Semi-martingales and rough paths theory

By Laure Coutin, Antoine Lejay. 2005

52 citations recorded by the source corpus.

We prove that the theory of rough paths, which is used to define path-wise integrals and path-wise differential equations, can be used with continuous semi-martingales. We provide then an almost sure theorem of type Wong-Zakai. Moreover, we show that the conditions UT and UCV, used to prove that one can interchange limits and Ito or Stratonovich integrals, provide the same result when one uses the rough paths theory.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0c121293aa0ef7e41884fa0f9c007e088fff6ad7

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Citations in the corpus

Rough path analysis via fractional calculus (2006, methodology): “We refer, for instance, to [1], [3], [4] and [7] for recent applications of the rough path analysis to the stochastic calculus with respect to the Wiener process.”
Discretely sampled signals and the rough Hoff process (2013, background): “2) is the Itô integral not the Stratonovich integral as might be expected from the classical Wong-Zakai Theorem (see the original paper [30] by Wong and Zakai, or the rough path versions in [5, 27]).”
Differential equations driven by rough paths with jumps (2017, methodology): “For the sake of completeness, we note that continuous semimartingales as continuous rough path were studied in [4, 14, 12].”
Pathwise stochastic integrals for model free finance (2013, background): “While this seems intuitively clear, the only other proof that we know of is somewhat involved: it relies on a strong version of the Burkholder–Davis–Gundy inequality, a time change, and Kolmogorov’s continuity criterion; see [6] or Chapter 14 of [20].”
Yet another introduction to rough paths (2009, methodology): “Note that however, especially when dealing with stochastic processes, some results in the rough paths theory are dependent from the choice of a dyadic partition (see for example [CL05]).”
Canonical RDEs and general semimartingales as rough paths (2017, background): “2 for the definition of UCV, and also [12] for some links to continuous semimartingale rough paths.”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “Rough path regularity of enhanced martingales and certain convergence results were first established in [27].”