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We obtain a sufficient condition of smoothness for the distribution density of a multidimensional Ornstein–Uhlenbeck process with Levy noise, i.e., for the solution of a linear stochastic differential equation with Levy noise.

For L p convergence rates of a time homogeneous Markov process, sufficient conditions are given in terms of an exponential ϕ -coupling. This provides sufficient conditions for L p convergence rates and related spectral and functional properties (spectral gap and Poincare inequality) in terms of appropriate combination of ‘local mixing’ and ‘recurrence’ conditions on the initial process, typical in the ergodic theory of Markov processes. The range of applications of the approach includes processes that are not time-reversible. In particular, sufficient conditions for the spectral gap property for the Levy driven Ornstein–Uhlenbeck process are established.

General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the functionals on the probability space generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form ${\mathbb{R}}^{+}\times{\mathbb{U}}$ , and its intensity measure to equal dt Π(du). We introduce the family of time stretching transformations of the configurations of the point measure. Sufficient conditions for absolute continuity and convergence in variation are given in terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDEs driven by Poisson point measures, including SDEs with non-constant jump rate.

For additive functionals defined on a sequence of Markov chains that approximate a Markov process, we establish the convergence of functionals under the condition of local convergence of their characteristics (mathematical expectations).

Mild sufficient conditions for exponential ergodicity of a Markov process defined as the solution to a SDE with jump noise are given. These conditions include three principal claims: recurrence condition R , topological irreducibility condition S and non-degeneracy condition N , the latter formulated in terms of a certain random subspace of R m , associated with the initial equation. Examples are given, showing that, in general, none of the principal claims can be removed without losing ergodicity of the process. The key point in the approach developed in the paper is that the local Doeblin condition can be derived from N and S via the stratification method and a criterium for the convergence in variation of the family of induced measures on R m .

Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with L\'evy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced.

The truncated local limit theorem is proved for difference approximations of multidimensional diffusions. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities uniformly convergent to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain nontrivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times for multidimensional diffusions are presented.

The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distribution. This condition is formulated only in terms of coefficients of the equation and does not require any regularity properties of the Levy measure of the noise.

For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given.

We determine generalized diffusion coefficients and describe the structure of local times for a process defined as a solution of a one-dimensional stochastic differential equation with singular drift coefficient.

We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process.

We study the classes C(α, β) and CH(α, β) of logarithmically convex measures that are a natural generalization of the notion of Boltzmann measure to an infinite-dimensional case. We prove a theorem on the characterization of these classes in terms of finite-dimensional projections of measures and describe some applications to the theory of random series.

We investigate the distribution properties of the fractional Levy motion defined by the Mandelbrot-Van Ness representation: $$\displaystyle{Z_{t}^{H}:=\int _{ \mathbb{R}}f(t,s)dZ_{s},}$$ where Z s , \(s \in \mathbb{R}\), is a (two-sided) real-valued Levy process, and $$\displaystyle{f(t,s):= \frac{1} {\varGamma (H + 1/2)}\left [(t - s)_{+}^{H-1/2} - (-s)_{ +}^{H-1/2}\right ],\quad t,s \in \mathbb{R}.}$$ We consider separately the cases 0 < H < 1∕2 (short memory) and 1∕2 < H < 1 (long memory), where H is the Hurst parameter, and present the asymptotic behaviour of the distribution density of the process. Some examples are provided, in which it is shown that the behaviour of the density in the cases 0 < H < 1∕2 and 1∕2 < H < 1 is completely different.

We consider the Fisher-Snedecor diffusion; that is, the Kolmogorov-Pearson diffusion with the Fisher-Snedecor invariant distribution. In the nonstationary setting, we give explicit quantitative rates for the convergence rate of respective finite-dimensional distributions to that of the stationary Fisher-Snedecor diffusion, and for the $\beta$-mixing coefficient of this diffusion. As an application, we prove the law of large numbers and the central limit theorem for additive functionals of the Fisher-Snedecor diffusion and construct $P$-consistent and asymptotically normal estimators for the parameters of this diffusion given its nonstationary observation., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ453 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We give a sequence of stochastic integro-differential equations that approximates a stochastic differential equation with an anticipating initial condition and localized Skorokhod stochastic integral. A sequence of solutions of these equations is obtained. The convergence of this sequence to a certain process implies that this process is a solution (generally speaking, local) of the original equation.

Nous presentons une methode de renormalisation pour construire certains modeles de potentiels aleatoires dans un nuage Poissonnien qui sont physiquement plus realistes. Nous obtenons le mouvement brownien dans un potentiel aleatoire renormalise et les modeles d’Anderson paraboliques associes. Par exemple, avec cette renormalisation, nous pouvons construire rigoureusement des modeles consistants avec la loi de la gravitation de Newton.

We give the complete description of the set of partial limits for a large class of sequences of weighted sums of independent random variables with triangular matrices of coefficients.

In (Chen and Kulik, 2009), a method of renormalization was proposed for constructing some more physically realistic random potentials in a Poisson cloud. This paper is devoted to the detailed analysis of the asymptotic behavior of the annealed negative exponential moments for the Brownian motion in a renormalized Poisson potential. The main results of the paper are applied to studying the Lifshitz tails asymptotics of the integrated density of states for random Schrödinger operators with their potential terms represented by renormalized Poisson potentials.

A version of the saddle point method is developed, which allows one to describe exactly the asymptotic behavior of distribution densities of Levy driven stochastic integrals with deterministic kernels. Exact asymptotic behavior is established for (a) the transition probability density of a real-valued Levy process; (b) the transition probability density and the invariant distribution density of a Levy driven Ornstein-Uhlenbeck process; (c) the distribution density of the fractional Levy motion.

General statement of the problem of testing two hypotheses Let the trajectory \(x(\cdot)\) of a stochastic process \(\{X(t), t\in[0,T]\}\) be observed.

Consider a complete filtration \({\mathcal{F}}_t\), t∈[0,T]} and an m-dimensional Wiener process {W(t), t ∈ [0,T]} with respect to it. By definition, a stochastic differential equation (SDE) is an equation of the form with X 0=ξ, where ξ is an \({\mathcal{F}}_0\)-measurable random vector, \(b=b(t,x): [0,T]\times \mathbb{R}^n\rightarrow \mathbb{R}^n\), and \(\sigma=\sigma(t,x): [0,T]\times \mathbb{R}^n\rightarrow \mathbb{R}^{n\times m}\) are measurable functions.

The following theorem shows that all the finite-dimensional distributions for the process with independent increments on

The optimal stopping problem can be considered for nearly any stochastic process, and its formulation will be similar in each case. But its solution will be relatively simple only for a few processes. One class of such processes consists of discrete-time Markov chains.

In this chapter we consider random functions with the phase space being either real line \(\Bbb{R}\) or complex plane \(\Bbb{C}\).

Let \({\Bbb T} \subset \mathbb{R}\), \((\Omega, {\rm{F}}, \{{\rm{F}}_t\}_{t\in {\Bbb T}}, {\mathsf{P}})\) be a probability space with complete filtration. Let \(X = \{X(t), t\in{\Bbb T}\}\) be an adapted stochastic process taking values in some metric space \(({\Bbb X}, {\frak X})\), which sometimes is called the phase space of the process X.

If a random function is stochastically continuous (continuous with probability one, continuous in the L p sense) at every point of the parametric set

Let \(\{W(t),t\in \mathbb{R}^+\}\) be a Wiener process, and \(\{g(t),{\mathcal{F}}_t^W,t\in \mathbb{R}^+\}\) a stochastic process (recall that the previous notation means that g is adapted to a natural filtration \(\{{\mathcal{F}}_t^W\}\) of the Wiener process). Let \({\mathcal{F}}^W=\sigma\{W_t, t\geq 0\}.\) A process g is said to belong to the class \(\hat{\mathcal{L}}_2([a,b])\) if it is measurable and \({{\mathsf{E}}}\int_a^b g^2(s) d s

In this chapter we consider complex-valued stochastic processes.Let us recall the definition of covariance for complex-valued random variables X, Y

Let phase space \(\Bbb{X}\) of a random sequence \(\{X_n,\,n\in\Bbb{Z}^+\}\) be enumerable. The sequence \(\{X_n,\,n\in\Bbb{Z}^+\}\) is called a Markov chain if $$\forall n\in\Bbb{N}\ \forall i_1,\ldots,i_n,i_{n+1}\in \Bbb{X} \ \forall t_1\leq\cdots\leq t_n\leq t_{n+1}\in\Bbb{Z}_+:$$ $$\mathsf{P}(X_{t_{n+1}}=i_{n+1} / X_{t_{1}}=i_{1},\ldots,X_{t_{n}}=i_{n})=\mathsf{P}(X_{t_{n+1}}=i_{n+1} / X_{t_{n}}=i_{n}).$$

Below in this chapter we denote the discrete-time processes as X n , M n , and so on (with the lower index as time) and continuous-time processes, as before, are denoted as X(t), M(t), and so on (the time index is inside the parentheses).

Consider a model of a financial market with a finite number of periods (i.e., of the moments of time) at which it is possible to trade, consume, spend, or receive money or other valuables. The model consists of the following components.

Hereinafter the mapping X(t, ) is denoted as X(t). According to commonly accepted terminology it is a random element taking values in X. The definition introduced above is obviously equivalent to the following one.

In this chapter, we consider random elements taking values in metric spaces and their distributions. The definition of a random element taking values in \({{\Bbb X}}\) involves the predefined \(\sigma\)-algebra \({{\rm X}}\) of subsets of \({{\Bbb X}}\). The following statement shows that in a separable metric space, in fact, the unique natural choice for the \(\sigma\)-algebra \({{\rm X}}\) is the Borel \(\sigma\)-algebra \({{\rm B}}({{\Bbb X}})\).

Mathematical foundations of investigating of the risk process in insurance were created by Swedish mathematician Filip Lundberg in 1903–1909. For a long time this theory had been developed by mostly Nordic mathematicians, such as Cramer, Segerdal, Teklind, and others. Later on risk theory started to develop not only with connection to insurance but also as the method of solving different problems in actuarial and financial mathematics, econometrics. In the second half of the twentieth century the applied area of risk theory was expanded significantly.