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Kantorovich-Rubinstein Maximum Principle in the Stability Theory of Markov Semigroups

Henryk Gacki (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

A new sufficient condition for the asymptotic stability of a locally Lipschitzian Markov semigroup acting on the space of signed measures s i g is proved. This criterion is applied to the semigroup of Markov operators generated by a Poisson driven stochastic differential equation.

Karhunen-Loève expansions of α-Wiener bridges

Mátyás Barczy, Endre Iglói (2011)

Open Mathematics

We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation d X t ( α ) = - α T - t X t ( α ) d t + d B t , t [ 0 , T ) , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function...

Kermack-McKendrick epidemic model revisited

Josef Štěpán, Daniel Hlubinka (2007)

Kybernetika

This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale N t that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size N t . Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer...

Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's

Michal Vyoral (2005)

Applications of Mathematics

We consider a stochastic process X t x which solves an equation d X t x = A X t x d t + Φ d B t H , X 0 x = x where A and Φ are real matrices and B H is a fractional Brownian motion with Hurst parameter H ( 1 / 2 , 1 ) . The Kolmogorov backward equation for the function u ( t , x ) = 𝔼 f ( X t x ) is derived and exponential convergence of probability distributions of solutions to the limit measure is established.

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