Mean stochastic comparison of diffusions

Summary Stochastic bounds are derived for one dimensional diffusions (and somewhat more general random processes) by dominating one process pathwise by a convex combination of other processes. The method permits comparison of diffusions with different diffusion coefficients. One interpretation of the bounds is that an optimal control is identified for certain diffusions with controlled drift and diffusion coefficients, when the reward function is convex. An example is given to show how the bounds and the Liapunov function technique can be applied to yield bounds for multidimensional diffusions.This work was supported by the Office of Naval Research under Contract N00014-82-K-0359 and the U.S. Army Research Office under Contract DAAG29-82-K-0091 (administered through the University of California at Berkeley).     

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.     

Singularly Perturbed Switching Diffusions: Rapid Switchings and Fast Diffusions

Motivated by many problems in optimization and control, this paper is concerned with singularly perturbed systems involving both diffusions and pure jump processes. Two models are treated. In the first model, the jump process changes very rapidly by comparison with the diffusion processes. In the second model, the diffusions change rapidly in comparison with the jump process. Asymptotic expansions are developed for the transition density vectors via a constructive method; justification of the asymptotic expansions and analysis of the remainders are provided.     

Exit identities for diusion processes observed at Poisson arrival times

For diffusion processes, we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process. The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.     

Strong Averaging Along Foliated Lévy Diffusions with Heavy Tails on Compact Leaves

This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed Lévy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order \(p\geqslant 2\), perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of Lévy rotations on the unit circle subject to perturbations by a planar Lévy-Ornstein-Uhlenbeck process is carried out in detail.     

Semimartingales on rays,Walsh diffusions,and related problems of control and stopping

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of stochastic control with discretionary stopping involving Walsh diffusions.     

On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem in the bounded case. The approach is illustrated through the optimal stopping problem of a $d$-dimensional Wiener process with a positive definite quadratic form reward function.     

Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces

We give equivalent characterizations for off-diagonal upper bounds of the heat kernel of a regular Dirichlet form on the metric measure space, in two settings: for the upper bounds with the polynomial tail (typical for jump processes) and for the upper bounds with the exponential tail (for diffusions). Our proofs are purely analytic and do not use the associated Hunt process.     

Some Notes on Large Deviations of Markov Processes

In this paper we shall characterize the large deviation principles (abbreviated to LDP) of Donsker-Varadhan of a Markov process both for the weak convergence topology and for the τ-topology, by means of a hyper-exponential recurrence property. A Lyapunov criterion for this type of recurrence property is presented. These results are applied to countable Markov chains, unidimensional diffusions, elliptic or hypoelliptic diffusions on Rienmannian manifolds. Several counter-examples are equally presented. Received July 20, 1998, Accepted March 25, 1999     

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.     

The partial malliavin calculus and its application to non-linear filtering

In this article, the Malliavin calculus is used to derive regularity properties of the conditional distribution of one ltd process given a second Ito process. The relation between the processes involved is the usual one assumed in the study of filtering theory. The non-degeneracy which we require is stated in terms of Malliavins covariance matrix in Theorem (3.15). More practical conditions are given in Lemma (3.19) for general Tto processes and in Lemma (3.29) for diffusions. Finally, in Theorem (4.6) a “localized” version of these results is given for diffusions.     

A Class of Infinitesimal Generators and Time Reversal for the Corresponding One-Dimensional Markov Processes

A class of infinitesimal generators A of strongly continuous nonnegative contraction semigroups in a subspace of C[0, 1] is introduced. It contains the class of generators of regular gap diffusions. A construction of the Markov process X generated by A gives some stochastic interpretations of the integral term which appears in A. The infinitesimal generator of the time reversal of X (with respect to its life time) is explicitly given. It belongs to the introduced class of generators too. Thus, the considered class is invariant under this transformation. Two examples, the time reversal of gap diffusions with nonlocal boundary conditions and the time reversal of processes with Levy-measure, complete the note.     

Interacting diffusions in a random medium: comparison and longtime behavior

We consider a collection of linearly interacting diffusions (indexed by a countable space) in a random medium. The diffusion coefficients are the product of a space–time dependent random field (the random medium) and a function depending on the local state. The main focus of the present work is to establish a comparison technique for systems in the same medium but with different state dependence in the diffusion terms. The technique is applied to generalize statements on the longtime behavior, previously known only for special choices of the diffusion function.One of these special choices, which we employ as a reference model, is that of interacting Fisher–Wright diffusions in a catalytic medium where duality was used to obtain detailed results. The other choice is that of interacting Feller's branching diffusions in a catalytic medium which is itself an (autonomous) branching process and where infinite divisibility was used as the main tool.     

Heavy-tailed fractional Pearson diffusions

We define heavy-tailed fractional reciprocal gamma and Fisher–Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher–Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher–Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation.     

ARONSON'S ESTIMATES AND CONDITIONAL DIFFUSION PROCESSES

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Time reversal of infinite-dimensional point processes

Time reversal is considered for an infinite-dimensional point process with Markov intensity. An infinite-dimensional duality equation is derived, which is the point-process counterpart of the classical duality equation for diffusions.     

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.     

On nonlinear coupled diffusions in competition systems

A class of reaction-diffusion systems modeling plant growth with spatial competition in saturated media is presented. We show, in this context, that standard diffusion can not lead to pattern formation (Diffusion Driven Instability of Turing). Degenerated nonlinear coupled diffusions inducing free boundaries and exclusive spatial diffusions are proposed. Local and global existence results are proved for smooth approximations of the degenerated nonlinear diffusions systems which give rise to long-time pattern formations. Numerical simulations of a competition model with degenerate/non degenerate nonlinear coupled diffusions are performed and we carry out the effect of the these diffusions on pattern formation and on the change of basins of attraction.     

Certain properties related to well posedness of switching diffusions

This work is devoted to switching diffusions that have two components (a continuous component and a discrete component). Different from the so-called Markovian switching diffusions, in the setup, the discrete component (the switching) depends on the continuous component (the diffusion process). The objective of this paper is to provide a number of properties related to the well posedness. First, the differentiability with respect to initial data of the continuous component is established. Then, further properties including uniform continuity with respect to initial data, and smoothness of certain functionals are obtained. Moreover, Feller property is obtained under only local Lipschitz continuity. Finally, an example of Lotka–Volterra model under regime switching is provided as an illustration.