Euler estimates for rough differential equations

By Peter K. Friz, Nicolas Victoir. 2006

74 citations recorded by the source corpus.

Abstract We consider controlled ordinary differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A.M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. Such Euler estimates have powerful applications. By a simple limit argument they apply to rough path differential equations (RDEs) in the sense of T. Lyons and hence also to stochastic differential equations driven by Brownian motion or other random rough paths with sufficient integrability. In the context of the latter, we obtain strong remainder estimates in stochastic Taylor expansions a la Azencott, Ben Arous, Castell and Platen. Although our findings appear novel even in the case of driving Brownian motion our main insight is the genuine rough path nature of (quantitative) remainder estimates in stochastic Taylor expansions. There are several other applications of which we discuss in detail L q -convergence in Lyons' Universal Limit Theorem and moment control of RDE solutions.

Subfields: Computational Mechanics, Demography, Finance

View Paper: https://www.semanticscholar.org/paper/0ff0d5666997330d238eec672bd5deb2bc06faaa

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In Computational Mechanics

1. Ismaël F. Bailleul
2. Peter K. Friz
3. Huilin Zhang
4. Nicolas Victoir

In Demography

1. Ismaël F. Bailleul
2. Peter K. Friz
3. Huilin Zhang
4. Nicolas Victoir

In Finance

1. Peter K. Friz
2. Ismaël F. Bailleul
3. Nicolas Victoir
4. Huilin Zhang
5. T. Cass
6. Martin Hairer
7. Terry Lyons
8. Samy Tindel

Citations in the corpus

• Differential equations driven by rough paths with jumps (2017, methodology): “As a first application, we discuss discrete-time approximations (“higher-order Euler schemes”) to rough differential equations first discussed by Davie [6] in the level-2 setting and then [15] (see also [12]) in case of geometric prough paths, always continuous.”
• Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, methodology): “2 improves on earlier results by the authors obtained in [63].”
• Flows driven by rough paths (2012, methodology): “It is a nice feature of our approach that Taylor expansion of solution flows to rough differential equations come almost for free, providing an alternative and simple proof of similar results due to Friz and Victoir [12].”
• The Signature of a Rough Path: Uniqueness (2014, unknown): “Victoir [25] via discrete approximation.”