zbMATH — the first resource for mathematics

Bismut-Elworthy-Li-type formulae for stochastic differential equations with jumps. (English) Zbl 1202.60092
The author considers jump-type stochastic differential equations with drift, diffusion and jump terms and studies the logarithmic derivatives of densities. The approach is based upon the Kolmogorov backward equation. This study is very useful in mathematical finance.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
60J75 Jump processes (MSC2010)
Full Text: DOI arXiv
[1] Bally, V., Bavouzet, M.-P., Messaoud, M.: Integration by parts formula for locally smooth laws and applications to sensitivity computations. Ann. Appl. Probab. 17, 33–66 (2007) · Zbl 1139.60025 · doi:10.1214/105051606000000592
[2] Bichteler, K., Gravereaux, J.-B., Jacod, J.: Malliavin Calculus for Processes with Jumps. Gordon and Breach, New York (1987) · Zbl 0706.60057
[3] Bismut, J.M.: Calcul des variations stochastique et processus de saut. Z. Wahrscheinlichkeiststheor. Verw. Geb. 63, 147–235 (1983) · Zbl 0494.60082 · doi:10.1007/BF00538963
[4] Bismut, J.M.: Large Deviations and the Malliavin Calculus. Birkhäuser, Boston (1984) · Zbl 0537.35003
[5] Cass, T.R., Friz, P.K.: The Bismut–Elworthy–Li formula for jump-diffusions and applications to Monte Carlo methods in finance. arXiv:math/0604311 [math:PR] (2007)
[6] Davis, M.H.A., Johansson, M.P.: Malliavin Monte Carlo Greeks for jump diffusions. Stoch. Process. Appl. 116, 101–129 (2006) · Zbl 1081.60040 · doi:10.1016/j.spa.2005.08.002
[7] El-Khatib, Y., Privault, N.: Computations of Greeks in a market with jumps via the Malliavin calculus. Finance Stoch. 8, 161–179 (2004) · Zbl 1098.91050 · doi:10.1007/s00780-003-0111-6
[8] Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125, 252–286 (1994) · Zbl 0813.60049 · doi:10.1006/jfan.1994.1124
[9] Fournié, E., Lasry, J.M., Lebuchoux, J., Lions, P.L., Touzi, N.: Applications of Malliavin calculus to Monte-Carlo methods in finance. Finance Stoch. 3, 391–412 (1999) · Zbl 0947.60066 · doi:10.1007/s007800050068
[10] Fujiwara, T., Kunita, H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25, 71–106 (1985) · Zbl 0575.60065
[11] Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Springer, New York (1972) · Zbl 0242.60003
[12] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1989) · Zbl 0684.60040
[13] Ishikawa, Y., Kunita, H.: Malliavin calculus on the Wiener–Poisson space and its applications to canonical SDE with jumps. Stoch. Process. Appl. 116, 1743–1769 (2006) · Zbl 1107.60028 · doi:10.1016/j.spa.2006.04.013
[14] Kawai, R., Takeuchi, A.: Greeks formulae for an asset price dynamics model with gamma processes, Math. Finance (2010, to appear) · Zbl 1186.60044
[15] Komatsu, T., Takeuchi, A.: On the smoothness of pdf of solutions to SDE of jump type. Int. J. Differ. Equ. Appl. 2, 141–197 (2001) · Zbl 1045.60054
[16] Komatsu, T., Takeuchi, A.: Generalized Hörmander theorem for non-local operators. In: Albeverio, S., et al. (eds.) Recent Developments in Stochastic Analysis and Related Topics, pp. 234–245. World Scientific, Singapore (2004) · Zbl 1082.60045
[17] Kunita, H.: Representation of martingales with jumps and applications to mathematical finance. In: Kunita, H., et al. (eds.) Stochastic Analysis and Related Topics, pp. 209–232. Mathematical Society of Japan, Tokyo (2004) · Zbl 1059.60059
[18] Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006) · Zbl 1099.60003
[19] Picard, J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996) · Zbl 0853.60064 · doi:10.1007/BF01191910
[20] Takeuchi, A.: Remarks on logarithmic derivatives of densities for degenerate stochastic differential equations with jumps (2010, in preparation)
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.