Donsker's Delta Research Articles

Characterization and analysis of generalized grey incomplete gamma noise

The grey incomplete gamma distributions was established by one of the authors in a previous publication. In this article we use the Kondratiev characterization theorem to identify those via a suitable Laplace transform with holomorphic functions with suitable properties. We establish theorems for the integration and convergence of sequences of these distributions. As direct applications of these analytic tools we give the examples of Donsker's delta function, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein–Uhlenbeck process.

The Donsker delta function and local time for McKean–Vlasov processes and applications

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean–Vlasov (mean-field) stochastic differential equation.If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon–Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process.For some particular McKean–Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.

Mean-variance asset-liability management with inside information

This paper studies an asset-liability management (ALM) problem under mean-variance criterion with inside information. The asset-liability manager is allowed to invest in a financial market composed of a bond and a stock. The stock price process is governed by a diffusion process with random parameters. The uncontrolled liability process is described by a general diffusion process with hedgeable risks and unhedgeable risks. We model the inside information by a general random variable related to the future values of financial assets and liabilities. By using the Donsker Delta functional technique and the BSDE method, we derive the analytic expressions of efficient strategy and efficient frontier. To illustrate the general result, an example is provided in which the efficient strategy and efficient frontier are obtained in closed form. The comparison of efficient frontiers with and without inside information demonstrates that taking advantage of inside information can improve the efficient frontier.

Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

Donsker’s Delta Functional of Stochastic Processes with Memory

A class of stochastic processes with memory within the framework of the Hida calculus was studied. It was proved that the Donsker delta functionals of the processes are Hida distributions. Furthermore, the probability density function of the processes and the chaos decomposition of the Donsker delta functional were derived. As an application, the existence of the renormalized local times in an arbitrary dimension of the Riemann-Liouville fractional Brownian motion as a white noise generalized function was proved.

Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

A white noise approach to optimal insider control of systems with delay

We use a white noise approach to study the problem of optimal insider control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system. These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the logarithmic utility optimal insider portfolio in a generalized Black-Scholes type financial market. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable, i.e. the maximal utility is infinite. We consider two extensions to delay of this result and prove the following:•If the risky asset price is given by a stochastic delay equation, the resulting insider market is still not viable.•If, however, there is delay in the information flow to the insider, the market becomes viable.

Pricing European Option When the Stock Price Process Is Being Driven by Geometric Brownian Motion

This report is about modeling a European Option in general when the stock price process is being driven by geometric Brownian motion (gBm). The volatility parameter is used as an example of a basic estimator and simulated values of geometric Brownian motion hence exploring some of the properties that improve the accuracy of an estimator. The theory is then extended to estimate the volatility from real data by using the Roger-Satchell Estimator. Hence the estimated volatility is used in the model developed in calculating the value on European option using the Donsker Delta Function approach and is compared with that of the Black-Scholes formula. A unique finding is an observation that the value of an option obtained from using the Donsker Delta Function approach is more of the European Put Option than European Call Option which uses the Black-Scholes formula, then this roughly leads to the conclusion that the Donsker Delta Function approach computes a European Put Option.

English

english

Optimal insider control and semimartingale decompositions under enlargement of filtration

ABSTRACTWe combine stochastic control methods, white noise analysis, and Hida–Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples.

Stochastic differential games with inside information

We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida–Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.

Mittag-Leffler analysis II: Application to the fractional heat equation

Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker's delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green's function to the time-fractional heat equation can be constructed using generalized grey Brownian motion (ggBm) by extending the fractional Feynman–Kac formula from Schneider. Moreover we analyse ggBm, show its differentiability in a distributional sense and the existence of corresponding local times.

Erratum to: A Donsker Delta Functional Approach to Optimal Insider Control and Applications to Finance

Erratum to: A Donsker Delta Functional Approach to Optimal Insider Control and Applications to Finance

A Donsker Delta Functional Approach to Optimal Insider Control and Applications to Finance

We study optimal insider control problems, i.e., optimal control problems of stochastic systems where the controller at any time t, in addition to knowledge about the history of the system up to this time, also has additional information related to a future value of the system. Since this puts the associated controlled systems outside the context of semimartingales, we apply anticipative white noise analysis, including forward integration and Hida–Malliavin calculus to study the problem. Combining this with Donsker delta functionals, we transform the insider control problem into a classical (but parametrised) adapted control system, albeit with a non-classical performance functional. We establish a sufficient and a necessary maximum principle for such systems. Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by Ito–Levy processes. Finally, in the Appendix, we give a brief survey of the concepts and results we need from the theory of white noise, forward integrals and Hida–Malliavin calculus.

A white noise analysis of Volterra processes

In this paper we present a realization of Volterra processes within the white noise analysis framework. We show that Donsker's delta functions of Volterra processes are elements from the space of Hida distributions. An explicit expression for the corresponding chaos decomposition in terms of Wick tensor powers of white noise is also given.

Mittag-Leffler analysis I: Construction and characterization

We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space.

A white noise approach to the Feynman integrand for electrons in random media

Using the Feynman path integral representation of quantum mechanics it is possible to derive a model of an electron in a random system containing dense and weakly coupled scatterers [see F. Edwards and Y. B. Gulyaev, “The density of states of a highly impure semiconductor,” Proc. Phys. Soc. 83, 495–496 (1964)]. The main goal of this paper is to give a mathematically rigorous realization of the corresponding Feynman integrand in dimension one based on the theory of white noise analysis. We refine and apply a Wick formula for the product of a square-integrable function with Donsker's delta functions and use a method of complex scaling. As an essential part of the proof we also establish the existence of the exponential of the self-intersection local times of a one-dimensional Brownian bridge. As a result we obtain a neat formula for the propagator with identical start and end point. Thus, we obtain a well-defined mathematical object which is used to calculate the density of states [see, e.g., F. Edwards and Y. B. Gulyaev, “The density of states of a highly impure semiconductor,” Proc. Phys. Soc. 83, 495–496 (1964)].

Applications of white noise calculus to the computation of Greeks

We apply white noise calculus to the computation of Greeks for contingent claims priced both in a diusion model and in a pure jump model. More precisely, we show how the option price can be represented in terms of the Donsker delta function. Using this representation we prove how one can explicitly compute Greeks. In particular we give proofs for .

WHITE NOISE LÉVY–MEIXNER PROCESSES THROUGH A TRANSFER PRINCIPAL FROM ONE-MODE TO ONE-MODE TYPE INTERACTING FOCK SPACES

Consider the Lévy–Meixner one-mode interacting Fock space {ΓLM, 〈 ⋅, ⋅ 〉LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉LM, we define scalar products 〈 ⋅, ⋅ 〉LM , nin symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces [Formula: see text] naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μr; r > 0}. The Fourier transform in generalized joint eigenvectors of a family [Formula: see text] of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘LMbetween [Formula: see text] and the so-called Lévy–Meixner white noise space [Formula: see text]. We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.

DELTA FUNCTION FOR AN AFFINE SUBSPACE

The Kubo–Yokoi and Donsker delta functions are well known generalized functions in infinite dimensional distribution theory. In this paper we develop the delta function for an affine subspace and show that it is a generalization of the Kubo–Yokoi and Donsker delta functions. The Wiener– Itˆo expansion of the delta function for an affine subspace is also given.