Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures

2010 ◽  
Vol 85 (11) ◽  
pp. 1365-1389 ◽  
Author(s):  
Nitin Agarwal ◽  
N. R. Aluru
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Measures ◽  
Uniform Probability ◽  
Smolyak Algorithm

scholarly journals On stochastic relaxed control for partially observed diffusions

1984 ◽  
Vol 93 ◽  
pp. 71-108 ◽  
Author(s):  
W. H. Fleming ◽  
M. Nisio
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Control Region ◽  
The State ◽  
Probability Measures ◽  
Control Problems ◽  
Relaxed Control ◽  
Observation Process ◽  
Partially Observed ◽  
Partially Observed Diffusions

In this paper we are concerned with stochastic relaxed control problems of the following kind. Let X(t), t ≥ 0, denote the state of a process being controlled, Y(t), t ≥ 0, the observation process and p(t, ·) a relaxed control, that is a process with values probability measures on the control region Г. The state and observation processes are governed by stochastic differential equationsandwhere B and W are independent Brownian motions with values in Rn and Rm respectively, (put m = 1 for simplicity).

scholarly journals Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Michał Branicki ◽  
Kenneth Uda
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Periodic Orbits ◽  
Linear Response ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Climate Modelling ◽  
Small Perturbations ◽  
Asymptotic Probability ◽  
Time Periodic

AbstractWe consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.

scholarly journals Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps

2021 ◽  
Vol 24 (01) ◽  
pp. 2150006
Author(s):  
Huijie Qiao ◽  
Jiang-Lun Wu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Recent Work ◽  
Probability Measures ◽  
Brownian Motions ◽  
Additive Functionals ◽  
Path Independence ◽  
Independent Property

In this paper, the path independent property of additive functionals of McKean–Vlasov stochastic differential equations with jumps is characterized by nonlinear partial integro-differential equations involving [Formula: see text]-derivatives with respect to probability measures introduced by Lions. Our result extends the recent work16 by Ren and Wang where their concerned McKean–Vlasov stochastic differential equations are driven by Brownian motions.

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law

scholarly journals Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

2020 ◽  
Vol 53 (2) ◽  
pp. 2220-2224
Author(s):  
William M. McEneaney ◽  
Hidehiro Kaise ◽  
Peter M. Dower ◽  
Ruobing Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Solution ◽  
Solution Existence
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