44 research outputs found
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 be a Brownian motion, and let be the space of all continuous periodic functions with period 1. It is shown that the set of all such that the stochastic convolution , 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 be the transition semigroup of the Markov family
defined by SDE where is a system of independent real-valued L\'evy processes.
Using the Malliavin calculus we establish the following gradient formula where the random field does not depend on . Sharp
estimates on when are -stable
processes, , 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