W-symmetries of Ito stochastic differential equations

Abstract

We discuss W-symmetries of Ito stochastic differential equations, introduced in a recent paper by Gaeta and Spadaro [J. Math. Phys. 2017]. In particular, we discuss the general form of acceptable generators for continuous (Lie-point) W-symmetry, arguing they are related to the (linear) conformal group, and how W-symmetries can be used in the integration of Ito stochastic equations along Kozlov theory for standard (deterministic or random) symmetries. It turns out this requires, in general, to consider more general classes of stochastic equations than just Ito ones. *

Symmetries of stochastic differential equations using Girsanov transformations

“…In this setting, a natural problem is the comparison between the set of Lie point symmetries of Kolmogorov (or Fokker-Planck) equation and the set of symmetries of the corresponding SDE (for the importance of Lie point symmetries of Kolmogorov equation in integrating the associated diffusion process see [11,9,10,28,29]). In the case of Brownian-motion-driven SDEs all the different notions of symmetry introduced for SDEs (strong symmetries, W symmetries, weak symmetries) have been proved to be also symmetries of the corresponding Kolmogorov equation, but the converse is false (see [16,21,24,31,33]). In this paper we provide an appropriate extension of the notion of symmetry of an SDE so that the converse of the previous result holds.…”

Integration of the stochastic logistic equation via symmetry analysis

“…The basic result for the use of standard symmetries was provided by Kozlov [22][23][24]. Here we quote it from [12], see Proposition 3 in there. and hence is readily integrated as…”

“…Copyright: the authors Random symmetries were introduced in [17] and studied in a number of other papers [13,14,25,26]; W-symmetries were introduced in [17] and further studied in [12].…”

“…These are discussed in the general case in the literature, see e.g. [11,12,17]; here we will just consider the case of interest here, i.e. that of scalar equations.…”

“…As already mentioned, we refer e.g. to [11,12,17] for details on the derivation of these determining equations and for their version in arbitrary dimension.…”