochastic differential equations: theory, examples and application to finance

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a “quantitative” value to this weak information.     

Smoothness of Densities of Generalized Locally Non-Degenerate Wiener Functionals

Several criteria for existence of smooth densities of Wiener functionals are known in the framework of Malliavin calculus. In this article, we introduce the notion of generalized locally non-degenerate Wiener functionals and prove that they possess smooth densities. The result presented here unifies the earlier works by Shigekawa and Florit-Nualart. As an application, we prove that the law of the strong solution to a stochastic differential equation driven by Brownian motion admits a smooth density without an assumption of Lipschitz continuity for dispersion coefficients.     

广义布朗单的容度估计

研究了既没有平稳增量性,也没有scaling性质的N指标d维广义布朗单的容度问题．证明了广义布朗单“好象”一个局部平稳增量过程,应用Cairoli极大不等式和多参数鞅的方法得到了广义布朗单的碰撞概率与容度之间的关系,给出了其碰撞概率的确切容度估计．所得结果包含了布朗单和可加布朗运动的相应结果．     

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.     

带干扰的双Cox风险模型的破产概率

对于受布朗运动干扰的双Cox模型,用鞅方法得到了估计其破产概率上界的Lundberg不等式,并对Lundberg不等式进行了推广.     

Regularity of local times of random fields

In this paper, we study the fractional smoothness of local times of general processes starting from the occupation time formula, and obtain the quasi-sure existence of local times in the sense of the Malliavin calculus. This general result is then applied to the local times of N-parameter d-dimensional Brownian motions, fractional Brownian motions and the self-intersection local time of the 2-dimensional Brownian motion, as well as smooth semimartingales.