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Symmetries of Stochastic Differential Equations

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Symmetries of Integro-Differential Equations

Part of the book series: Lecture Notes in Physics ((LNP,volume 806))

Abstract

This chapter deals with applications of the group analysis method to stochastic differential equations. These equations are often obtained by including random fluctuations in differential equations, which have been deduced from phenomenological or physical view. In contrast to deterministic differential equations, only few attempts to apply group analysis to stochastic differential equations can be found in the literature. It is worth to note that this theory is still developing.

Before defining an admitted symmetry for stochastic differential equations an introduction into the theory of this type of equations is given. The introduction includes the discussion of a stochastic integration, a stochastic differential and a change of the variables (Itô formula) in stochastic differential equations. Applications of the Itô formula are considered in the next section which deals with the linearization problem. The Itô formula and the change of time in stochastic differential equations are the main tools of defining admitted transformations for them. After introducing an admitted Lie group and supporting material of the introduced definition, some examples of applications of the given definition are studied.

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Notes

  1. 1.

    A stochastic process depends on two variables X=X(t,ω); the second variable ω usually is omitted.

  2. 2.

    The proofs can be found, for example, in [2].

  3. 3.

    Summation over the repeated indices is assumed.

  4. 4.

    Notice that the function η(t,x,b) also depends on x and b.

  5. 5.

    The proof of the theorem is similar to [24] and can be found in [29].

  6. 6.

    Criteria for a linear parabolic equation to be equivalent to one of the Lie canonical equations can be found in [14, 16, 18].

  7. 7.

    Details and references can be found therein.

  8. 8.

    Considering h=h(t,x,b) this generalization can be extended including the Brownian motion B t into the transformation.

  9. 9.

    In [5, 13, 25] only particular cases were studied. In [33] one of sufficient conditions is missing. The present section complete this niche.

  10. 10.

    Unfortunately one of sufficient conditions is missing in [33].

  11. 11.

    Details and references one can find in [21].

  12. 12.

    These considerations are similar to the constructions applied in [29, 30]. It should be noted that there is no change of variables in the integrands as the authors of [8] misleadingly state.

  13. 13.

    The admitted Lie algebra of (5.5.20) is infinite dimensional.

  14. 14.

    Here signs and scaling these constants are chosen for further convenience.

  15. 15.

    In [8] the authors also tried to correct [29, 30]. Their attempt led to the strong restriction: all possible admitted transformations are fiber preserving.

  16. 16.

    Other terms of the expansion are cumbersome and not presented here.

  17. 17.

    In [8] the constant c 0 is mistakenly kept.

  18. 18.

    Using symbolic calculations on computer we checked equations \(\frac{\partial ^{k}S}{\partial a^{k}}(t,x,0)=0\) (k≤7). It is likely that this identity holds for all large k.

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Authors and Affiliations

  1. Inst. Computational Technologies, Russian Academy of Sciences, Pr. Lavrentjeva 6, 630090, Novosibirsk, Russia

    Prof. Yurii N. Grigoriev

  2. Dept. Mathematics & Science, Blekinge Institute of Technology, 371 79, Karlskrona, Sweden

    Prof. Nail H. Ibragimov

  3. Inst. Mathematical Modelling, Russian Academy of Sciences, Miusskaya Square 4A, 125047, Moscow, Russia

    Dr. Vladimir F. Kovalev

  4. School of Mathematics, Institute of Science, Suranaree University of Technology (SUT), 30000, Nakhon Ratchasima, Thailand

    Sergey V. Meleshko

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  1. Prof. Yurii N. Grigoriev
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  2. Prof. Nail H. Ibragimov
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  3. Dr. Vladimir F. Kovalev
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  4. Sergey V. Meleshko
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Correspondence to Yurii N. Grigoriev .

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Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V. (2010). Symmetries of Stochastic Differential Equations. In: Symmetries of Integro-Differential Equations. Lecture Notes in Physics, vol 806. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3797-8_5

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