Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov eq...

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Hlavní autoři: Bou-Rabee, Nawaf, Vanden-Eijnden, Eric
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2018
Vydání:1
Edice:Memoirs of the American Mathematical Society
Témata:
Random walks (Mathematics)
Stochastic differential equations
Stochastic differential equations > Numerical solutions
parabolic partial differential equation
Stochastic differential equations
stochastic simulation algorithm
discrete maximum principle
Markov jump process
invariant measure
Fokker-Planck equation
stochastic Lyapunov function
geometric ergodicity
Kolmogorov equation
non-symmetric diffusions
monotone operator
ISBN:9781470431815, 1470431815
ISSN:0065-9266, 1947-6221
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Popis
Shrnutí:This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the infinitesimal generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension makes it possible to quantify the computational cost of the approximation. Using a stochastic representation of the global error, we show that the approximation is (weakly) accurate in representing finite and infinite-time expected values, with an order of accuracy identical to the order of accuracy of the infinitesimal generator of the approximation. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.
Bibliografie:Includes bibliographical references (p. 117-124)
November 2018, volume 256, number 1228 (fourth of 6 numbers)
ISBN:9781470431815
1470431815
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1228