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Hiding a constant drift

Vilmos Prokaj, Miklós Rásonyi, Walter Schachermayer (2011)

Annales de l'I.H.P. Probabilités et statistiques

The following question is due to Marc Yor: Let B be a brownian motion and St=t+Bt. Can we define an -predictable process H such that the resulting stochastic integral (H⋅S) is a brownian motion (without drift) in its own filtration, i.e. an -brownian motion? In this paper we show that by dropping the requirement of -predictability of H we can give a positive answer to this question. In other words, we are able to show that there is a weak solution to Yor’s question. The original question, i.e.,...

Hitting time of a corner for a reflected diffusion in the square

F. Delarue (2008)

Annales de l'I.H.P. Probabilités et statistiques

We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way...

Infinite system of Brownian balls with interaction: the non-reversible case

Myriam Fradon, Sylvie Rœlly (2007)

ESAIM: Probability and Statistics

We consider an infinite system of hard balls in d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.

Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations

I. Stojkovic, O. van Gaans (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider a stochastic delay differential equation with exponentially stable drift and diffusion driven by a general Lévy process. The diffusion coefficient is assumed to be locally Lipschitz and bounded. Under a mild condition on the large jumps of the Lévy process, we show existence of an invariant measure. Main tools in our proof are a variation-of-constants formula and a stability theorem in our context, which are of independent interest.

Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz (2002)

Annales Polonici Mathematici

We consider the stochastic differential equation (1) d u ( t ) = a ( u ( t ) , ξ ( t ) ) d t + ∫ Θ σ ( u ( t ) , θ ) p ( d t , d θ ) for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup P t t ≥ 0 corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup P t t ≥ 0 describing the evolution of measures along trajectories and vice versa.

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...

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