Displaying similar documents to “A note on one-dimensional stochastic equations”

Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Johanna Dettweiler, J.M.A.M. van Neerven (2006)

Czechoslovak Mathematical Journal

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Let A = d / d θ denote the generator of the rotation group in the space C ( Γ ) , where Γ denotes the unit circle. We show that the stochastic Cauchy problem d U ( t ) = A U ( t ) + f d b t , U ( 0 ) = 0 , ( 1 ) where b is a standard Brownian motion and f ∈ C ( Γ ) is fixed, has a weak solution if and only if the stochastic convolution process t ↦ ( f * b ) t has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution...

Geometry of non-holonomic diffusion

Simon Hochgerner, Tudor S. Ratiu (2015)

Journal of the European Mathematical Society

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We study stochastically perturbed non-holonomic systems from a geometric point of view. In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry of the constraint distribution. For G -Chaplygin systems, this yields a stochastic criterion for the existence of a smooth preserved measure. As an application of our results we consider the motion planning problem for the noisy two-wheeled robot and the noisy snakeboard. ...

Strong solutions for stochastic differential equations with jumps

Zenghu Li, Leonid Mytnik (2011)

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

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General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.

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

Michal Vyoral (2005)

Applications of Mathematics

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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.