zbMATH — the first resource for mathematics

Weak reflection principle for Lévy processes. (English) Zbl 1330.60064
Summary: In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the classical reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the Lévy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in financial mathematics, computational methods and inverse problems.

60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
60J60 Diffusion processes
91G20 Derivative securities (option pricing, hedging, etc.)
91G80 Financial applications of other theories
45Q05 Inverse problems for integral equations
Full Text: DOI Euclid arXiv
[1] Bachelier, L. (1901). Théorie mathématique de jeu. Annales Scientifiques de l’É. N. S. 18 143-201. · JFM 32.0225.01
[2] Baeumer, B., Meerschaert, M. and Naber, M. (2010). Stochastic models for relativistic diffusion. Phys. Rev. 2 1-5.
[3] Bensoussan, A. and Lions, J.-L. (1982). Contrôle Impulsionnel et Inéquations Quasi-Variationnelles . Gauthier-Villars, Paris. · Zbl 0491.93002
[4] Bertoin, J. (1996). Lévy Processes . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003
[5] Carmona, R., Masters, W. C. and Simon, B. (1990). Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91 117-142. · Zbl 0716.35006 · doi:10.1016/0022-1236(90)90049-Q
[6] Carr, P. and Chou, A. (1997). Breaking barriers. Risk 7 45-49.
[7] Carr, P. and Lee, R. (2009). Put-call symmetry: Extensions and applications. Math. Finance 19 523-560. · Zbl 1184.91198 · doi:10.1111/j.1467-9965.2009.00379.x
[8] Carr, P. and Nadtochiy, S. (2011). Static hedging under time-homogeneous diffusions. SIAM J. Financial Math. 2 794-838. · Zbl 1247.91182 · doi:10.1137/100818303
[9] Cont, R. and Volotchkova, E. (2005). Integro-differential equations for option prices in exponential Lévy models. Finance Stoch. 9 299-325. · Zbl 1096.91023 · doi:10.1007/s00780-005-0153-z
[10] Davies, B. (2002). Integral Transforms and Their Applications , 3rd ed. Springer, New York. · Zbl 0996.44001
[11] Eberlein, E. and Glau, K. (2014). Variational solutions of the pricing PIDEs for European options in Lévy models. Appl. Math. Finance 21 417-450. · Zbl 1395.91497 · doi:10.1080/1350486X.2014.886817
[12] Hilber, N., Reichmann, O., Schwab, C. and Winter, C. (2013). Computational Methods for Quantitative Finance : Finite Element Methods for Derivative Pricing . Springer, Heidelberg. · Zbl 1275.91005 · doi:10.1007/978-3-642-35401-4
[13] Kuznetsov, A. (2010). Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 1801-1830. · Zbl 1222.60038 · doi:10.1214/09-AAP673 · arxiv:1011.1790
[14] Kuznetsov, A., Kyprianou, A. E., Pardo, J. C. and van Schaik, K. (2011). A Wiener-Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Probab. 21 2171-2190. · Zbl 1245.65005 · doi:10.1214/10-AAP746 · arxiv:0912.4743
[15] Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2013). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II. Lecture Notes in Math. 2061 97-186. Springer, Heidelberg. · Zbl 1261.60047 · doi:10.1007/978-3-642-31407-0_2 · arxiv:1104.1280
[16] Lévy, P. (1940). Sur certains processus stochastiques homogénes. Compos. Math. 7 283-339. · JFM 65.1346.02
[17] Molchanov, I. and Schmutz, M. (2010). Multivariate extension of put-call symmetry. SIAM J. Financial Math. 1 396-426. · Zbl 1200.91292 · doi:10.1137/090754194
[18] Rheinländer, T. and Schmutz, M. (2014). Quasi-self-dual exponential Lévy processes. SIAM J. Financial Math. 5 656-684. · Zbl 1312.60058 · doi:10.1137/110859555
[19] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press, Cambridge. · Zbl 0973.60001
[20] Tehranchi, M. R. (2009). Symmetric martingales and symmetric smiles. Stochastic Process. Appl. 119 3785-3797. · Zbl 1177.60044 · doi:10.1016/j.spa.2009.07.007
[21] Titchmarsh, E. C. (1946). Eigenfunction Expansions Associated with Second-Order Differential Equations . Clarendon Press, Oxford. · Zbl 0061.13505
[22] Widder, D. V. (1941). The Laplace Transform . Princeton Univ. Press, Princeton, NJ. · Zbl 0063.08245
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.