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Register a new userSymmetries of Stochastic Differential Equations using Girsanov transformations
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Added by
Francesco Carlo De Vecchi
Publication date
2019
fields
Physics
and research's language is
English
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Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models.
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Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs.
A reduction procedure for stochastic differential equations based on stochastic symmetries including Girsanov random transformations is proposed. In this setting, a new notion of reconstruction is given, involving the expectation values of functionals of solution to the SDE and a reconstruction theorem for general stochastic symmetries is proved. Moreover, the notable case of reduction under the closed subclass of quasi Doob transformations is presented. The theoretical results are applied to stochastic models relevant in the applications.
We obtain a dimensional reduction result for the law of a class of stochastic differential equations using a supersymmetric representation first introduced by Parisi and Sourlas.
A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.
We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by systems from each of the above classes and subclasses. We also show how equivalence transformations and Lie symmetries can be used for reduction of order of such systems and their integration. As an illustrative example of using the theory developed, we solve the complete group classification problems for all these classes in the case of two dependent variables.
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