Time-fractional telegraph equations and telegraph processes with brownian time

We study the fundamental solutions to time-fractional telegraph equations of order 2. We are able to obtain the Fourier transform of the solutions for any and to give a representation of their inverse, in terms of stable densities. For the special case =1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.This research has been partially supported by the NATO grant No. SA (PST.CLG.976361) 5437.     

On the definition of an admitted Lie group for stochastic differential equations

A new definition of an admitted Lie group of transformations for stochastic differential equations involving Brownian motion is presented. The transformation of the dependent variables involves time as well, and it is proved that Brownian motion is transformed to Brownian motion. Applications to a variety of stochastic differential equations are presented.     

On the well-posedness of coupled forward–backward stochastic differential equations driven by Teugels martingales

We deal with a class of fully coupled forward–backward stochastic differential equations (FBSDEs), driven by Teugels martingales associated with a general Lévy process. Under some assumptions on the derivatives of the coefficients, we prove the existence and uniqueness of a global solution on an arbitrarily large time interval. Moreover, we establish stability and comparison theorems for the solutions of such equations. Note that the present work extends known results proved for FBSDEs driven by a Brownian motion, by using martingale techniques related to jump processes, to overcome the lack of continuity.     

On the Moments of the Modulus of Continuity of Itô Processes

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.     

Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type

Processes of Ornstein-Uhlenbeck type on $\text{R}$^d are analogues of the Ornstein-Uhlenbeck process on $\text{R}$^d with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator-selfdecomposable distributions of Urbanik is characterized as the class of limit distributions of such processes. Continuity of the correspondence is proved. Integro-differential equations for operator-selfdecomposable distributions are established. Examples are given for null recurrence and transience of processes of Ornstein-Uhlenbeck type on $\text{R}$¹.     

The mean of the running maximum of an integrated Gauss–Markov process and the connection with its first-passage time

We find explicit formulae for the mean of the running maximum of conditional and unconditional Brownian motion; they are used to obtain the mean, a(t), of the running maximum of an integrated Gauss–Markov process. Then, we deal with the connection between the moments of its first-passage-time and a(t). As explicit examples, we consider integrated Brownian motion and integrated Ornstein–Uhlenbeck process.     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

Brownian Processes for Monte Carlo Integration on Compact Lie Groups

This article proposes a Monte Carlo approach for the evaluation of integrals of smooth functions defined on compact Lie groups. The approach is based on the ergodic property of Brownian processes in compact Lie groups. The article provides an elementary proof of this property and obtains the following results. It gives the rate of almost sure convergence of time averages along with a “large deviations” type upper bound and a central limit theorem. It derives probability of error bounds for uniform approximation of the paths of Brownian processes using two numerical schemes. Finally, it describes generalization to compact Riemannian manifolds.     

Asymptotically One-Dimensional Diffusion on the Sierpinski Gasket and Multi-Type Branching Processes with Varying Environment

Asymptotically one-dimensional diffusions on the Sierpinski gasket constitute a one parameter family of processes with significantly different behaviour to the Brownian motion. Due to homogenization effects they behave globally like the Brownian motion, yet locally they have a preferred direction of motion. We calculate the spectral dimension for these processes and obtain short time heat kernel estimates in the Euclidean metric. The results are derived using branching process techniques, and we give estimates for the left tail of the limiting distribution for a supercritical multi-type branching process with varying environment.     

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.     

Karhunen–Loève expansion for additive Brownian motions

For Gaussian random fields defined as additive processes based on standard Brownian motions and Brownian bridges, we find their Karhunen–Loève expansions and make connections with related mean centered processes in distribution. Moreover, Pythagorean type distribution identities are established for additive Brownian motions and Brownian bridges. As applications, the corresponding Laplace transform and small deviation estimates are given.     

维纳和布朗运动

布朗运动,作为一种特殊的随机过程,在随机过程理论处于一个中心地位.布朗运动理论在其他许多领域也有重要应用.在布朗运动理论的发展和完善过程中,布朗,爱因斯坦和维纳等人都作出了重要贡献.通过解读原始文献,考察了维纳建立布朗运动数学理论的过程.揭示了维纳在布朗运动的数学理论严格化进程中的重要作用.     

Pathwise Taylor schemes for random ordinary differential equations

Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.     

Fractional Lévy Processes as a Result of Compact Interval Integral Transformation

Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Lévy processes are defined by integrating the infinite interval kernel w.r.t. a general Lévy process. In this article we define fractional Lévy processes using the com pact interval representation.

We prove that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Lévy processes and analyze the properties of such processes. A financial example is introduced as well.     

Small Ball Estimates for Gaussian Processes under Sobolev Type Norms

A sharp small ball estimate under Sobolev type norms is obtained for certain Gaussian processes in general and for fractional Brownian motions in particular. New method using the techniques in large deviation theory is developed for small ball estimates. As an application the Chung's LIL for fractional Brownian motions is given in this setting.     

Characterizing anomalous diffusion by studying displacements

Subordinated Lévy processes provide very diverse conceptual models for mass transport, beside other paradigms (e.g., fractional Brownian motion) generalizing Brownian motion. Some of that many models exhibit similar empirical Mean Squared Displacements growing non-linearly with time, while their increments have very different characteristic functions. In many media, such functionals can be directly measured, but accurate inversion methods adapted to them and to subordinated processes are still lacking. We show that each such process is associated to an operator that transforms the deviation from 1 of the characteristic function of its increments into a quantity that does not depend on the wave-number. We build an inversion method based on this property: it deduces the individual identity of each subordinated Lévy process from data sampling the characteristic functions of its increments.     

Logarithmic Level Comparison for Small Deviation Probabilities

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L² norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm. An erratum to this article can be found at     

Symmetry and singularity properties of a system of ordinary differential equations arising in the analysis of the nonlinear telegraph equations

We investigate the symmetry and similarity properties of a system of equations arising in the analysis of the nonlinear telegraph equations. The system of two equations can be decoupled and integrated. Although one of equations is linear in one of the dependent variables, we are able to perform successfully a singularity analysis. We are able to interpret the results of the singularity analysis in terms of the possibility of the existence of a subsidiary solution.     

On the Local Time of the Weighted Bootstrap and Compound Empirical Processes

This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.