Dirichlet Forms and Analysis on Wiener Space

Bouleau, Nicolas; Hirsch, Francis

doi:10.1515/9783110858389

Cited by 457 publications

(594 citation statements)

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“…It is determined by X φ dΓ(u, u) = E(u, φu) − 1 2 E(u 2 , φ) and called energy measure; see also [11]. The energy measure satisfies the Leibniz rule, dΓ(u · v, w) = udΓ(v, w) + vdΓ(u, w), as well as the chain rule…”

Section: Assumptions and Basic Propertiesmentioning

confidence: 99%

Sch’nol’s theorem for strongly local forms

Monvel¹,

Lenz

Stollmann

2009

Isr. J. Math.

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Section: Assumptions and Basic Propertiesmentioning

confidence: 99%

Sch’nol’s theorem for strongly local forms

Monvel¹,

Lenz

Stollmann

2009

Isr. J. Math.

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“…We refer to [Kat66,Kat80] for a detailed description of the general theory and to [BH91] [FOT94] for the theory of Dirichlet forms. Throughout we assume that X is a locally compact σ-compact metric space and µ a positive Radon measure with supp µ = X.…”

Section: Preliminariesmentioning

confidence: 99%

“…If ξ ∈ B(E) + then E ξ is a Markovian form which satisfies the bounds 0 ≤ E ξ (ϕ) ≤ ξ ∞ E(ϕ) for all ϕ ∈ B(E) (see [BH91], Proposition I.4.1.1). Therefore the quadratic form E ξ extends by continuity to a Markovian form on D(E) although the identity (5) is not necessarily valid for the extension.…”

Section: IImentioning

confidence: 99%

On extensions of local Dirichlet forms

Robinson

2016

J. Evol. Equ.

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Let E be a Dirichlet form on L 2 (X ; µ) where (X, µ) is locally compact σ-compact measure space. Assume E is inner regular, i.e. regular in restriction to functions of compact support, and local in the sense that E(ϕ, ψ) = 0 for all ϕ, ψ ∈ D(E) with ϕ ψ = 0. We construct two Dirichlet forms E m and E M such that E m ≤ E ≤ E M . These forms are potentially the smallest and largest such Dirichlet forms. InWe analyze the family of local, inner regular, Dirichlet forms F which extend E and satisfy E m ≤ F ≤ E M . We prove that the latter bounds are valid if and only ifAs an application we show that if E and F are strongly local then the Ariyoshi-Hino set-theoretic distance is the same for each of the forms E, E M and F . If in addition E m is strongly local then it also defines the same distance. Finally we characterize the uniqueness condition E M = E m by capacity estimates.

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“…An alternative method that we will use here is based on potential theory (see Albeverio [1], Bouleau and Hirsch [5] and Fukushima et al [17]) suggested by Bouleau [6]. We assume that the uncertainties are small random variables added to the true values of the parameters.…”

Section: Introductionmentioning

confidence: 99%

Asset Pricing under Uncertainty

Mirakhor¹,

Krichene²

2012

Introductory Mathematics and Statistics for Islamic Finance

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We study the effect of parameter uncertainty on a stochastic diffusion model, in particular the impact on the pricing of contingent claims, using methods from the theory of Dirichlet forms. We apply these techniques to hedging procedures in order to compute the sensitivity of SDE trajectories with respect to parameter perturbations. We show that this analysis can justify endogenously the presence of a bid-ask spread on the option prices. We also prove that if the stochastic differential equation admits a closed form representation then the sensitivities have closed form representations.We examine the case of log-normal diffusion and we show that this framework leads to a smiled implied volatility surface coherent with historical data.

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Dirichlet Forms and Analysis on Wiener Space

Cited by 457 publications

References 0 publications

Sch’nol’s theorem for strongly local forms

Sch’nol’s theorem for strongly local forms

On extensions of local Dirichlet forms

Asset Pricing under Uncertainty

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