Optimal investment and consumption with forward preferences and uncertain parameters

Abstract

We introduce and analyze a class of forward performance criteria in incomplete markets in the presence of model ambiguity. Incompleteness stems from general investment constraints, while model uncertainty is represented by a convex and compact set of plausible model parameter processes. Following the max-min criteria in traditional (backward) robust control, we formulate similar criteria for the robust forward performance processes and focus on the rich class of time-monotone processes. We provide a novel PDE characterization and a semi-explicit saddle-point construction of the robust forward performance criteria and their optimal policies. Furthermore, we present additional results within the class of homothetic constant relative risk aversion (CRRA) processes. Within this class, we investigate the relationship between forward performance processes on wealth and those on consumption, establishing an interesting dominance through time.

Keywords

Robust forward performance criteria / Portfolio constraints / Consumption / Saddle points / Constant relative risk aversion

References

[1] Angoshtari, B., Predictable forward performance processes in complete markets, Probability, Uncertainty and Quantitative Risk, 2023, 8(2): 141-176.

[2] Angoshtari, B., Zariphopoulou, T. and Zhou, X., Predictable forward performance processes: The binomial case, SIAM Journal on Control and Optimization, 2020, 58(1): 327-347.

[3] Anthropelos, M., Geng, T. and Zariphopoulou, T., Competition in fund management and forward relative performance criteria, SIAM Journal on Financial Mathematics, 2022, 13(4): 1271-1301.

[4] Avanesyan, L., Shkolnikov, M. and Sicar, R., Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem, Finance and Stochastics, 2020, 24: 981-1011.

[5] Berrier, F. P. Y. S. and Tehranchi, M. R., Forward utility of investment and consumption, Working papers, Statistical Laboratory, University of Cambridge, Cambridge, 2008.

[6] Biagini, S. and Pınar, M. Ç., The robust Merton problem of an ambiguity averse investor, Mathematics and Financial Economics, 2017, 11(1): 1-24.

[7] Bo, L., Capponi, A. and Zhou, C., Power forward performance in semimartingale markets with stochastic integrated factors, Mathematics of Operations Research, 2023, 48(1): 288-312.

[8] Bordigoni, G., Matoussi, A. and Schweizer, M., A stochastic control approach to a robust utility maximization problem, In: BenthF. E., Di NunnoG., LindstrømT., ØksendalB. and ZhangT. (eds.), Stochastic Analysis and Applications, Abel Symposia, Springer, Berlin, Heidelberg, 2007, 2: 125-151.

[9] Cheridito, P. and Hu, Y., Optimal consumption and investment in incomplete markets with general constraints, Stochastics and Dynamics, 2011, 11(2): 283-299.

[10] Chong, W. F., Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences, Insurance: Mathematics and Economics, 2019, 88: 93-107.

[11] Chong, W. F., Hu, Y., Liang, G. and Zariphopoulou, T., An ergodic BSDE approach to forward entropic risk measures: Representation and large-maturity behavior, Finance and Stochastics, 2019, 23(1): 239-273.

[12] Chong, W. F. and Liang, G., Optimal forward investment and consumption with general drift and volatility uncertainties: Non-zero volatility case through randomization, Forthcoming, 2024.

[13] Choulli, T., Stricker, C. and Li, J., Minimal Hellinger martingale measures of order q, Finance and Stochastics, 2007, 11: 399-427.

[14] El Karoui, N., Hillairet, C. and Mrad, M., Consistent utility of investment and consumption: A forward/backward SPDE viewpoint, Stochastics, 2018, 90(6): 927-954.

[15] El Karoui, N., Hillairet, C. and Mrad, M., Ramsey rule with forward/backward utility for long-term yield curves modeling, Decisions in Economics and Finance, 2022, 45: 375-414.

[16] El Karoui, N. and Mrad, M., An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE, SIAM Journal on Financial Mathematics, 2023, 4(1): 697-736.

[17] El Karoui, N. and Mrad, M., Recover dynamic utility from observable process: Application to the economic equilibrium, SIAM Journal on Financial Mathematics, 2021, 12(1): 189-225.

[18] Fouque, J. P., Pun, C. S. and Wong, H. Y., Portfolio optimization with ambiguous correlation and stochastic volatilities, SIAM Journal on Control and Optimization, 2016, 54(5): 2309-2338.

[19] He, X. D., Strub, M. S. and Zariphopoulou, T., Forward rank-dependent performance criteria: Time-consistent investment under probability distortion, Mathematical Finance, 2021, 31(2): 683-721.

[20] Henderson, V., Valuing the option to invest in an incomplete market, Mathematics and Financial Economics, 2007, 1(2): 103-128.

[21] Henderson, V. and Hobson, D., Horizon-unbiased utility functions, Stochastic Processes and their Applications, 2007, 117(11): 1621-1641.

[22] Hernández-Hernández, D. and Schied, A., Robust utility maximization in a stochastic factor model, Statistics and Decisions, 2006, 24(1): 109-125.

[23] Hernández-Hernández, D. and Schied, A., A control approach to robust utility maximization with logarithmic utility and time-consistent penalties, Stochastic Processes and their Applications, 2007, 117(8): 980-1000.

[24] Hu, Y., Imkeller, P. and Müller, M., Utility maximization in incomplete markets, The Annals of Applied Probability, 2005, 15(3): 1691-1712.

[25] Hu, Y., Liang, G. and Tang, S., Systems of ergodic BSDEs arising in regime switching forward performance processes, SIAM Journal on Control and Optimization, 2020, 58(4): 2503-2534.

[26] Källblad, S., Black’s inverse investment problem and forward criteria with consumption, SIAM Journal on Financial Mathematics, 2020, 11(2): 494-525.

[27] Källblad, S., Obłój, J. and Zariphopoulou, T., Dynamically consistent investment under model uncertainty: The robust forward criteria, Finance and Stochastics, 2018, 22(4): 879-918.

[28] Liang, G., Strub, M. and Wang, Y., Predictable forward performance processes: Infrequent evaluation and applications to human-machine interactions, Mathematical Finance, 2023, 33(4): 1248-1286.

[29] Liang, G. and Zariphopoulou, T., Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE, SIAM Journal on Financial Mathematics, 2017, 8(1): 344-372.

[30] Lin, Q. and Riedel, F., Optimal consumption and portfolio choice with ambiguous interest rates and volatility, Economic Theory, 2021, 71: 1189-1202.

[31] Lin, Q., Sun, X. and Zhou, C., Horizon-unbiased investment with ambiguity, Journal of Economic Dynamics and Control, 2020, 114: 103896.

[32] Matoussi, A., Possamaï D. and Zhou, C., Robust utility maximization in nondominated models with 2BSDE: The uncertain volatility model, Mathematical Finance, 2015, 25(2): 258-287.

[33] Musiela, M. and Zariphopoulou, T., Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model, Advances in Mathematical Finance, 2007: 303-334.

[34] Musiela, M. and Zariphopoulou, T., Optimal asset allocation under forward exponential performance criteria, In: EthierS. N., FengJ. and StockbridgeR. H., Markov Processes and Related Topics: A Festschrift for T. G. Kurtz, Institute of Mathematical Statistics Collections, 2008, 4: 285-300.

[35] Musiela, M. and Zariphopoulou, T., Derivative pricing, investment management and the term structure of exponential utilities:The case of binomial model, In: CarmonaR. (ed.), Indifference Pricing, Princeton University Press, 2009, 3-41.

[36] Musiela, M. and Zariphopoulou, T., Portfolio choice under dynamic investment performance criteria, Quantitative Finance, 2009, 9(2): 161-170.

[37] Musiela, M. and Zariphopoulou, T., Portfolio choice under space-time monotone performance criteria, SIAM Journal on Financial Mathematics, 2010, 1: 326-365.

[38] Musiela, M. and Zariphopoulou, T., Stochastic partial differential equations and portfolio choice, In: ChiarellaC. and NovikovA. (eds.), Contemporary Quantitative Finance, Springer, Berlin, Heidelberg, 2010, 195-215.

[39] Nadtochiy, S. and Tehranchi, M., Optimal investment for all time horizons and Martin boundary of space-time diffusions, Mathematical Finance, 2017, 27(2): 438-470.

[40] Nadtochiy, S. and Zariphopoulou, T., A class of homothetic forward investment performance processes with non-zero volatility, In: KabanovY., RutkowskiM. and ZariphopoulouT. (eds.), Inspired by Finance, Springer, 2014, 475-505.

[41] Ng, K. T. H. and Chong, W. F., Optimal investment in defined contribution pension schemes with forward utility preferences, Insurance: Mathematics and Economics, 2024, 114: 192-211.

[42] Schied, A., Robust optimal control for a consumption-investment problem, Mathematical Methods of Operations Research, 2008, 67(1): 1-20.

[43] Shkolnikov, M., Sircar, R. and Zariphopoulou, T., Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations, SIAM Journal on Financial Mathematics, 2016, 7: 588-618.

[44] Strub, M. and Zhou, X., Evolution of the Arrow-Pratt measure of risk-tolerance for predictable forward utility processes, Finance and Stochastics, 2021, 25(2): 331-358.

[45] Tevzadze, R., Toronjadze, T. and Uzunashvili, T., Robust utility maximization for a diffusion market model with misspecified coefficients, Finance and Stochastics, 2013, 17(3): 535-563.

[46] Yang, Z., Liang, G. and Zhou, C., Constrained portfolio-consumption strategies with uncertain parameters and borrowing costs, Mathematics and Financial Economics, 2018, 13(3): 393-427.

[47] Zariphopoulou, T. and Žitković G., Maturity-independent risk measures, SIAM Journal on Financial Mathematics, 2010, 1: 266-288.

[48] Žitković G., A dual characterization of self-generation and exponential forward performances, The Annals of Applied Probability, 2009, 19(6): 2176-2210.