Canonical RDEs and general semimartingales as rough paths
By Ilya Chevyrev, Peter K. Friz. Annals of Probability, 2017
53 citations recorded by the source corpus.
In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of L\'epingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.
Subfields: Finance, Management Science and Operations Research
View Paper: https://www.semanticscholar.org/paper/12c6956a43825d59ae7189b757958b017c7a5777
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In Finance
- Huilin Zhang
- Peter K. Friz
- Nicolas Victoir
- T. Cass
- Martin Hairer
- Terry Lyons
- Samy Tindel
- Archil Gulisashvili
In Management Science and Operations Research
Citations in the corpus
- Differential equations driven by rough paths with jumps (2017, methodology): “For p ∈ [2, 3) this just means that the state space can be identified with R ⊕ (Rd)⊗2, dropping the well-known geometricity (a.”