Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation

By A. Davie. 2007

232 citations recorded by the source corpus.

driving path x(t) is nondifferentiable, has recently been developed by Lyons. I develop an alternative approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. I also give some other examples showing that the main results are reasonably sharp.

Subfields: Finance

View Paper: https://www.semanticscholar.org/paper/d04570a818fafb1ec595dc9ac95c240ab341e31b

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Citations in the corpus

Pathwise stochastic integrals for model free finance (2013, background): “Here we show that Davie’s criterion implies our assumption (RIE).”
Convergence rates for the full Gaussian rough paths (2011, background): “[4], [9]) shows that the step-N Euler scheme(3) for an RDE driven by a 1/p-Hölder rough path with step size 1/k (i.”
A theory of regularity structures (2013, background): “2 In the particular case of functions of one variable, this point of view has been advocated by Gubinelli in [54,55] (and to some extent by Davie in [31]) as a way of interpreting Lyons’s theory of rough paths.”
Yet another introduction to rough paths (2009, methodology): “Davie proved in [Dav07] that there exists a unique solution of gi are of class C(2), and that the solution may not be unique of gi has only Hölder continuous derivatives.”
Euler estimates for rough differential equations (2006, methodology): “When the Rd-valued driving signal x and the vector fields are Lipschitz continuous then there exists a unique solution for every starting point y0. Building on recent work by A. M. Davie [ 7 ], we control the H¨older norm of y in terms of suited H¨older norms of x and its iterated integrals and also”
Existence and Uniqueness of the Solution of Stochastic Differential Equation Involving Wiener Process and Fractional Brownian Motion with Hurst Index H > 1/2 (2011, unknown): “Numerical methods for pathwise stochastic differential equations with fBm were considered in [11, 12, 2, 4].”
Stochastic Normalizing Flows (2020, methodology): “By instead taking a ﬁrst-order approximation we arrive at the Davie scheme (Davie, 2008) where X s,t represents the “integral” t s X r d X r .”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “(cid:0) (cid:1) Motivated by the discussion in [10] and [6], we can then consider the solution to the RDE over [ s, t ] by the solution of the ODE driven by x s,t ; the advantage in doing this is that (3.2) ensures that the ⌊ p ⌋ th -order Euler approximations to these two solutions agree, and…”
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.”
Controlled differential equations as Young integrals: A simple approach (2010, methodology): “In addition, we are able to deal with the case where f is not bounded, unlike the articles relying on the Picard iteration (however, a global existence result under similar conditions is stated in [7]).”
A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “8)) will essentially follow Davie’s approach to rough systems ([11]).”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, methodology): “Our presentation builds on [63] and combines Davie’s approach [33] with geometric ideas.”
Geometric versus non-geometric rough paths (2012, unknown): “…where we sum over all vectors a = (a 1 ,... ,a n ) ∈ { 1 ,... ,d } n and we use the shorthand for linear trees If we use the injection map ι : T ( R d ) → H deﬁned in Remark 2.7, then we have which coincides with the standard Davie solution [12,18], deﬁned in the case of a geometric rough path .”
Flows driven by rough paths (2012, methodology): “[5] Davie, A. M. Differential equations driven by rough paths: an approach via discrete approximation.”
A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion (2010, methodology): “To obtain a convergent numerical method Davie proposed in [9] a scheme of Milstein type.”