Thaleia Zariphopoulou (The University of Texas, Austin)

Optimal portfolio choice: the backward and forward investment performance criteria

ABSTRACT:

These lectures will first provide a general overview of theoretical models for optimal portfolio management, the related stochastic optimization problems, methods for their analysis, and properties of the optimal policies. They will then give an introduction to the forward investment performance approach, discuss the associated optimization problems, and present various classes of solutions and their optimal policies in discrete and continuous time. The forward criterion in the presence of model ambiguity will be also discussed as well as its connection to the (fractional) Kelly criterion.

Lecture 1. FUNDAMENTALS OF OPTIMAL PORTFOLIO THEORY

(Tuesday May 10, 10:00-12:00, CDT Lecture Room 2)

The lecture will first provide an introduction to the modeling elements of the stochastic optimization problems arising in maximal expected utility models. It will cover background results for the Hamilton-Jacobi-Bellman equation, feedback optimal policies and their robustness, multi-scale cases, and some open problems. It will continue with a critique of the classical setting, and, in turn, introduce the forward investment performance criterion.

Lecture 2. TIME-MONOTONE FORWARD CRITERIA AND APPLICATIONS

(Wednesday May 11, 14:30-16:30, CDT Lecture Room 2)

The second lecture will first provide an introduction to the forward stochastic PDE, and, in turn, give a detailed analysis of the time-monotone case in Ito-diffusion markets. Long-term (turnpike) results will be presented and compared to their classical counterparts. The lecture will conclude with the cases of partial information and model ambiguity.

Lecture 3. FORWARD CRITERIA IN DISCRETE AND CONTINUOUS-TIME

(Thursday May 12, 10:00-12:00, CDT Lecture Room 2)

In the first part, forward performance process with non-zero volatility will be discussed. The family of homothetic forward processes will be, in turn, presented and its connection with ergodic and infinite-horizon BSDE, and with the long-term limit of the classical expected utility, will be discussed. In the second part, a discrete-time family of forward processes will be presented, and results for predictable forward processes will be given for completely monotonic utility data.

REFERENCES

1. El Karoui and M.rad, M.: An exact connection between two solvable SDE and a non-linear utility stochastic PDE, SIAM J. Financial Mathematics 4(1), 697-736, 2014.
2. Fouque, J.-P., Sircar, R. and Zariphopoulou, T.: Portfolio optimization and volatility asymptotics, Mathematical Finance, 2016.
3. Liang, G. and Zariphopoulou, T.: Representation of homothetic forward performance processes vie ergodic and infinite-horizon quadratic BSDE in stochastic factor models, hhtp://arxiv.org/abs/1511.04863.
4. Kallblad, S.: Topic in portfolio choice: qualitative properties, time consistency and investment under model uncertainty. Ph.D. Thesis, University of Oxford, 2014.
5. Kallblad, S., Obloj, J. and Zariphopoulou, T.: Time-consistent investment under model uncertainty: the robust forward criteria, submitted for publication, 2015.
6. Kallbald, S. and T. Zariphopoulou, T.: Qualitative analysis of optimal investment strategies in log-normal markets, SSRN 2373587, 2014.
7. Monin, P. and Zariphopoulou, T.: On the optimal wealth process in a log-normal market: applications to risk management, Journal of Financial Engineering, 1(1), 2014.
8. Musiela, M. and Zariphopoulou, T.: Stochastic partial differential equations in port.io choice, Contemporary Quantitative Finance: Essays in Honour of E. Platen, C. Chiarella and A. Novikov, Eds., 195-215, 2010.
9. Musiela, M. and Zariphopoulou, T.: Portfolio choice under space-time monotone performance criteria, SIAM Journal on Financial Mathematics, 326-365, 2010.
10. Natodchiy, S. and Tehranchi, M.: Optimal investment for all time horizons and Martin boundary of space-time diffusions, Mathematical Finance, published on line, 2015.
11. Nadtochiy, S. and Zariphopoulou, T.: A class of homothetic forward investment performance processes with non-zero volatility, Inspired by Finance: The Musiela Festschrift, Y. Kabanov et al. Eds., 475-505, 2013.
12. Shkolnikov, M., Sircar, R. and Zariphopoulou, T.: Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations, submitted for publication, 2016.
13. Zariphopoulou, T.: Stochastic modeling and methods in optimal portfolio construction, Proceedings of the International Congress of Mathematicians, Seoul, 2014 (Vol. IV, 1163-1186).
14. Zariphopoulou, T. and Zitkovic, G.: Maturity-independent risk measures, SIAM Journal on Financial Mathematics, 1, 266-288 (2010).
15. Zitkovic, G.: A dual characterization of self-generation and exponential forward performances, Annals of Applied Probability, 19(6), 2176-2210 (2009).