BSDEs and log-utility maximization for Lévy processes
Volume 6, Issue 4 (2019), pp. 479–494
Pub. online: 28 October 2019
Type: Research Article
Open Access
Open Access
Received
15 February 2019
15 February 2019
Revised
2 October 2019
2 October 2019
Accepted
3 October 2019
3 October 2019
Published
28 October 2019
28 October 2019
Abstract
In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.
References
Becherer, D.: Bounded solutions to backward sdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16(4), 2027–2054 (2006). MR2288712. https://doi.org/10.1214/105051606000000475
Cvitanić, J., Karatzas, I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab., 2(4), 767–818 (1992). MR1189418
Cohen, S., Elliott, R.J.: Stochastic Calculus and Applications. Birkhäuser (2015). MR3443368. https://doi.org/10.1007/978-1-4939-2867-5
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(1), 463–520 (1994). MR1304434. https://doi.org/10.1007/BF01450498
Di Tella, P., Engelbert, H.-J.: The chaotic representation property of compensated-covariation stable families of martingales. Ann. Probab. 44(6), 3965–4005 (2016). MR3572329. https://doi.org/10.1214/15-AOP1066
Di Tella, P., Engelbert, H.-J.: The predictable representation property of compensated-covariation stable families of martingales. Teor. Veroâtn. Primen. 60, 99–130 (2016). MR3568759. https://doi.org/10.1137/S0040585X97T98748X
Goll, T., Kallsen, J.: Optimal portfolios for logarithmic utility. Stoch. Process. Appl., 89(1), 31–48 (2000), MR1775225. https://doi.org/10.1016/S0304-4149(00)00011-9
Goll, T., Kallsen, J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13(2), 774–799 (2003). MR1970286. https://doi.org/10.1214/aoap/1050689603
He, S., J., W., Yan, J.: Semimartingale Theory and Stochastic Calculus. Taylor & Francis (1992). MR1219534
Hu, Y., P., I., Müller, M.: Utility maximization in incomplete markets. Ann. Appl. Probab. 15(3), 1691–1712 (2005). MR2152241. https://doi.org/10.1214/105051605000000188
Izumisawa, M., Sekiguchi, T., Shiota, Y.: Remark on a characterization of BMO-martingales. Tohoku Math. J. (2) 31(3), 281–284 (1979). MR0547642. https://doi.org/10.2748/tmj/1178229795
Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Springer (1979). MR0542115
Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, Springer (2003). MR1943877. https://doi.org/10.1007/978-3-662-05265-5
Kallsen, J.: Optimal portfolios for exponential Lévy processes. Math. Methods Oper. Res., 51(3), 357–374 (2000), MR1778648. https://doi.org/10.1007/s001860000048
Morlais, M.-A.: Utility maximization in a jump market model. Stochastics 81(1), 1–27 (2009). MR2489997. https://doi.org/10.1080/17442500802201425
Morlais, M.-A.: A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem. Stoch. Process. Appl. 120(10), 1966–1995 (2010). MR2673984. https://doi.org/10.1016/j.spa.2010.05.011
Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stoch. Process. Appl. 90(1), 109–122 (2000). MR1787127. https://doi.org/10.1016/S0304-4149(00)00035-1
Nualart, D., Schoutens, W.: Backward stochastic differential equations and Feynman–Kac formula for Lévy processes, with applications in finance. Bernoulli 7(5), 761–776 (2001). MR1867081. https://doi.org/10.2307/3318541
Rouge, R., El Karoui, N.: Pricing via utility maximization and entropy. Math. Finance 10(2), 259–276 (2000). MR1802922. https://doi.org/10.1111/1467-9965.00093
Wang, J.-G.: Some remarks on processes with independent increments. Sémin. Probab. XV. Lect. Notes Math. 15(850), 627–631 (1981). MR0622593
