On Comparison Results for Neutral Stochastic Differential Equations of Reaction-Diffusion Type in L2(ℝd)

2018 ◽  
pp. 351-395
Author(s):  
Oleksandr M. Stanzhytskyi ◽  
Viktoria V. Mogilova ◽  
Alisa O. Tsukanova
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Reaction Diffusion ◽  
Diffusion Type ◽  
Comparison Results

Propagation of bursting oscillations

2009 ◽  
Vol 367 (1908) ◽  
pp. 4863-4875 ◽  
Author(s):  
Benjamin Ambrosio ◽  
Jean-Pierre Françoise
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Time Evolution ◽  
Global Attractor ◽  
Reaction Diffusion ◽  
Functional Space ◽  
Excitable Cells ◽  
Diffusion Type ◽  
Bursting Oscillations

We investigate a system of partial differential equations of reaction–diffusion type which displays propagation of bursting oscillations. This system represents the time evolution of an assembly of cells constituted by a small nucleus of bursting cells near the origin immersed in the middle of excitable cells. We show that this system displays a global attractor in an appropriated functional space. Numerical simulations show the existence in this attractor of recurrent solutions which are waves propagating from the central source. The propagation seems possible if the excitability of the neighbouring cells is above some threshold.

Markov Dilation of Diffusion Type Processes and its Application to the Financial Mathematics

1999 ◽  
Vol 6 (4) ◽  
pp. 363-378
Author(s):  
R. Tevzadze
Keyword(s):  
Differential Equations ◽  
Integral Representation ◽  
Stochastic Differential Equations ◽  
Markov Processes ◽  
Contingent Claim ◽  
Financial Mathematics ◽  
Diffusion Type ◽  
Replicating Portfolio ◽  
Infinitesimal Operators

Abstract The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.

scholarly journals $${{\varvec{L}}}^{{\varvec{p}}}$$-Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

2020 ◽  
Author(s):  
Stefan Kremsner ◽  
Alexander Steinicke
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Backward Stochastic Differential Equations ◽  
Time Dependent ◽  
Unified Approach ◽  
General Growth ◽  
Comparison Results ◽  
Osgood Condition

AbstractWe present a unified approach to $$L^p$$ L p -solutions ($$p > 1$$ p > 1 ) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.

scholarly journals Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

2020 ◽  
Vol 28 (4) ◽  
pp. 197-222
Author(s):  
Christian Beck ◽  
Fabian Hornung ◽  
Martin Hutzenthaler ◽  
Arnulf Jentzen ◽  
Thomas Kruse
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Applied Mathematics ◽  
Nonlinear Partial Differential Equations ◽  
Reaction Diffusion ◽  
Curse Of Dimensionality ◽  
Computational Effort ◽  
Approximation Schemes ◽  
Partial Differential ◽  
Diffusion Type

AbstractOne of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

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