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Large deviation principle for stochastic differential equations driven by stochastic integrals

Ryoji Takano

TL;DR

This work establishes a large deviation framework for one-dimensional SDEs driven by stochastic integrals using a minimally invasive rough-path approach. It introduces the $α$-Uniformly Exponentially Tightness concept and proves an LDP for stochastic integrals on Hölder spaces, then transfers this to SDEs via rough-path lifts, yielding a comprehensive LDP with rate functions expressed through Cameron–Martin-type controls. The results cover small-noise and short-time regimes for rough volatility models (notably rough SABR-type dynamics) and enable asymptotic formulas for implied volatility. Practically, the findings provide rigorous, general asymptotics for option pricing and implied volatility in rough volatility settings, with a unified treatment across short-time horizons and model-generality in the volatility function and kernel operators.

Abstract

In this paper, we prove the large deviation principle (LDP) for stochastic differential equations driven by stochastic integrals in one dimension. The result can be proved with a minimal use of rough path theory, and this implies the LDP for many class of rough volatility models, and it characterizes the asymptotic behavior of implied volatility. First, we introduce a new concept called $α$-Uniformly Exponentially Tightness, and prove the LDP for stochastic integrals on Hölder spaces. Second, we apply this type of LDP to deduce the LDP for stochastic differential equations driven by stochastic integrals in one dimension. Finally, we derive the asymptotic behavior of the implied volatility as an application of main results.

Large deviation principle for stochastic differential equations driven by stochastic integrals

TL;DR

This work establishes a large deviation framework for one-dimensional SDEs driven by stochastic integrals using a minimally invasive rough-path approach. It introduces the -Uniformly Exponentially Tightness concept and proves an LDP for stochastic integrals on Hölder spaces, then transfers this to SDEs via rough-path lifts, yielding a comprehensive LDP with rate functions expressed through Cameron–Martin-type controls. The results cover small-noise and short-time regimes for rough volatility models (notably rough SABR-type dynamics) and enable asymptotic formulas for implied volatility. Practically, the findings provide rigorous, general asymptotics for option pricing and implied volatility in rough volatility settings, with a unified treatment across short-time horizons and model-generality in the volatility function and kernel operators.

Abstract

In this paper, we prove the large deviation principle (LDP) for stochastic differential equations driven by stochastic integrals in one dimension. The result can be proved with a minimal use of rough path theory, and this implies the LDP for many class of rough volatility models, and it characterizes the asymptotic behavior of implied volatility. First, we introduce a new concept called -Uniformly Exponentially Tightness, and prove the LDP for stochastic integrals on Hölder spaces. Second, we apply this type of LDP to deduce the LDP for stochastic differential equations driven by stochastic integrals in one dimension. Finally, we derive the asymptotic behavior of the implied volatility as an application of main results.
Paper Structure (14 sections, 15 theorems, 147 equations, 1 table)

This paper contains 14 sections, 15 theorems, 147 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Large deviation principle for stochastic integrals
  4. Large deviation principle for stochastic differential equations driven by stochastic integrals in one dimension
  5. An application for mathematical finance
  6. Small noise asymptotics for rough volatility models \ref{['rSABR']}
  7. Short time asymptotics for rough volatility models \ref{['rSABR']}
  8. Short time asymptotics for put/call options and implied volatility for rough volatility models \ref{['rSABR']}
  9. Proof of main theorems
  10. Proof of Theorem \ref{['main1']} and \ref{['main2']}
  11. Proof of Theorem \ref{['main4']} and \ref{['main5']}
  12. Proof of Theorem \ref{['short']}
  13. Some properties for $\mathcal{K}^\epsilon$
  14. Proof of Theorem \ref{['putcall']}

Key Result

Lemma 2.3

Let $\{X^{\epsilon} \}_{\epsilon > 0}$ be a uniformly exponentially tight family of cadlag adapted semi-martingales on $\mathbb{R}$ and $\{ A^{\epsilon} \}_{\epsilon > 0}$ be a family of real valued cadlag adapted processes. Assume that $\{ (A^{\epsilon},X^{\epsilon}) \}_{\epsilon > 0}$ satisfies th where $\textbf{BV}$ is the set of bounded variation and $a\cdot x$ means the Riemann-Stieltjes inte

Theorems & Definitions (49)

  • Definition 2.1: Definition 1.1 Ga08
  • Definition 2.2: Section 1.2 DeZe
  • Lemma 2.3: Theorem 1.2 Ga08
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Theorem 2.9
  • proof
  • ...and 39 more