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Strangely sweeping one-dimensional diffusion

Ryszard Rudnicki (1993)

Annales Polonici Mathematici

Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that l i m s u p t → ∞ t - 1 ∫ 0 t p ( s ) d s = 1 and l i m i n f t → ∞ t - 1 ∫ 0 t p ( s ) d s = 0 .

Strong solutions for stochastic differential equations with jumps

Zenghu Li, Leonid Mytnik (2011)

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

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.

Superposition rules and stochastic Lie–Scheffers systems

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega (2009)

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

This paper proves a version for stochastic differential equations of the Lie–Scheffers theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie–Scheffers result. We show that the stochastic analog...

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