A weak Local Linearization scheme for stochastic differential equations with multiplicative noise

Jiménez, Juan Carlos; Mora, Carlos M.; Selva, M.

doi:10.1016/j.cam.2016.09.013

Cited by 9 publications

(2 citation statements)

References 22 publications

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“…Using the local linearization method, [14] introduced an exponential scheme for (1.1) with d = 1. The article [50] develops an integrator of Euler-exponential type for multidimensional SDEs with multiplicative noise (see also, e.g., [22,35,37,59,60]), and [32] provides a numerical method based on the computation of the conditional mean and the square root of the conditional covariance matrix of a local linearization approximation to (1.1). Schemes adapted to specific SDEs are given, for instance, in [10,13,15,50].…”

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confidence: 99%

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

Mora¹,

Mardones²,

Jiménez³

et al. 2017

SIAM J. Numer. Anal.

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We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution Xt by integrating the system of coupled SDEs that describes the evolution of the norm of Xt and its projection on the unit sphere. This allows us to develop an explicit scheme for stiff SDEs with multiplicative noise that shows a solid performance in various numerical experiments. Under general conditions, the new integrator preserves the almost sure stability of the solutions for any step-size, as well as the property of being distant from 0. The scheme also has linear rate of weak convergence for a general class of SDEs with locally Lipschitz coefficients, and one-half strong order of convergence.

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confidence: 99%

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

Mora¹,

Mardones²,

Jiménez³

et al. 2017

SIAM J. Numer. Anal.

Self Cite

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“…In [30,44,50] the starting step-size is given by the user. In the framework of the continuous-discrete estimation problem of the filtering theory, [24] develops an adaptive filter of minimum variance that uses, between consecutive observation times, the weak local linearization scheme given in [25] (see also [9]), together with an adaptive strategy controlling the predictions for the first two conditional moments of the continuous state equation that does not involve sampling random variables.…”

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Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

Mora¹,

Jiménez²,

Selva³

2020

Preprint

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We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the matching between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce three variable step-size Euler schemes derived from three different discrepancy functions, as well as four variable step-size higher order weak schemes are proposed. Third, to compute the expectation of functionals of diffusion processes a general procedure for designing adaptive schemes with variable step-size and sample-size is presented, which combines a conventional Monte Carlo technique for estimating the total number of simulations with the new variable step-size weak schemes. Finally, a variety of numerical simulations are presented to show the potential of the introduced variable stepsize strategies and adaptive schemes to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.

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Pathwise methods for the integration of a stochastic SVIR model

Muñoz,

de la Cruz,

Mora

2023

Math Methods in App Sciences

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We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed.

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A weak Local Linearization scheme for stochastic differential equations with multiplicative noise

Cited by 9 publications

References 22 publications

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

Pathwise methods for the integration of a stochastic SVIR model

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