Abstract
In an L 2-framework, we study various aspects of stochastic calculus with respect to the centered doubly stochastic Poisson process. We introduce an orthogonal basis via multilinear forms of the value of the random measure and we analyze the chaos representation property. We review the structure of non-anticipating integration for martingale random fields and in this framework we study non-anticipating differentiation. We present integral representation theorems where the integrand is explicitly given by the non-anticipating derivative.
Stochastic derivatives of anticipative nature are also considered: The Malliavin type derivative is put in relationship with another anticipative derivative operator here introduced. This gives a new structural representation of the Malliavin derivative based on simple functions. Finally we exploit these results to provide a Clark–Ocone type formula for the computation of the nonanticipating derivative.
Mathematics Subject Classification (2010). 60H07, 60H05.
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References
F.E. Benth, G. Di Nunno, A. Løkka, B. Øksendal, and F. Proske, Explicit representation of the minimal variance portfolio in markets driven by Lévy processes. Mathematical Finance, 13 (1) (2003), 55–72.
T. Bielecki, M. Jeanblanc, and M. Rutkowski, Hedging of Defaultable Claims. Paris- Princeton Lectures on Mathematical Finance, 2004.
P. Billingsley, Probability and Measure. Wiley, 1995.
R. Boel, P. Varaiya, and E. Wong, Martingales on jump processes. I. Representation results. SIAM Journal on control, 13 (5) (1975), 999–1021.
P. Brémaud, Point Processes and Queues – Martingale Dynamics. Springer, 1981.
R. Cairoli and J. Walsh, Stochastic integrals in the plane. Acta Mathematica, 134 (1975), 111–183.
P. Carr, H. German, D. Madan, and M. Yor, Stochastic volatility for Lévy processes. Mathematical Finance, 13 (3) (2003), 345–282.
D.R. Cox, Some statistical methods connected with series of events. Journal of the Royal Statistical Society, Series B (Methodological), 17 (2) (1955), 129–164.
D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes I. Springer, 2003.
M.H.A. Davis, The representation of martingales of jump processes. SIAM Journal on Control and Optimization, 14 (4) (1976), 623–638.
M.H.A. Davis, Martingale representation and all that. In Advances in Control, Communication Networks, and Transportation Systems, E.H. Abed (editor), Systems & Control: Foundations & Applications, Birkhäuser Boston, 2005, 57–68.
G. Di Nunno, Stochastic integral representation, stochastic derivatives and minimal variance hedging. Stochastics and Stochastics Reports, 73 (2002), 181–198.
G. Di Nunno, Random fields evolution: non-anticipating integration and differentiation. Theory of Probability and Mathematical Statistics, AMS, 66 (2003), 91–104.
G. Di Nunno, On orthogonal polynomials and the Malliavin derivative for Lévy stochastic measures. SMF Séminaires et Congr`es, 16 (2007), 55–69.
G. Di Nunno, Random fields: non-anticipating derivative and differentiation formulas. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 10 (2007), 465–481.
G. Di Nunno and I. Baadshaug Eide, Minimal-variance hedging in large financial markets: random fields approach. Stochastic Analysis and Applications, 28 (2010), 54–85.
G. Di Nunno and S. Sjursen, Backwards stochastic differential equations driven by the doubly stochastic Poisson process. Manuscript, 2012.
D. Duffie and K. Singleton, Credit Risk: Pricing, Measurement, and Management. Princetown University Press, 2003.
J. Grandell, Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics, 529, Springer-Verlag, 1976.
J. Jacod, Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales. Probability Theory and Related Fields, 31 (3) (1975), 235–253.
O. Kallenberg, Random Measures. Berlin, Akademie-Verlag, 1986.
O. Kallenberg. Foundations of Modern Probability. Springer, 1997.
D. Lando, On Cox processes and credit risky securities. Review of Derivatives Research, 2 (1998), 99–120.
D. Nualart, The Malliavin Calculus and Related Topics. Springer, 1995.
B. Øksendal, Stochastic Differential Equations. Springer, 2005.
N. Privault, Generalized Bell polynomials and the combinatorics of Poisson central moments. The Electronic Journal of Combinatorics, 18 (1) (2011), 1–10.
M. Winkel, The recovery problem for time-changed Lévy processes. Research Report MaPhySto 2001-37, October 2001.
E. Wong and M. Zakai, Martingales and stochastic integrals for processes with a multi-dimensional parameter. Probability Theory and Related Fields, 29 (1974), 109– 122.
A.L. Yablonski, The Malliavin calculus for processes with conditionally independent increments. In Stochastic Analysis and Applications, The Abel Symposium 2005, 2, Berlin: Springer, 2007, 641–678.
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Nunno, G.D., Sjursen, S. (2013). On Chaos Representation and Orthogonal Polynomials for the Doubly Stochastic Poisson Process. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VII. Progress in Probability, vol 67. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0545-2_2
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DOI: https://doi.org/10.1007/978-3-0348-0545-2_2
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