White Noise of Poisson Random Measures

We develop a white noise theory for Poisson random measures associated with a pure jump Lévy process. The starting point of this theory is the chaos expansion of Itô. We use this to construct the white noise of a Poisson random measure, which takes values in a certain distribution space. Then we show, how a Skorohod/Itô integral for point processes can be represented by a Bochner integral in terms of white noise of the random measure and a Wick product. Further, based on these concepts we derive a generalized Clark–Haussmann–Ocone theorem with respect to a combination of Gaussian noise and pure jump Lévy noise. We apply this theorem to obtain an explicit formula for partial observation minimal variance portfolios in financial markets, driven by Lévy processes. As an example we compute the “closest” hedge to a binary option.

[1]  D. Sondermann.  Hedging of non-redundant contingent claims , 1985 .

[2]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[3]  Yoshifusa Ito.  Generalized Poisson Functionals , 1988 .

[4]  S. Albeverio,et al.  The vacuum of the Høegh-Krohn model as a generalized white noise functional , 1989 .

[5]  M. Zakai,et al.  Generalized stochastic integrals and the malliavin calculus , 1986 .

[6]  Nobuaki Obata,et al.  White Noise Calculus and Fock Space , 1994 .

[7]  Yoshifusa Ito.  On a generalization of non-linear Poisson functionals , 1980 .

[8]  Jorge A. León,et al.  On Lévy processes, Malliavin calculus and market models with jumps , 2002, Finance Stochastics.

[9]  I. Karatzas,et al.  A generalized clark representation formula, with application to optimal portfolios , 1991 .

[10]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .

[11]  Nicolas Privault.  An extension of stochastic calculus to certain non-Markovian processes , 1997 .

[12]  R. Elliott,et al.  A General Fractional White Noise Theory And Applications To Finance , 2003 .

[14]  Bernt Øksendal,et al.  Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach , 1996 .

[15]  Frank Proske,et al.  Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes , 2003 .

[16]  Fred Espen Benth,et al.  A Remark on the Equivalence between Poisson and Gaussian Stochastic Partial Differential Equations , 1998 .

[17]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[18]  WHITE NOISE ANALYSIS AND APPLICATIONS , 1994 .

[19]  Jean Picard,et al.  On the existence of smooth densities for jump processes , 1996 .

[20]  Jürgen Potthoff,et al.  On a dual pair of spaces of smooth and generalized random variables , 1995 .

[21]  Bernt Øksendal,et al.  White Noise Analysis for Lévy Processes. , 2004 .

[22]  A. Üstünel.  An Introduction to Analysis on Wiener Space , 1995 .

[23]  S. Albeverio,et al.  The Wightman Axioms and the Mass Gap for Strong Interactions of Exponential Type in Two-Dimensional Space-Time , 1974 .

[24]  Nicolas Bouleau,et al.  Dirichlet Forms and Analysis on Wiener Space , 1991 .

[25]  P. Protter.  Stochastic integration and differential equations , 1990 .

[26]  H. Pham.  Mean-Variance Hedging For Partially Observed Drift Processes , 2001 .

[27]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[28]  Kiyosi Itô,et al.  SPECTRAL TYPE OF THE SHIFT TRANSFORMATION OF DIFFERENTIAL PROCESSES WITH STATIONARY INCREMENTS( , 1956 .

[29]  D. Nualart,et al.  Chaotic and predictable representation for L'evy Processes , 2000 .

[30]  B. Øksendal,et al.  FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE , 2003 .

[31]  Generalized Appell Systems , 1999, math/9908032.

[32]  Bernt Øksendal,et al.  A White Noise Approach to Stochastic Differential Equations Driven by Wiener and Poisson Processes , 1998 .

[33]  Hui-Hsiung Kuo,et al.  White noise distribution theory , 1996 .

[34]  Bernt Øksendal,et al.  White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance , 2000, Finance Stochastics.

[35]  S. Thangavelu.  Lectures on Hermite and Laguerre expansions , 1993 .

[36]  A. Üstünel.  Representation of the distributions on Wiener space and stochastic calculus of variations , 1987 .

[37]  Anticipative calculus for Lévy processes and stochastic differential equations , 2004 .

[38]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .