Distributions of functionals of diffusions with jumps

The paper deals with methods of computing the distributions of functionals of a process that is a diffusion with jumps occurring according to a compound Poisson process. For symmetric processes, some exact formulas for distributions related to the first exit time are derived. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 27–41.     

Distributions of functionals of diffusions with jumps

The paper deals with a generalization of diffusion with jumps. One of the main points is that values of jumps depend on positions of the diffusion before the jump. The next generalization concerns moments of jumps. These moments occur in accordance with the compound Poisson process or with jumping moments constructed by inverse integral functionals of the diffusion. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 15–36.     

Transformations of diffusions with jumps

The paper deals with some transformations of diffusions with jumps. We consider the class of diffusions with jumps that is closed with respect to composition with invertible, twice continuously differentiable functions. A special random time change gives us again a diffusion with jumps. A result on transformation of a measure is valid for this class of diffusions with jumps. Bibliographty: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 79–100.     

Forward equations for reflected diffusions with jumps

In this paper we obtain the forward equations associated with the evolution of the density, if it exists, of reflected diffusions on the positive orthant with jumps which form a marked point process whose random jump measure possesses a stochastic intensity. These results generalize the so-called generalized Dynkin equations for piecewise deterministic jump processes due to Davis. We then consider the stationary case where the existence of a stochastic intensity is not needed. The techniques are based on local times and the use of random jump measures. We discuss the application of these results to problems arising in queuing and storage processes as well as stationary distributions of diffusions with delayed and jump reflections at the origin.This research was supported in part by the Quebec-France Cooperative Research Program and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP 0042024.     

Distributions of functionals of diffusions with jumps stopped at the location of the maximum or minimum

The paper deals with methods of computation of distributions of integral functionals of diffusions with jumps at time moments at which the maximal and minimal values of diffusions are achieved. As an example, we obtain closed-form expressions for the Laplace transform of joint locations of the minimum and maximum of a process that equals the sum of a Brownian motion and the compound Poisson process. Bibliography: 7 titles.     

On parabolic inequalities for generators of diffusions with jumps

We prove absolute continuity of space-time probabilities satisfying certain parabolic inequalities for generators of diffusions with jumps. As an application, we prove absolute continuity of transition probabilities of singular diffusions with jumps under minimal conditions that ensure absolute continuity of the corresponding diffusions without jumps.     

Ergodic control of reflected diffusions with jumps

We discuss ergodicity properties of a controlled jumps diffusion process reflected from the boundary of a bounded domain. The control parameters act on the drift term and on a first-order-type jump density. The controlled process is generated via a Girsanov change of probability, and a long-run average criterion is optimized. An optimal stationary feedback is constructed by means of the Hamilton-Jacobi-Bellman equation.     

Optimal impulse control problems for degenerate diffusions with jumps

Optimal stopping and impulse control problems for degenerate diffusion with jumps are studied in this paper. Lipschitzian coefficients for the diffusion process, data with polynomial growth, and evolution in the whole space are the main assumptions on the models. Several characterizations of the optimal cost functions are given. Existence of optimal policies is obtained.This research has been supported in part by Army Research Office Contract DAAG29-83-K-0014 and by National Science Foundation Grant DMS-8601998.     

Large deviations for Markov bridges with jumps

In this paper, we consider a family of Markov bridges with jumps constructed from truncated stable processes. These Markov bridges depend on a small parameter ?>0$?>0$, and have fixed initial and terminal positions. We propose a new method to prove a large deviation principle for this family of bridges based on compact level sets, change of measures, duality and various global and local estimates of transition densities for truncated stable processes.     

Distributions of the location of maximuma and minimuma for diffusions with jumps

The paper deals with methods of computation of distributions of location for maxima and minima for diffusions with jumps. As an example, we obtain explicit formulas for distributions of location for the maximum of the process which is equal to the sum of a Brownian motion and the compound Poisson process. Bibliography: 8 titles.     

Ergodic behavior of diffusions with random jumps from the boundary

We consider a diffusion process on D⊂R^d$D\subset {\mathbb{R}}^d$, which upon hitting ∂D$\partial D$, is redistributed in D$D$ according to a probability measure depending continuously on its exit point. We prove that the distribution of the process converges exponentially fast to its unique invariant distribution and characterize the exponent as the spectral gap for a differential operator that serves as the generator of the process on a suitable function space.     

Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of Arapostathis et al. (2019) to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks.We provide a full characterizations of optimality via the Hamilton–Jacobi–Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s. pathwise optimal. Lastly, we show that one can fix a stable control outside a compact set and obtain near-optimal solutions by solving the HJB on a sufficiently large bounded domain. This is useful for constructing asymptotically optimal scheduling policies for multiclass parallel server networks.     

A simplified proof of the representation of functionals of diffusions

A simple proof is given for the stochastic integral representation of a Fréchet differentiable functional of the paths of a given diffusion process. The proof begins when the functional depends on one coordinate, then passes to (by appropriate conditioning) a perhaps more difficult case—when the functional depends on a finite number of coordinates—and finally, by approximation, to the general case.     

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], α∈R, (Bt)_t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,

     

Divergence,convergence and moments of some integral functionals of diffusions

Summary In this paper we study some integral functionals of diffusions. We obtain criteria for their divergence and their convergence and we investigate the existence of their moments.     

Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation

Let X _t be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X _t are asymptotically equivalent in probability to some deterministic process v _t . Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v _t , extending some recent results for ergodic diffusions.      

On the distribution of exponential functionals for Lévy processes with jumps of rational transform

We derive explicit formulas for the Mellin transform and the distribution of the exponential functional for Lévy processes with rational Laplace exponent. This extends recent results by Cai and Kou [3] on the processes with hyper-exponential jumps.     

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.