Donsker's Delta Research Articles

ON EXPLICIT STRONG SOLUTION OF ITÔ–SDE'S AND THE DONSKER DELTA FUNCTION OF A DIFFUSION

We determine a new explicit representation of strong solutions of Itô-diffusions and elicit its correspondence to the general stochastic transport equation. We apply this formula to deduce an explicit Donsker delta function of a diffusion.

The Donsker delta function of a Lévy process with application to chaos expansion of local time

We give an explicit formula for the Donsker delta function of a certain class of Lévy processes in the Lévy–Hida distribution space. As an application we use the Donsker delta function to derive an explicit chaos expansion of local time for Lévy processes, in terms of iterated integrals with respect to the associated compensated Poisson random measure. On donne une formule explicite pour la fonction delta de Donsker d'une classe de processus de Lévy dans l'espace des distributions de Lévy–Hida. A titre d'application on donne le développement en chaos du temps local d'un processus de Lévy, sous la forme d'intégrales itérées de la mesure de Poisson compensée.

The Donsker delta function of a Lévy process with application to chaos expansion of local time

We give an explicit formula for the Donsker delta function of a certain class of Lévy processes in the Lévy–Hida distribution space. As an application we use the Donsker delta function to derive an explicit chaos expansion of local time for Lévy processes, in terms of iterated integrals with respect to the associated compensated Poisson random measure.

An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter

We consider fractional Brownian motions B t H with arbitrary Hurst coefficients 0< H<1 and prove the following results: (i) An integral representation of the fractional white noise as generalized Wiener integral; (ii) an Itô formula for generalized functionals of B t H ; (iii) an analogue of Tanaka's formula; (iv) a Clark–Ocone formula for Donsker's delta function of B t H ; (v) an integral representation of the local time of B t H .

Donsker's Delta Function and the Covariance between Generalized Functionals

A classical space and a nonstandard space are used as models for white noise. The expectation of the product of certain generalized functionals is first proved to be finite. Then the covariance and the correlation between Donsker's delta functions are calculated and given meaning.

Using the Donsker Delta Function to Compute Hedging Strategies

We use white noise calculus and the Donsker Delta Function to find explicit formulas for the replicating portfolios in a Black–Scholes market for a class of contingent T-claims.

A generalized Clark-Ocone formula

We extend the Clark-Ocone formula to a suitable class of generalized Brownian function- als. As an example we derive a representation of Donsker's delta function as (limit of) a stochastic integral.

Donsker's Delta Function of Lévy Process

Let X={X(t):t∈R} be a Levy process and β a non-decreasing, right continuous, bounded function with β(−∞)=0 (((1+u2)/u2)dβ(u) is the Levy measure). In this paper we define the Donsker delta function δ(X(t)−a), t>0 and a∈R, as a generalized Levy functional under the condition that β(0)−β(0−)>0. This leads us to define F(X(t)) for any tempered distribution F, and as an application, we derive an Ito formula for F(X(t)) when β has jumps at 0 and 1.

Law of large numbers and central limit theorem for Donsker's delta function of diffusions I

Limit theorems in the space of Hida distributions, similar to the law of large numbers and the central limit theorem, are shown for composites of the Dirac distribution with solutions of one-dimensional, non-degenerate Itô equations.

Stochastic levi sums

AbstractAn analogy of Levi sums is considered for a class of stochastic partial differential equations to construct their stochastic fundamental solutions. These notions are shown to coincide with Donsker's delta functions, typical generalized Wiener functionals, which have been studied in the frame of the Malliavin calculus. © 1994 John Wiley &amp; Sons, Inc.

Fractional order Sobolev spaces on Wiener space

Fractional order Sobolev spaces are introduced on an abstract Wiener space and Donsker's delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Holder continuity of a class of conditional expectations is obtained.

Law of large numbers and central limit theorem for donsker's delta function

It is shown that limit theorems similar to the law of large numbers and the central limit theorem hold for (certain versions of) Donsker's delta function strongly in the space of Hida distributions . Furthermore, the law of large numbers is shown to hold weakly in the Meyer-Watanabe space .

Regularity property of Donsker's delta function

LetL be the space of rapidly decreasing smooth functions on ℝ andL * its dual space. Let (L 2)+ and (L 2)− be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL * with standard Gaussian measure. The Donsker delta functionδ(B(t)−x) is in (L 2)− and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH n is the Hermite polynomial of degreen. It is shown that forφ in (L 2)+,g t,φ(x)≡〈δ(B(t)−x), φ〉 is inL and the linear map takingφ intog t,φ is continuous from (L 2)+ intoL. This implies that forf inL * is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL *, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where∂ * is the adjoint of $$\dot B(s)$$ -differentiation operator∂ s .