Formulae for the Derivatives of Heat Semigroups

Elworthy, K. D.; Li, Xue Mei

doi:10.1006/jfan.1994.1124

Cited by 264 publications

(241 citation statements)

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“…Known methods to prove pointwise gradient estimates include the Li-Yau method ( [63], see also [79] for generalizations), as well as coupling ( [26]), and other probabilistic methods (see for instance [76]) including the derivation of Bismut type formulae which enable one to estimate the logarithmic derivative of the heat kernel as in (1.11) (see for instance [39], [95], [96] and references therein). Unless one assumes non-negativity of the curvature, all these methods are limited so far to small time, more precisely they yield the crucial factor 1 √ t only for small time.…”

Section: About Our Assumptionsmentioning

confidence: 99%

Riesz transform on manifolds and heat kernel regularity

Auscher

Coulhon

Duong

et al. 2004

Annales Scientifiques de l’École Normale Supérieure

219

413

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On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p , pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d'exposants p.One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.MSC numbers 2000: 58J35, 42B20

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Section: About Our Assumptionsmentioning

confidence: 99%

Riesz transform on manifolds and heat kernel regularity

Auscher

Coulhon

Duong

et al. 2004

Annales Scientifiques de l’École Normale Supérieure

219

413

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“…Take ϕ : S → R to be a bounded Borel-measurable function. The BismutElworthy-Li formula [EL94] yields (after a simple substitution) for the Fréchet derivative of P s ε 2 ϕ in the direction h:…”

Section: Preliminary Estimatesmentioning

confidence: 99%

Multiscale Expansion of Invariant Measures for SPDEs

Blömker

Hairer

2004

Commun. Math. Phys.

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We derive the first two terms in an ε-expansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ε 2 . This result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions. The error between the approximate and the true invariant measure is bounded in both the Wasserstein and the total variation distance.

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“…We remark that the probabilistic representation (4) of the space derivative of the solution to the associated Kolmogorov equation is also referred to as Bismuth-Elworthy-Li type formula in the literature due to [13], [6]. The strength of (4) is that the Delta is expressed again as an expectation of the pay-off multiplied by the so-called Malliavin weight T 0 a(t) σ −1 (X x t ) Z t dB t .…”

Section: Introductionmentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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A well-known application of Malliavin calculus in Mathematical Finance is the probabilistic representation of option price sensitivities, the socalled Greeks, as expectation functionals that do not involve the derivative of the pay-off function. This allows for numerically tractable computation of the Greeks even for discontinuous pay-off functions. However, while the pay-off function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example excludes already simple regime switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, whereat we here focus on the computation of the Delta, which is the option price sensitivity with respect to the initial value of the underlying. To this purpose we first show existence, Malliavin differentiability, and (Sobolev) differentiability in the initial condition of strong solutions of Itô diffusions with drift coefficients that can be decomposed into the sum of a bounded but merely measurable and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivatives in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding Deltas for European and path-dependent options. We conclude with a small simulation study of several regime-switching examples.

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Formulae for the Derivatives of Heat Semigroups

Abstract: Proof: Let T > 0. Parabolic regularity ensures that Itô's formula can be applied to (t,for t ∈ [0, T ). Taking the limit as t → T , we have:

Cited by 264 publications

References 13 publications

Riesz transform on manifolds and heat kernel regularity

Riesz transform on manifolds and heat kernel regularity

Multiscale Expansion of Invariant Measures for SPDEs

Computing deltas without derivatives

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