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The Trotter-Kato Approximation in Utility Maximization Open Access


Other title
utility maximization
Type of item
Degree grantor
University of Alberta
Author or creator
Miao, Lingyu
Supervisor and department
Frei, Christoph (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Frei, Christoph (Department of Mathematical and Statistical Sciences)
Cadenillas, Abel (Department of Mathematical and Statistical Sciences)
Schmuland, Byron (Department of Mathematical and Statistical Sciences)
Melnikov, Alexander (Department of Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Mathematical Finance
Date accepted
Graduation date
Master of Science
Degree level
This thesis deals with the Trotter-Kato approximation in utility maximization. The Trotter-Kato approximation is a method to split a differential equation into two parts, which are then solved iteratively over small time intervals. In the context of utility maximization, this procedure was introduced by Nadtochiy and Zariphopoulou [11] for partial differential equations (PDEs) in a Markovian setting, which we revisit in the first part of this thesis. We then study what the Trotter-Kato approximation can mean for backward stochastic differential equations (BSDEs), which do not need Markovian assumptions and allow for a probabilistic interpretation. We also discuss how the Trotter- Kato approximation can be implemented numerically in both the PDE and the BSDE case.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
C. Frei. Convergence results for the indifference value based on the sta- bility of BSDEs. Stochastics, 85:464–488, 2013.Y. Hu P. Imkeller and M. Mu ̈ller. Utility maximization in incomplete markets. Annals of Applied Probability, 15:1691–1712, 2005.T. Kato. Trotters product formula for an arbitrary pair of self-adjoint contraction semigroups. Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York, pages 185–195, 1978.N. Kazamaki. Continuous Exponential Martingales and BMO, volume 1579 of Lecture Notes in Mathematics. Springer, 1994.M. Kobylanski. Backward stochastic differential equations and partial dif- ferential equations with quadratic growth. Annals of Probability, 28:558– 602, 2000.B. Øksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, 6th edition, 2003.M. Mania and M. Schweizer. Dynamic exponential utility indifference valuation. Annals of Applied Probability, 15:2113–2143, 2005.R. Merton. Lifetime portfolio selection under uncertainty: the continuous- time case. The review of Economics and Statistics, 51:247–257, 1969.M. Morlais. Quadratic BSDEs driven by a continuous martingale and ap- plications to the utility maximization problem. Finance and Stochastics, 13:121–150, 2009.M. Musiela and T. Zariphopoulou. An example of indifference prices under exponential preferences. Finance and Stochastics, 8:229–239, 2004.S. Nadtochiy and T. Zariphopoulou. An approximation scheme for so- lution to the optimal investment problem in incomplete markets. SIAM Journal on Financial Mathematics, 4:494–538, 2013.N. Touzi. Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, volume 29 of Fields Institute Monographs. Springer, 2013.H.F. Trotter. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10:545–551, 1959.

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