Random walks and Lévy processes as rough paths

Abstract

We consider random walks and Lévy processes in a homogeneous group G. For all p > 0, we completely characterise (almost) all G-valued Lévy processes whose sample paths have finite p-variation, and give sufficient conditions under which a sequence of G-valued random walks converges in law to a Lévy process in p-variation topology. In the case that G is the free nilpotent Lie group over R d , so that processes of finite p-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes.

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

“…The main result of [9] gives sufficient conditions for convergence in distribution of a random walk on (R ) to a Lévy process on (R ) in a suitable rough path topology. While [5] and [9] are also generalizations of theorem 3 from [8], they do not apply to the class of processes described in our theorem 1.1, as the increments of these processes are not necessarily i.i.d. Moreover, none of these results is concerned by the study of the area anomaly, as we will see that it is trivial when the discrete process has i.i.d.…”

Optimal Execution with Rough Path Signatures

“…Lévy processes. More generally, certain Lévy processes can also be lifted into pgeometric rough paths [FS17,Che18] and they are thus included in this framework.…”

Differential equations driven by rough paths with jumps

“…d(X n τn , X n τn+hn ) → 0 in probability, for any sequence of bounded F n -stopping times τ n and any sequence of positive numbers h n (see Theorem 4.8.2 in [28]). For example, one has the following criterion according to [3]. Proposition 6.16.…”

“…Proposition 6.16. (Corollary 4.9 in [3]) Suppose (X n ) n≥1 a sequence of cádlág strong Markov processes defined as above. Assume that (1) for any rational h ∈ [0, T ], (X n h ) n is tight.…”

“…Concerning the history of rough differential equations with jumps: the Young case p ∈ [1,2) was already studied in [44]. Rough integration against càdlàg rough path, p ∈ [2, 3), was introduced in [16], as was the notion of càdlàg geometric p-rough path, any p < ∞. In the latter setting, limit theorems and applicability to semimartingales (canonical solutions in the sense of Marcus) was obtained in [2].…”