SOLUTIONS TO BSDES DRIVEN BY BOTH FRACTIONAL BROWNIAN MOTIONS AND THE UNDERLYING STANDARD BROWNIAN MOTIONS

The local existence and uniqueness of the solutions to backward stochastic differential equations（BSDEs, in short） driven by both fractional Brownian motions with Hurst parameter H ∈ （1/2, 1） and the underlying standard Brownian motions are studied. The generalization of the It6 formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.

A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

Let B={B^H(t)}t≥0 be a d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1).Consider the functionals of k independent d-dimensional fractional Brownian motions 1/√n∫0^ent1⋯∫0^entk f(B^H,1(s1)+⋯+B^H,k(sk))ds1⋯dsk,where the Hurst index H=k/d.Using the method of moments,we prove the limit law and extending a result by Xu\cite{xu}of the case k=1.It can also be regarded as a fractional generalization of Biane\cite{biane}in the case of Brownian motion.

Influence of Brownian Motion, Thermophoresis and Magnetic Effects on a Fluid Containing Nanoparticles Flowing over a Stretchable Cylinder

The influence of Brownian motion and thermophoresis on a fluid containing nanoparticles flowing over a stretchable cylinder is examined.The classical Navier-Stokes equations are considered in a porous frame.In addition,the Lorentz force is taken into account.The controlling coupled nonlinear partial differential equations are transformed into a system of first order ordinary differential equations by means of a similarity transformation.The resulting system of equations is solved by employing a shooting approach properly implemented in MATLAB.The evolution of the boundary layer and the growing velocity is shown graphically together with the related profiles of concentration and temperature.The magnetic field has a different influence(in terms of trends)on velocity and concentration.

Nonparametric Estimation of the Trend Function for Stochastic Processes Driven by Fractional Brownian Motion of the Second Kind

The present paper deals with the problem of nonparametric kernel density estimation of the trend function for stochastic processes driven by fractional Brownian motion of the second kind.The consistency,the rate of convergence,and the asymptotic normality of the kernel-type estimator are discussed.Besides,we prove that the rate of convergence of the kernel-type estimator depends on the smoothness of the trend of the nonperturbed system.

On a Compound Poisson Risk Model Perturbed by Brownian Motion with Variable Premium and Tail Dependence between Claims Amounts and Inter-Claim Time

This paper considers the compound Poisson risk model perturbed by Brownian motion with variable premium and dependence between claims amounts and inter-claim times via Spearman copula. It is assumed that the insurance company’s portfolio is governed by two classes of policyholders. On the one hand, the first class where the amount of claims is high, and on the other hand, the second class where the amount of claims is low, this difference in claim amounts has significant implications for the insurance company’s pricing and risk management strategies. When policyholders are in the first class, they pay an insurance premium of a constant amount c1 and when they are in the second class, the premium paid is a constant amount c2 such that c1 > c2. The nature of claims (low or high) is measured via random thresholds . The study in this work will focus on the determination of the integro-differential equations satisfied by Gerber-Shiu functions and their Laplace transforms in the risk model perturbed by Brownian motion with variable premium and dependence between claims amounts and inter-claim times via Spearman copula. .

MULTI-DIMENSIONAL GEOMETRIC BROWNIANMOTIONS, ONSAGER-MACHLUP FUNCTIONS, AND APPLICATIONS TO MATHEMATICAL FINANCE

The solutions of the following bilinear stochastic differential equation are studied [GRAPHICS] where A(t)(k), B-t are (deterministic) continuous matrix-valued functions of t and w(1) (t),..., w(m) (t) are m independent standard Brownian motions. Conditions are given such that the solution is positive if the initial condition is positive. The equation the most probable path must satisfy is also derived and applied to a mathematical finance problem.

Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.

Smoothness for the collision local times of bifractional Brownian motions

Let B^Hi,Ki ={ Bt^Hi,Ki, t ≥ 0}, i= 1, 2 be two independent bifractional Brownian motions with respective indices Hi ∈ （0, 1） and K∈ E （0, 1]. One of the main motivations of this paper is to investigate f0^Tδ（Bs^H1 ,K1 - the smoothness of the collision local time, introduced by Jiang and Wang in 2009, IT = f0^T δ（Bs^H1,K1）ds, T 〉 0, where 6 denotes the Dirac delta function. By an elementary method, we show that iT is smooth in the sense of the Meyer-Watanabe if and only if min{H-1K1, H2K2} 〈-1/3.

On the Collision Local Time of Fractional Brownian Motions

In this paper, the existence and smoothness of the collision local time are proved for two independent fractional Brownian motions, through L^2 convergence and Chaos expansion. Furthermore, the regularity of the collision local time process is studied.

Solutions to BSDEs Driven by Both Standard and Fractional Brownian Motions

The backward stochastic differential equations driven by both standard and fractional Brownian motions （or, in short, SFBSDE） axe studied. A Wick-It6 stochastic integral for a fractional Brownian motion is adopted. The fractional It6 formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.

Upper tail probabilities of integrated Brownian motions

We obtain new upper tail probabilities of m-times integrated Brownian motions under the uniform norm and the Lp norm. For the uniform norm, Talagrand's approach is used, while for the Lp norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities(large ball probabilities) for general Gaussian random variables in Banach spaces. As applications,explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of m-times integrated Brownian motions are presented as well.

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R^d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L^b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p^b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L^b, C_c~∞(R^d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.

Stationary distributions for two-dimensional sticky Brownian motions:Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions,which find applications in many areas including queueing theory and mathematical finance.In this paper,we focus on stationary distributions for sticky Brownian motions.Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions.The kernel method,copula concept and extreme value theory are the main tools used in our analysis.

ON THE MULTIPLE TIME SET OF BROWNIAN MOTIONS

Let Sdp be the p-multiple time set of the Brownian motion in d dimensions. In this paper , the Hausdorff measure function for S32 is proved to be , and the Hausdorff measuure problem for S2p is also discussed. As a result, a conjecture suggested by J. Rosen is partially proved.

ABSOLUTE CONTINUITY FOR INTERACTING MEASURE-VALUED BRANCHING BROWNIAN MOTIONS

Exponential trichotomy theory is developed and the Fredholm Alternative Lemma is proved for the system with exponential trichotomies. An application of thesetheories is also given to obtain the persistence condition for heteroclinic orbits connecting nonhyperbolic equilibria, which extends the corresponding result of .

OPTIMAL TRACKING FOR BILINEAR STOCHASTIC SYSTEM DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

This paper discusses a problem of optimal tracking for a linear control system driven by fractional Brownian motion.An equation is obtained for the linear Markov feedback control.The existence and uniqueness of the solution to the equation are also studied.

Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in R

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ（·）. We focus on the functional ergodic limits for the occupation time processes of the models in IR. It is proved that the limiting process has the form of λζ（·）, where A is the Lebesgue measure on R and ζ（·） is a real-valued process which is non-degenerate if and only if cr is integrable. When ζ（·） is non-degenerate, it is strictly positive for t 〉 0. Moreover, ζ converges to 0 in finite-dimensional distributions if the integral of a tends to infinity.

Maximal Coupling of Euclidean Brownian Motions

We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting from single points and discuss the connection between the uniqueness of maximalMarkovian coupling of Brownian motions and certain mass transportation problems.

HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING

In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.

Millisecond dynamics of colloidal suspension studied by X-ray photon correlation spectroscopy at the Shanghai Synchrotron Radiation Facility

X-ray photon correlation spectroscopy(XPCS)has emerged as a powerful tool for probing the nanoscale dynamics of soft condensed matter and strongly correlated materials owing to its high spatial resolution and penetration capabilities.This technique requires high brilliance and beam coherence,which are not directly available at modern synchrotron beamlines in China.To facilitate future XPCS experiments,we modified the optical setup of the newly commissioned BL10U1 USAXS beamline at the Shanghai Synchrotron Radiation Facility(SSRF).Subsequently,we performed XPCS measurements on silica suspensions in glycerol,which were opaque owing to their high concentrations.Images were collected using a high frame rate area detector.A comprehensive analysis was performed,yielding correlation functions and several key dynamic parameters.All the results were consistent with the theory of Brownian motion and demonstrated the feasibility of XPCS at SSRF.Finally,by carefully optimizing the setup and analyzing the algorithms,we achieved a time resolution of 2 ms,which enabled the characterization of millisecond dynamics in opaque systems.