scholarly journals Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations

2018 ◽  
Vol 334 ◽  
pp. 39-57 ◽  
Author(s):  
Cónall Kelly ◽  
Alexandra Rodkina ◽  
Eeva Maria Rapoo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonlinear Stochastic Differential Equations ◽  
Pathwise Stability ◽  
Adaptive Timestepping

A variational approach to nonlinear stochastic differential equations with linear multiplicative noise

2019 ◽  
Vol 25 ◽  
pp. 71
Author(s):  
Viorel Barbu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Parabolic Equations ◽  
Optimization Problem ◽  
Multiplicative Noise ◽  
Generalized Solution ◽  
Convex Optimization Problem ◽  
Nonlinear Stochastic Differential Equations ◽  
Infinite Dimensional ◽  
Wiener Processes

One introduces a new concept of generalized solution for nonlinear infinite dimensional stochastic differential equations of subgradient type driven by linear multiplicative Wiener processes. This is defined as solution of a stochastic convex optimization problem derived from the Brezis-Ekeland variational principle. Under specific conditions on nonlinearity, one proves the existence and uniqueness of a variational solution which is also a strong solution in some significant situations. Applications to the existence of stochastic total variational flow and to stochastic parabolic equations with mild nonlinearity are given.

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