Abstract
We describe the integrand in the martingale (or stochastic integral) representation of a square integrable functional F of a Lévy process in terms of (a derivative or difference operator acting on) a map ß F introduced in Rajeev and Fitzsimmons (Stochastics 81, 467–476, 2009). The kernels in the chaos expansion of F are also described in terms of the iterated derivative and difference operators.
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References
Dellacherie C. and Meyer P.A. (1982). Probabilities and Potential B. North Holland.
Di Nunno G., Øksendal B. and Proske F. (2004). White Noise Analysis for Lévy Processes. J. Funct. Anal. 206, 109–148.
Di Nunno G. (2007). Randon Fields: non anticipating derivatives and differentiation formulas. Infinite Dimensional Analysis and Quantum Probability 10.
Di Nunno G., Øksendal B. and Proske F. (2009). Malliavin Calculus for Lévy processes with Applications to Finance. Springer.
Émery M. (2006). Chaotic Representation Property of certain Azema martingales. Ill. J. Math. 50, 2, 395–411.
He S.-W., Wang J.-G. and Yan J.-A. (1992). Semi- Martingale Theory and Stochastic Calculus. Science Press, Beijing.
Itô K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169.
Itô K. (1956). Spectral type of the shift transformation of differential processes with stattionary. increments. Trans. Am. Math. Soc. 81, 252–263.
Kallenberg O. (2002). Foundations of Modern Probability. Springer.
Privault N. (2009). Stochastic Analysis in Discrete and Continuous Settings. Springer.
Nualart D. and Schoutens W. (2000). Chaotic and predictable representation for Lévy processes. Stoch. Process. Appl. 90, 109–122.
Rajeev B. and Fitzsimmons P. (2009). A new approach to Martingale representation. Stochastics 81, 467–476.
Rajeev B. 2009 Stochastic Integrals and Derivatives, Bulletin of Kerala Mathematics Association. October 2009, special issue, p. 105–127.
Solé J.L., Utzet F. and Vives J. (2007). Chaos expansions and Malliavin Calculus for Lévy processes, Benth F. E. et al. (eds.),. Proc. 2nd Abel Symposium, Springer 1987.
Stroock D.W. (1987). Homogenous chaos revisited. Séminaire de Probabilités 21, 1–7.
Wiener N. (1938). The homogeneous chaos. Amer. J. Math. 60, 897–936.
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Rajeev, B. Martingale Representations for Functionals of Lévy Processes. Sankhya A 77, 277–299 (2015). https://doi.org/10.1007/s13171-015-0073-8
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DOI: https://doi.org/10.1007/s13171-015-0073-8
Keywords and phrases
- Martingale representation
- Stochastic integral representation
- Lévy processes
- Chaos expansion
- Stochastic derivative