[Paper Review] Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths

Qi Feng, Xuejing Zhang|arXiv (Cornell University)|Sep 20, 2012

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper establishes Taylor expansions and Castell-type tail estimates for solutions of stochastic differential equations (SDEs) driven by rough paths, including fractional Brownian motion with Hurst parameter $ H > 1/4 $ and continuous centered Gaussian processes with finite 2D $ \rho $-variation. It proves convergence of the Taylor series under analyticity of vector fields and derives exponential decay estimates for remainder terms using pathwise rough path techniques and iterated integral bounds.

ABSTRACT

We study the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p>2$. We prove a general theorem concerning the convergence of the Taylor expansion on a nonempty interval provided that the vector fields are analytic on a ball centered at the initial point. We also derive criteria that enable us to study the rate of convergence of the Taylor expansion. Finally and this is also the main and the most original part of this paper, we prove Castell expansions and tail estimates with exponential decays for the remainder terms of the solutions of the stochastic differential equations driven by continuous centered Gaussian process with finite $2D~ρ-$variation and fractional Brownian motion with Hurst parameter $H>1/4$.

Motivation & Objective

• To extend stochastic Taylor expansion convergence results to SDEs driven by $ p $-rough paths with $ p > 2 $, generalizing prior work on Brownian motion and fractional Brownian motion.
• To establish quantitative criteria for the convergence radius of the Taylor expansion based on iterated integral estimates and vector field regularity.
• To derive Castell-type expansions and exponential tail estimates for remainder terms in SDEs driven by continuous centered Gaussian processes with finite 2D $ \rho $-variation and fractional Brownian motion with $ H > 1/4 $.
• To provide a pathwise deterministic framework for remainder term estimation using rough path theory, avoiding reliance on probabilistic bounds in the main derivation.
• To verify and extend previous claims—particularly in [7]—by proving exponential tail decay for the remainder in SDEs driven by fBm with $ H > 1/2 $.

Proposed method

• Uses the rough path framework to define iterated integrals $ \int_{\triangle^k[0,t]} dx^I $ for $ p $-rough paths with $ p > 2 $, enabling the construction of Taylor series for solutions of rough differential equations.
• Applies a general convergence theorem for Taylor expansions under the assumption that vector fields are analytic on a ball around the initial point, ensuring local convergence on a non-empty interval.
• Employs estimates from [5, Theorem 7.16] and [23] on the $ p $-variation norm of iterated integrals to derive quantitative bounds on the convergence radius and rate.
• Adapts the Azencott–Castell approach by introducing approximating sequences for fractional Brownian motion and Gaussian processes with finite 2D $ \rho $-variation.
• Uses the $ \omega(\alpha, c, \zeta) $ family of processes to control the growth of coefficients in the Taylor expansion and remainder terms, applying inductive estimates and boundedness arguments.
• Applies pathwise estimates and recurrence relations to show that remainder terms $ r_{N+1} $ and associated processes $ M_{N+1} $ belong to the $ \omega(\alpha, c, \zeta) $ class, enabling tail probability bounds with exponential decay.

Experimental results

Research questions

• RQ1 Under what conditions does the Taylor expansion of the solution to a rough differential equation converge on a non-empty time interval?
• RQ2 How can the rate of convergence of the Taylor expansion be quantitatively characterized in terms of iterated integral bounds and vector field regularity?
• RQ3 Can Castell-type expansions and exponential tail estimates be established for SDEs driven by rough paths with $ H > 1/4 $, including fractional Brownian motion and general Gaussian processes?
• RQ4 What is the role of analyticity of vector fields in ensuring the convergence and decay properties of the Taylor series remainder?
• RQ5 How do the pathwise estimates and $ \omega(\alpha, c, \zeta) $-type bounds contribute to proving exponential tail decay for remainder terms in SDEs driven by rough paths?

Key findings

• The Taylor expansion of the solution to a rough differential equation driven by $ p $-rough paths with $ p > 2 $ converges on a non-empty interval when the vector fields are analytic on a neighborhood of the initial point.
• The convergence radius of the Taylor series is quantitatively characterized via bounds on iterated integrals, enabling explicit estimation of the interval of convergence.
• For SDEs driven by fractional Brownian motion with $ H > 1/4 $, the paper proves a Castell expansion with exponential tail decay for the remainder term, extending prior results to the full range $ H > 1/4 $.
• The remainder terms of the solution are shown to satisfy $ \mathbb{P}(\sup_{t \in [0,\tau]} \|P_{N+1}(\varepsilon,t)\| \geq \xi) \leq c e^{-\alpha \xi} $ for some $ \alpha, c > 0 $, indicating exponential tail decay.
• The method confirms the claim in [7] regarding exponential tail decay for $ H > 1/2 $, now extended to $ H > 1/4 $, using a deterministic rough path approach.
• The use of the $ \omega(\alpha, c, \zeta) $ family of processes allows inductive control of coefficient growth, ensuring that remainder terms and associated processes remain bounded with exponential tail behavior.