Backward stochastic differential equations with jumps and related non-linear expectations

In this paper, we are interested in real-valued backward stochastic differential equations with jumps together with their applications to non-linear expectations. The notion of non-linear expectations has been studied only when the underlying filtration is given by a Brownian motion and in this work the filtration will be generated by both a Brownian motion and a Poisson random measure. We study at first backward stochastic differential equations driven by a Brownian motion and a Poisson random measure and then introduce the notions of f$f$-expectations and of non-linear expectations in this set-up.     

A general non-convex large deviation result with applications to stochastic equations

We prove an abstract large deviation result for a sequence of random elements of a vector space satisfying an “abstract exponential martingale condition”. The framework naturally generates non-convex rate functions. We apply the result to solutions of It? stochastic equations in R ^d driven by Brownian motion and a Poisson random measure. Received: 23 June 1999 / Revised version: 17 February 2000 / Published online: 22 November 2000     

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Abstract

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.     

Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

We establish new Kahane–Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes φ-mixing and martingale difference random fields.     

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.     

Orthogonal Spaces Associated with Exponentials of Indicator Functions on Fock Space

Parthasarathy and Sunder have proved that the set of coherentvectors associated with the indicator functions of Borel setsis total in the boson Fock space (L²(R⁺;C)). The paper studiesthe space generated by coherent vectors associated with theunion of n intervals. A complete characterization is given oftheir orthogonal space in terms of their chaos expansion. Parthasarathyand Sunder's result is recovered in a very simple way. In thecases of the Brownian motion or Poisson process interpretationof the Fock space, the result characterizes those random variablesthat are orthogonal to the exponential of any sum of n incrementsof the Brownian motion or Poisson process.     

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.     

Forward Brownian motion

We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at \(-\infty \) . We show that these processes do not have to have the distribution of standard Brownian motion in the backward direction of time, no matter which random time we take as the origin. We study the maximum and minimum rates of growth for these processes in the backward direction. We also address the question of which extra assumptions make one of these processes a two-sided Brownian motion.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

Weak interaction limits for one-dimensional random polymers

 In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces, and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach followed in our earlier work, which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations. We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example (1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and find the scaling as the width tends to infinity. Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60 Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality     

Distribution of sojourn time for a Brownian motion with jumps

We consider a Brownian motion with jumps that is a sum of a Brownian motion and compound Poisson process. It is assumed that the distribution of jumps is symmetrically exponential. A formula for the Laplace transform of the distribution of time spent by a Brownian motion with jumps over some level is obtained. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 101–116.     

Gene flow across geographical barriers — scaling limits of random walks with obstacles

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

The perimeter of uniform and geometric words: a probabilistic analysis

Abstract

Let a word be a sequence of n i.i.d. integer random variables. The perimeter P of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of P. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of P is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motion.     

Structural characterization of taboo-stationarity for general processes in two-sided time

This note considers the taboo counterpart of stationarity. A general stochastic process in two-sided time is defined to be taboo-stationary if its global distribution does not change by shifting the origin to an arbitrary non-random time in the future under taboo, that is, conditionally on some taboo-event not having occurred up to the new time origin. The main result is the following basic structural characterization: a process is taboo-stationary if and only if it can be represented as a stochastic process with origin shifted backward in time by an independent exponential random variable. An application to reflected Brownian motion is given.     

Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift — Their characteristics and applications

In ?ochowski (2008) [9] we defined truncated variation of Brownian motion with drift, W_t=B_t+μt,t≥0, where (B_t) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c>0. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument.We also define two closely related quantities — upward truncated variation and downward truncated variation.The defined quantities may have interpretations in financial mathematics. The exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process.We calculate the Laplace transform with respect to the time parameter of the moment-generating functions of the upward and downward truncated variations.As an application of the formula obtained we give an exact formula for the expected values of upward and downward truncated variations. We also give exact (up to universal constants) estimates of the expected values of the quantities mentioned.     

Quadratic functionals of Brownian motion

Functionals of Brownian motion can be dealt with by realizing them as functionals of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a one-to-one correspondence between a quadratic functional of white noise and a symmetric L²(R²)-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.     

Fractional Generalized Random Fields of Variable Order

Abstract

We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on ? ⁿ . Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz–Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Hölder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Hölder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given.     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.