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A note on one-dimensional stochastic equations

Hans-Jürgen Engelbert (2001)

Czechoslovak Mathematical Journal

We consider the stochastic equation X t = x 0 + 0 t b ( u , X u ) d B u , t 0 , where B is a one-dimensional Brownian motion, x 0 is the initial value, and b [ 0 , ) × is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients b , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on b ensuring the existence as well as the uniqueness in law of the solution.

A note on weak solutions to stochastic differential equations

Martin Ondreját, Jan Seidler (2018)

Kybernetika

We revisit the proof of existence of weak solutions of stochastic differential equations with continuous coeficients. In standard proofs, the coefficients are approximated by more regular ones and it is necessary to prove that: i) the laws of solutions of approximating equations form a tight set of measures on the paths space, ii) its cluster points are laws of solutions of the limit equation. We aim at showing that both steps may be done in a particularly simple and elementary manner.

A note on γ-radonifying and summing operators

Zdzisław Brzeźniak, Hongwei Long (2015)

Banach Center Publications

In this note, we discuss certain generalizations of γ-radonifying operators and their applications to the regularity for linear stochastic evolution equations on some special Banach spaces. Furthermore, we also consider a more general class of operators, namely the so-called summing operators and discuss the application to the compactness of the heat semi-group between weighted L p -spaces.

A relaxation theorem for partially observed stochastic control on Hilbert space

N.U. Ahmed (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we present a result on relaxability of partially observed control problems for infinite dimensional stochastic systems in a Hilbert space. This is motivated by the fact that measure valued controls, also known as relaxed controls, are difficult to construct practically and so one must inquire if it is possible to approximate the solutions corresponding to measure valued controls by those corresponding to ordinary controls. Our main result is the relaxation theorem which states that...

A second order SDE for the Langevin process reflected at a completely inelastic boundary

Jean Bertoin (2008)

Journal of the European Mathematical Society

It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.

A stochastic approach to relativistic diffusions

Ismaël Bailleul (2010)

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

A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced in the article of C. Chevalier and F. Debbasch (J. Math. Phys. 49 (2008) 043303), both in a heuristic and analytic way. A stochastic approach of these processes is proposed here, in the general framework of lorentzian geometry. In considering the dynamics of the random motion in strongly causal spacetimes, we are able to give a simple definition of the one-particle distribution function...

A stochastic min-driven coalescence process and its hydrodynamical limit

Anne-Laure Basdevant, Philippe Laurençot, James R. Norris, Clément Rau (2011)

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

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.

A stochastic model of symbiosis

Urszula Skwara (2010)

Annales Polonici Mathematici

We consider a system of stochastic differential equations which models the dynamics of two populations living in symbiosis. We prove the existence, uniqueness and positivity of solutions. We analyse the long-time behaviour of both trajectories and distributions of solutions. We give a biological interpretation of the model.

A stochastic symbiosis model with degenerate diffusion process

Urszula Skwara (2010)

Annales Polonici Mathematici

We present a model of symbiosis given by a system of stochastic differential equations. We consider a situation when the same factor influences both populations or only one population is stochastically perturbed. We analyse the long-time behaviour of the solutions and prove the asymptoptic stability of the system.

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