Bismut–Elworthy–Li-Type Formulae for Stochastic Differential Equations with Jumps

We consider jump-type stochastic differential equations with drift, diffusion, and jump terms. Logarithmic derivatives of densities for the solution process are studied, and Bismut–Elworthy–Li-type formulae are obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markov property of the process.     

Kolmogorov向前向后方程组的概率意义

研究了Kolmogorov向前向后方程组的概率意义,得到正规链满足Kolmogorov向前向后方程组的等价条件,并进一步得到不诚实但全稳定的转移函数对应的带“杀死”的Markov链满足Kolmogorov向前向后方程组的充分必要条件.     

The uniqueness of hierarchically extended backward solutions of the Wright–Fisher model

The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the Kolmogorov forward and backward equations, with a leading term that degenerates at the boundary. This degeneracy has the consequence that standard PDE tools do not apply, and solutions lack regularity properties. In this paper, we develop a regularizing blow-up scheme for the iteratively extended global solutions of the backward Kolmogorov equation presented in a previous paper, which are constructed from a known class of solutions, and establish their uniqueness for the stationary case. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the occurring singularities result from the loss of an allele. While in an analytical approach, this provides substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularizes the solution via a carefully constructed iterative transformation of the domain.     

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.     

Weak Euler Approximation for Itô Diffusion and Jump Processes

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.     

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING (F) OR (B)

This article is a continuation of [9]. Based on the discussion of random Kol-mogorov forward (backward) equations, for any given q-matrix in random environment,Q(θ) = (q(θ; x, y), x, y ∈ X), an infinite class of q-processes in random environments sat-isfying the random Kolmogorov forward (backward) equation is constructed. Moreover,under some conditions, all the q-processes in random environments satisfying the random Kolmogorov forward (backward) equation are constructed.     

The existence and uniqueness of the solution for nonlinear Kolmogorov equations

By means of backward stochastic differential equations, the existence and uniqueness of the mild solution are obtained for the nonlinear Kolmogorov equations associated with stochastic delay evolution equations. Applications to optimal control are also given.     

On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller?s seminal paper. In particular, this paper extends Feller?s results for continuous Q-functions to measurable Q-functions and provides additional results.     

Nonlocal transformations of Kolmogorov equations into the backward heat equation

We extend and solve the classical Kolmogorov problem of finding general classes of Kolmogorov equations that can be transformed to the backward heat equation. These new classes include Kolmogorov equations with time-independent and time-dependent coefficients. Our main idea is to include nonlocal transformations. We describe a step-by-step algorithm for determining such transformations. We also show how all previously known results arise as particular cases in this wider framework.     

Systems of equations of Kolmogorov type

We consider one class of degenerate parabolic systems of equations of the type of diffusion equation with Kolmogorov inertia. For systems whose coefficients may depend only on the time variable, we construct a fundamental matrix of solutions of the Cauchy problem and obtain estimates for this matrix and all its derivatives. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1650–1663, December, 2008.     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.     

L∞‐estimates for the Vlasov–Poisson–Fokker–Planck equation

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non‐linear diffusion process. In this way we derive new L^∞‐estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.     

A New Markov Selection Procedure for Degenerate Diffusions

A new selection principle is proposed for construction of a Feller solution to an ill-posed degenerate diffusion process. This is based on constructing the Feller transition kernel from the unique (under suitable conditions) continuous viscosity solutions to the backward Kolmogorov equation associated with the diffusion. The connection of this Feller process with the small-noise limits of the nondegenerate perturbations of the diffusion and several other related phenomena are also discussed.     

Density symmetries for a class of 2-D diffusions with applications to finance

We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models.     

On a class of elliptic singular perturbations with applications in population genetics

With the maximum principle for differential equations asymptotic estimates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to the stationary solution of the Kolmogorov backward equation in population genetics.     

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING （F） OR （B）

This article is a continuation of[9].Based on the discussion of random Kolmogorov forward（backward）equations,for any given q-matrix in random environment, Q（θ）=（q（θ;x,y）,x,y∈X）,an infinite class of q-processes in random environments satisfying the random Kolmogorov forward（backward）equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward（backward）equation are constructed.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

Kolmogorov equations in fractional derivatives for the transition probabilities of continuous-time Markov processes

A family of one-dimensional continuous-time Markov processes is considered, for which the author has earlier determined the transition probabilities by directly solving the Kolmogorov–Chapman equation; these probabilities have the form of single integrals. Analogues of the first and second Kolmogorov equations for the family of processes under consideration are obtained by using a procedure for obtaining integro-differential equations describing Markov processes with discontinuous trajectories. These equations turn out to be equations in fractional derivatives. The results are based on an asymptotic analysis of the transition probability as the start and end times of the transition approach each other. This analysis implies that the trajectories of a given Markov process are divided into two classes, depending on the interval in which they start. Some of the trajectories decay during a short time interval with a certain probability, and others are generated with a certain probability.     

Ergodicity for the 3D stochastic Navier–Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u_m of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.