Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Abstract

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general càdlàg semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries of SDEs, the new concepts of gauge and time symmetries for semimartingales on Lie groups are introduced. Markovian and non-Markovian examples of gauge and time symmetric processes are provided. The considered set of SDEs includes affine and Marcus type SDEs as well as smooth SDEs driven by Lévy processes. Non trivial invariance results concerning a class of iterated random maps are obtained as special cases.

Symmetry and integrability for stochastic differential equations

“…Let us consider an (Ito) stochastic differential equation (SDE) [4,9,10,16,17,24] dx i = f i (x, t) dt + σ i j (x, t) dw j ; (1) in (1) and below, t ∈ R, x ∈ R n (the x i can be thought as local coordinates on a smooth manifold, as we take care of distinguishing covariant and contravariant indices albeit never introducing explicitly a metric), f and σ are smooth vector and matrix functions of their arguments, and w j (j = 1, ..., m) are independent standard Wiener processes. For Ito equations, a geometrical interpretation of the equation is missing, and one is forced to resort to a purely algebraic notion of symmetry.…”

“…In order to have σ w = 0 we must require β ww = 1, i.e. β(t, w) = b(t) + [b (1) (t)]w + (1/2)w 2 ; with this, requiring f w = 0 enforces b (1)…”

Symmetries of stochastic differential equations using Girsanov transformations

“…where Ξ (2) denotes the second order prolongation of the vector field Ξ. If we rewrite (4.3) for a vector field of the form…”

“…Without claiming to be exhaustive, from the first studies on Brownian-motion-driven SDEs (see [7,3,24]) and Markov processes (see [26,27,36,8]) many different notions of symmetries for Brownian-motiondriven SDEs has been proposed (see the symmetries with random time change [44,42,20], W-symmetry in [21,23], weak symmetries in [15], random symmetries in [22,25,32] and symmetries of diffusions in [16]). Moreover, random transformations of semimartingales on Lie groups has been introduced (see [1]) and the notion of symmetry has been extended also to SDEs driven by general semimartingales (see [2,35]). Furthermore the idea of reduction and reconstruction of differential equations admitting a solvable Lie algebra of symmetries has been generalized from the deterministic to the stochastic setting (see [35,30] for strong symmetries and [14] for weak symmetries).…”

“…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 [34,39,43]). 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.…”