Stochastic Taylor expansions and heat kernel asymptotics

Baudoin, Fabrice

doi:10.1051/ps/2011107

Cited by 9 publications

(10 citation statements)

References 42 publications

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“…As a consequence, we obtain the following proposition which may be proved as in [6] (or [19]). This proposition may be used to understand the geometric meaning of the coefficients a k (x 0 ) of the small-time asymptotics p(t; x, x) = 1 t Hd a 0 (x) + a 1 (x)t 2H + · · · + a n (x)t 2nH + o(t 2nH ) .…”

Section: 3mentioning

confidence: 84%

Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

Baudoin

Ouyang

2011

Stochastic Processes and their Applications

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The goal of this paper is to show that under some assumptions, for a d-dimensional fractional Brownian motion with Hurst parameter H > 1/2, the density of the solution of the stochastic differential equationUnder the framework of this present work, the Laplace method can be obtained in general hypoelliptic case and without imposing the structure equations on vector fields in Theorem 1.1. These two assumptions are imposed to obtain the correct Riemannian distance in the kernel expansion.Remark 1.3. When H > 1/2, to obtain a short-time asymptotic formula for the density of solution to equation (1.1) but with drift, one need to work on a version of Laplace method with fractional powers of ε, which will be very heavy and tedious in computation.Remark 1.4. When the present work was almost completed, we noticed that a proof for the Laplace method for stochastic differential equation driven by fractional Brownian motion with Hurst parameter 1/3 < H <

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Section: 3mentioning

confidence: 84%

Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

Baudoin

Ouyang

2011

Stochastic Processes and their Applications

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“…Somewhat surprisingly, formula (1.1) and the stochastic area process (S t ) t≥0 appear in many different contexts. For instance, formula (1.1) has been used by Bismut [9,10] in a probabilistic approach to index theory and allows to construct explicit parametrices for the heat equation on vector bundles (see [2]). Also, the Mellin transform of S t is closely related to the Riemann zeta function (see [8]).…”

Section: The Lévy's Area Formulamentioning

confidence: 99%

Stochastic areas, winding numbers and Hopf fibrations

Baudoin

Wang

2016

Probab. Theory Relat. Fields

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We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces CP n and CH n . The characteristic functions of those processes are computed and limit theorems are obtained. In the case n = 1, we also study windings of the Brownian motion on those spaces and compute the limit distributions. For CP n the geometry of the Hopf fibration plays a central role, whereas for CH n it is the anti-de Sitter fibration.

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“…There are many approaches for approximations of heat kernels through certain asymptotic expansions: for instance, there are recent works such as Baudoin (2009), Gatheral, Hsu, Laurence, Ouyang and Wang (2009), Ben Arous and Laurence (2009), Ilhan, Jonsson and Sircar (2004), Takahashi, Takehara and Toda (2012) and Takahashi and Yamada (2012).…”

Section: Introductionmentioning

confidence: 99%

A Remark on Approximation of the Solutions to Partial Differential Equations in Finance

Takahashi

Yamada

2012

SSRN Journal

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This paper proposes a general approximation method for the solution to a second-order parabolic partial differential equation(PDE) widely used in finance through an extension of Léandre's approach (Léandre (2006(Léandre ( ,2008) and the Bismut identiy(e.g. chapter IX-7 of Malliavin (1997)) in Malliavin calculus. We present two types of its applications, approximations of derivatives prices and short-time asymptotic expansions of the heat kernel. In particular, we provide approximate formulas for option prices under local and stochastic volatility models. We also derive short-time asymptotic expansions of the heat kernel under general timehomogenous local volatility and local-stochastic volatility models in finance, which include Heston (Heston (1993)) and (λ-)SABR models (Hagan et.al. (2002), Labordere ( 2008)) as special cases. Some numerical examples are shown.

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Stochastic Taylor expansions and heat kernel asymptotics

Cited by 9 publications

References 42 publications

Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

Stochastic areas, winding numbers and Hopf fibrations

A Remark on Approximation of the Solutions to Partial Differential Equations in Finance

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