Taylor Approximations for Stochastic Partial Differential Equations

Stochastic Taylor Expansions
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Stochastic partial differential equations. Edition Second edition. Series Advances in applied mathematics.

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Taylor approximation of stochastic functional differential equations with the Poisson jump

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Find it at other libraries via WorldCat Limited preview. Pao Liu , author. Svishchuk and Kazmerchuk [ 10 ] studied the exponential stability of solutions of linear stochastic differential equations with Poisson jump [ 11 — 13 ] and Markovian switching [ 4 , 12 , 14 ]. In many applications, one assumes that the system under consideration is governed by a principle of causality, that is, the future states of the system are independent of the past states and are determined solely by the present.

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However, under closer scrutiny, it becomes apparent that the principle of causality is often only the first approximation to the true situation, and that a more realistic model would include some of the past states of the system. Stochastic functional differential equations [ 9 ] give a mathematical explanation for such a system. Unfortunately, in general, it is impossible to find the explicit solution for stochastic functional differential equations with the Poisson jump. Even when such a solution can be found, it may be only in an implicit form or too complicated to visualize and evaluate numerically.

Therefore, many approximate schemes were presented, for example, EM scheme, time discrete approximations, stochastic Taylor expansions [ 15 ], and so on.

Meanwhile, the rate of approximation to the true solution by the numerical solution is different for different numerical schemes. Jankovic et al.

In this paper, we develop approximate methods for stochastic differential equations driven by Poisson process, that is,. The rate of the L p -closeness between the approximate solution and the solution of the initial equation increases when the number of degrees in Taylor approximations of coefficients increases.

Although the Poisson jump is concerned, the rate of approximation to the true solution by the numerical solution is the same as the equation in [ 15 ].

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Even when the Poisson process is replaced by Poisson random measure, the rate is also the same. For the existence and uniqueness of the solutions of Eq. The explicit discrete approximation scheme is defined as follows:. Since the proof of the main result is very technical, to begin with, we present several lemmas which will play an important role in the subsequent section.

Then we obtain. Since x t and x n t satisfy the same initial condition, we can obtain. Denote that. Export Cancel. References [1] Burrage, K. Order conditions of stochastic Runge—Kutta methods by B -series.

Stochastic Manifolds for Nonlinear SPDEs I

This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor. CBMS-NSF Regional Conference Series in Applied Mathematics. Taylor Approximations for Stochastic Partial Differential Equations.

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