scholarly journals Optimal strong convergence rates of numerical methods for semilinear parabolic SPDE driven by Gaussian noise and Poisson random measure

2019 ◽  
Vol 77 (10) ◽  
pp. 2786-2803 ◽  
Author(s):  
Jean Daniel Mukam ◽  
Antoine Tambue
Keyword(s):  
Numerical Methods ◽  
Strong Convergence ◽  
Gaussian Noise ◽  
Convergence Rates ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Semilinear Parabolic ◽  
Strong Convergence Rates

scholarly journals Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

2020 ◽  
Author(s):  
Aurelien Junior Noupelah ◽  
Antoine Tambue
Keyword(s):  
Finite Element ◽  
Brownian Motion ◽  
Strong Convergence ◽  
Fractional Brownian Motion ◽  
Convergence Rates ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Element Discretization ◽  
Exponential Integrator ◽  
Strong Convergence Rates

AbstractIn this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac {1}{2}$ H > 1 2 and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order $\mathcal {O}(h^{2}+\varDelta t)$ O ( h 2 + Δ t ) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

scholarly journals First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200258
Author(s):  
Jianhai Bao ◽  
Christoph Reisinger ◽  
Panpan Ren ◽  
Wolfgang Stockinger
Keyword(s):  
Strong Convergence ◽  
Convergence Rates ◽  
Interacting Particle Systems ◽  
Particle System ◽  
First Order ◽  
Time Stepping ◽  
Vlasov Equations ◽  
Interacting Particle ◽  
Moment Stability ◽  
Strong Convergence Rates

In this paper, we derive fully implementable first-order time-stepping schemes for McKean–Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean–Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

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