On ergodicity of some Markov processes

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The investor problem based on the HJM model

We consider a consumption-investment problem (both on finite and infinite time horizon) in which the investor has an access to the bond market. In our approach prices of bonds with different maturities are described by the general HJM factor model. We assume that the bond market consists of entire family of rolling bonds and the investment strategy is a general signed measure distributed on all real numbers representing time to maturity specifications for different rolling bonds. In particular, we can consider portfolio of coupon bonds. The investor's objective is to maximize time-additive utility of the consumption process. We solve the problem by means of the HJB equation for which we prove required regularity of its solution and all required estimates to ensure applicability of the verification theorem. Explicit calculations for affine models are presented.Comment: v2 - 26 pages, detailed calculations of G2++ model, extended proof of theorem 4.1, two references added( [2] and [33]), v3 - 28 pages, revised version after reviews, (v4) - 30 pages, language corrections, (v5),(v6) - 29 pages, final correction

Ergodicity of Burgers' system

We consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a unique invariant measure is investigated. The paper generalizes the results of Da Prato, Debussche and Temam, and Da Prato and Gatarek, dealing with one equation describing the turbulent motion only.Comment: 18 page

Continuity of stochastic convolutions

summary:Let $B$ be a Brownian motion, and let $\mathcal C_{\mathrm p}$ be the space of all continuous periodic functions $f\:\mathbb{R}\rightarrow \mathbb{R}$ with period 1. It is shown that the set of all $f\in \mathcal C_{\mathrm p}$ such that the stochastic convolution $X_{f,B}(t)= \int _0^tf(t-s)\mathrm{d}B(s)$, $t\in [0,1]$ does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category

Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent L\'evy processes

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$d X= b(X) dt + d Z, \qquad X(0)=x,$$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued L\'evy processes. Using the Malliavin calculus we establish the following gradient formula $$\nabla P_tf(x)= \mathbb{E}\, f\left(X^x(t)\right) Y(t,x), \qquad f\in B_b(\mathbb{R}^d),$$ where the random field $Y$ does not depend on $f$. Sharp estimates on $\nabla P_tf(x)$ when $Z_1, \ldots , Z_d$ are $\alpha$-stable processes, $\alpha \in (0,2)$, are also given

Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes

Let (P-t) be the transition semigroup of the Markov family (X-x (t)) defined by SDEdX = b(X)dt+ dZ, X(0) = x,where Z = (Z(1),..., Z(d))* is a system of independent real-valued Levy processes. Using the Malliavin calculus we establish the following gradient formuladel P(t)f(x) = E f (X-x (t)) Y (t, x), f is an element of B-b(R-d),where the random field Y does not depend on f. Moreover, in the important cylindrical alpha-stable case alpha is an element of (0, 2), where Z(1),..., Z(d) are alpha-stable processes, we are able to prove sharp L-1-estimates for Y (t, x). Uniform estimates on del P(t)f(x) are also given

Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations

In this paper we prove the law of large numbers and central limit theorem for trajectories of a particle carried by a two dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier--Stokes system with a non-degenerate noise. The spectral gap property, with respect to Wasserstein metric, for such a system has been shown in [9]. In the present paper we show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. In consequence we conclude the law of large numbers and the central limit theorem for the tracer. The proof of the central limit theorem relies on the martingale approximation of the trajectory process