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On the Effects of Different Interpretations of Stochastic Differential Equations

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DOI: 10.4236/am.2019.1011063    226 Downloads   509 Views
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ABSTRACT

This paper addresses the problem of the interpretation of the stochastic differential equations (SDE). Even if from a theoretical point of view, there are infinite ways of interpreting them, in practice only Stratonovich’s and Itô’s interpretations and the kinetic form are important. Restricting the attention to the first two, they give rise to two different Fokker-Planck-Kolmogorov equations for the transition probability density function (PDF) of the solution. According to Stratonovich’s interpretation, there is one more term in the drift, which is not present in the physical equation, the so-called spurious drift. This term is not present in Itô’s interpretation so that the transition PDF’s of the two interpretations are different. Several examples are shown in which the two solutions are strongly different. Thus, caution is needed when a physical phenomenon is modelled by a SDE. However, the meaning of the spurious drift remains unclear.

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Floris​, C. (2019) On the Effects of Different Interpretations of Stochastic Differential Equations. Applied Mathematics, 10, 876-891. doi: 10.4236/am.2019.1011063.

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