Abril 2015 vol. 1 num. 1 - Congresso Nacional de Matemática Aplicada à Indústria

Artigo Completo - Open Access.

# A higher order and stable method for the numerical integration of Random Differential Equations

# A higher order and stable method for the numerical integration of Random Differential Equations

### Cruz, H. de la ; Jimenez, J.C. ;

##### Artigo Completo:

Over the last few years there has been a growing and renovated interest in the numerical study of Random Differential Equations (RDEs). On one hand it is motivated by the fact that RDEs have played an important role in the modeling of physical, biological, neurological and engineering phenomena, and on the other hand motivated by the usefulness of RDEs for the numerical analysis of Ito-stochastic differential equations (SDEs) -via the extant conjugacy property between RDEs and SDEs-, which allows to study stronger pathwise properties of SDEs driven by different kind of noises others than the Brownian. Since in most common cases no explicit solution of the equations is known, the construction of computational methods for the treatment and simulation of RDEs has become an important need. In this direction the Local Linearization (LL) approach is a successful technique that has been applied for defining numerical integrators for RDEs. However, a major drawback of the obtained methods is its relative low order of convergence; in fact it is only twice the order of the moduli of continuity of the driven stochastic process. The present work overcomes this limitation by introducing a new, exponential-based, high order and stable numerical integrator for RDEs. For this, a suitable approximation of the stochastic processes present in the random equation, together with the local linearization technique and an adapted Pade´ method with scaling and squaring strategy are conveniently combined. In this way a higher order of convergence can be achieved (independent of the moduli of continuity of the stochastic processes) while retaining the dynamical and numerical stability properties of the low order LL methods. Results on the convergence and stability of the suggested method and details on its efficient implementation are discussed. The performance of the introduced method is illustrated through computer simulations.

##### Artigo Completo:

Over the last few years there has been a growing and renovated interest in the numerical study of Random Differential Equations (RDEs). On one hand it is motivated by the fact that RDEs have played an important role in the modeling of physical, biological, neurological and engineering phenomena, and on the other hand motivated by the usefulness of RDEs for the numerical analysis of Ito-stochastic differential equations (SDEs) -via the extant conjugacy property between RDEs and SDEs-, which allows to study stronger pathwise properties of SDEs driven by different kind of noises others than the Brownian. Since in most common cases no explicit solution of the equations is known, the construction of computational methods for the treatment and simulation of RDEs has become an important need. In this direction the Local Linearization (LL) approach is a successful technique that has been applied for defining numerical integrators for RDEs. However, a major drawback of the obtained methods is its relative low order of convergence; in fact it is only twice the order of the moduli of continuity of the driven stochastic process. The present work overcomes this limitation by introducing a new, exponential-based, high order and stable numerical integrator for RDEs. For this, a suitable approximation of the stochastic processes present in the random equation, together with the local linearization technique and an adapted Pade´ method with scaling and squaring strategy are conveniently combined. In this way a higher order of convergence can be achieved (independent of the moduli of continuity of the stochastic processes) while retaining the dynamical and numerical stability properties of the low order LL methods. Results on the convergence and stability of the suggested method and details on its efficient implementation are discussed. The performance of the introduced method is illustrated through computer simulations.

**Palavras-chave:**
random differential equations, numerical integrators, exponential methods, Local linearization methods,
random differential equations, numerical integrators, exponential methods, Local linearization methods,

**Palavras-chave:**
,

**DOI:** 10.5151/mathpro-cnmai-0023

##### Referências bibliográficas

- [1] Arnold, L., Random Dynamical Systems, Springer-Verlag, Heidelberg, 1998.

- [2] Bharucha-Reid, A.T., Approximate Solution of Random Equations, North- Holland, New York and Oxford, 1979.

- [3] Carbonell F., Jimenez J.C, Biscay R.J, de la Cruz H., The Local Linearization method for numerical integration of random differential equations, BIT Num. Math, 45 (2005), 1-14.

- [4] De la Cruz H., Biscay R.J., Jimenez J.C, Carbonell F., Ozaki T., High Order Local Linearization methods: an approach for

- [5] constructing A-stable high order explicit schemes for stochastic differential equations with additive noise, BIT Num. Math, 50 (2010), 509-539.

- [6] [5] Golub G. H. and Van Loan C. F., Matrix Computations, The Johns Hopkins University Press, 2nd Edition, 1989.

- [7] [6] Grune, L. and Kloeden, P. E., Pathwise approximation of random ordinary differential equation, BIT, 41 (2001) 711–721.

- [8] [7] Hasminskii, R. Z., Stochastic Stability of Differential Equations, Sijthoff Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.

- [9] [8] Jentzen, A. , Kloeden P., Pathwise Taylor schemes for random ordinary differential equations. BIT Numer. Math., 49 (2009)

- [10] 113-140.

- [11] [9] Kloeden, P. E. and E. Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.

- [12] [10] Sobczyk, K., Stochastic differential equations with applications to Physics and Engineering, Kluwer, Dordrecht, 1991.

- [13] [11] Soong, T. T., Random Differential Equations in Science and Engineering, Academic Press, New York, 1974. 16 (1979) 1019- 1035.

- [14] [12] Vom Scheidt, J., Starkloff, H. J. and Wunderlich, R., Random transverse vibrations of a one-sided fixed beam and model reduction,

- [15] ZAMM Z. Angew. Math. Mech., 82 (2002) 847-859.

- [16] [13] Sidje, R. B., EXPOKIT: software package for computing matrix exponentials, AMC Trans. Math. Software, 24 (1998) 130-156.

- [17] [14] Tiwari, J. L. and Hobbie, J. E., Random differential equations as models of ecosystems: Monte Carlo simulation approach, Math.

- [18] Biosci., 28 (1976) 25-44.

- [19] [15] Tsokos, C. P. and Padgett, W. J., Random Integral Equations with Applications in Sciences and Engineering, Academic Press, New York, (1974).

- [20] [16] Wonham, W.M., Random Differential Equations in Control Theory, In: Probabilistic Methods in Applied Mathematics, Vol. 2,

- [21] A.T. Bharucha-Reid (Ed.), Academic Press, N.Y., (1971), 31-212.

##### Como citar:

Cruz, H. de la; Jimenez, J.C.; "A higher order and stable method for the numerical integration of Random Differential Equations", p. 103-110 . In: **Anais do Congresso Nacional de Matemática Aplicada à Indústria [= Blucher Mathematical Proceedings, v.1, n.1]**.
São Paulo: Blucher,
2015.

ISSN em b-reve,
DOI 10.5151/mathpro-cnmai-0023

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