A note on symmetries of diffusions within a martingale problem approach

Abstract

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second order geometry and Itô integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.2 Preliminaries: second order geometry and Itô integration on manifoldIn this section, also in order to fix notations, we briefly recall some basic facts about second order geometry and Itô integration on manifolds. The interested reader is referred to [5,16,19] for proofs and further details. Second order geometryGiven a smooth manifold M , we denote by C ∞ (M ) the set of real-valued smooth functions defined on M . If F is a bundle with base manifold B, we denote by S(F ) the set of smooth sections of F . Finally, if M ′ is a manifold and n ∈ N, we denote by J n (M, M ′ ) the bundle of n jets of n times differentiable functions defined on M and taking values in M ′ . Let M be a smooth manifold and u be a global coordinate defined on R. The subset u −1 (0) ⊂ J 2 (M, R) is a submanifold of J 2 (M, R) and actually a vector subbundle of J 2 (M, R). Definition 2.1 The submanifold u −1 (0) ⊂ J 2 (M, R) is called the bundle of codiffusors of the manifold M and is denoted by τ * M .

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

“…Comparing Eqs (43), (44) one the one hand, and ( 69), (70) on the other, we immediately observe a rather trivial (but useful) relation: if both f i and σ i k do not actually depend on the x variables (but possibly depend on t) and set R = 0, the symmetry coefficients ϕ i are also invariants for the Ito equation. (Note that such relation does not exist for R = 0, i.e., for proper W-symmetries [30].…”

“…We will consider in particular the approach which parallels the usual treatment of deterministic equations in the stochastic case (to which we gave several contributions in recent years); for a discussion of other approaches, including earlier attempts, see e.g., the review paper [22]. See also [66,67] for an approach relating symmetries to Girsanov theory, and more generally [68,69,70,71,72] for a related concurrent approach.…”

Symmetry of the isotropic Ornstein-Uhlenbeck process in a force field

“…Symmetry of Stochastic Differential Equations (SDEs) has been studied only in relatively recent years [21][22][23][24][25][26][27][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. By now, a sound formulation of the theory is available, and we know how to use symmetries to integrate -or reduce the order of -SDEs [29][30][31][32][33][34][35][36][37][38].…”

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.…”