A symmetry-adapted numerical scheme for SDEs

Abstract

We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry-adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the symmetry properties of the equation is introduced. Our algorithmic procedure is applied to the family of general linear SDEs for which two theoretical estimates of the numerical forward error are established.

Symmetries of stochastic differential equations using Girsanov transformations

“…L(k) = 0, is the same in both the sets of determining equations and, since µ i do not depend on u, Ξ(µ i ) = Y(µ i ) and equations ( 26) and ( 27) imply (15). Moreover, multiplying on the left (29) by σ, we immediately get (17). In order to prove that ( 16) holds, we consider the right multiplication of ( 28) by σ T and we get, using σ…”

“…Symmetries of SDEs arising from variational problems has been considered and Noether theorem and integration by quadratures have been generalized to variational stochastic problems (see [34,41,46]). Finally, a generalization of differential constraint method to SPDEs has been proposed in [12,13], while in [2,17] symmetries have been applied to the study of invariant numerical methods for SDEs.…”

Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

“…In contact geometry, a contact structure on a (2n + 1)-dimensional manifold M is a 1-form ζ, which is maximally non-integrable, namely, ζ ∧ (dζ) n = 0, and the contact transformations are the diffeomorphisms [58,59] for an introduction to the subject and [60] for an historical overview). This definition is satisfied by J 1 (M, R) with the 1-form ζ = κ defined in (15).…”

“…The development of a Lie symmetry analysis for stochastic differential equations (SDEs) and general random systems is relatively recent (see, e.g., [10][11][12][13][14][15][16][17][18][19][20] for some recent developments in the non-variational case). For stochastic systems arising from a variational framework, it is certainly interesting to study the relation between their symmetries and functionals that are conserved by their flow, and, in particular, to establish stochastic generalizations of the Noether theorem.…”

Noether theorem in stochastic optimal control problems via contact symmetries

“…The development of a Lie symmetry analysis for stochastic differential equations (SDEs) and general random systems is relatively recent (see, e.g., [1,2,17,18,19,20,26,25,27,33,38] for some recent developments in the non-variational case). For stochastic systems arising from a variational framework, it is certainly interesting to study the relation between their symmetries and functionals which are conserved by their flow, and, in particular, to establish stochastic generalizations of Noether theorem.…”