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Consumption in incomplete markets. (English) Zbl 1435.91179

Summary: We develop a method to find approximate solutions, and their accuracy, to consumption-investment problems with isoelastic preferences and infinite horizon, in incomplete markets where state variables follow a multivariate diffusion. We construct upper and lower contractions; these are fictitious complete markets in which state variables are fully hedgeable, but their dynamics is distorted. Such contractions yield pointwise upper and lower bounds for both the value function and the optimal consumption of the original incomplete market, and their optimal policies are explicit in typical models. Approximate consumption-investment policies coincide with the optimal one if the market is complete or utility is logarithmic.

MSC:

91G15 Financial markets
91G10 Portfolio theory
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[1] Barles, G.; Souganidis, P. E., Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal., 4, 271-283 (1991) · Zbl 0729.65077
[2] Bhatia, R., Positive Definite Matrices (2007), Princeton: Princeton University Press, Princeton · Zbl 1133.15017
[3] Bick, B.; Kraft, H.; Munk, C., Solving constrained consumption-investment problems by simulation of artificial market strategies, Manag. Sci., 59, 485-503 (2013)
[4] Campbell, J. Y.; Viceira, L. M., Who should buy long-term bonds?, Am. Econ. Rev., 91, 99-127 (2001)
[5] Campbell, J. Y.; Viceira, L. M., Strategic Asset Allocation: Portfolio Choice for Long-Term Investors (2002), New York: Oxford University Press, New York
[6] Castañeda-Leyva, N.; Hernández-Hernández, D., Optimal consumption-investment problems in incomplete markets with stochastic coefficients, SIAM J. Control Optim., 44, 1322-1344 (2005) · Zbl 1140.91381
[7] Cheridito, P.; Filipović, D.; Yor, M., Equivalent and absolutely continuous measure changes for jump-diffusion processes, Ann. Appl. Probab., 15, 1713-1732 (2005) · Zbl 1082.60034
[8] Cox, J. C.; Huang, C-f., Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory, 49, 33-83 (1989) · Zbl 0678.90011
[9] Cox, J. C.; Ingersoll, J. E.; Ross, S. A., A theory of the term structure of interest rates, Econometrica, 53, 385-407 (1985) · Zbl 1274.91447
[10] Cvitanić, J.; Karatzas, I., Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2, 767-818 (1992) · Zbl 0770.90002
[11] Duffie, D.; Fleming, W. H.; Soner, H. M.; Zariphopoulou, T., Hedging in incomplete markets with HARA utility, J. Econ. Dyn. Control, 21, 753-782 (1997) · Zbl 0899.90026
[12] Dybvig, P. H.; Rogers, L. C.G.; Back, K., Portfolio turnpikes, Rev. Financ. Stud., 12, 165-195 (1999)
[13] Feller, W., Two singular diffusion problems, Ann. Math., 54, 173-182 (1951) · Zbl 0045.04901
[14] Fleming, W. H.; Hernández-Hernández, D., An optimal consumption model with stochastic volatility, Finance Stoch., 7, 245-262 (2003) · Zbl 1035.60028
[15] Fleming, W. H.; Pang, T., An application of stochastic control theory to financial economics, SIAM J. Control Optim., 43, 502-531 (2004) · Zbl 1101.93085
[16] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1998), Berlin: Springer, Berlin · Zbl 0691.35001
[17] Goll, T.; Kallsen, J., A complete explicit solution to the log-optimal portfolio problem, Ann. Appl. Probab., 13, 774-799 (2003) · Zbl 1034.60047
[18] Guasoni, P.; Robertson, S., Portfolios and risk premia for the long run, Ann. Appl. Probab., 22, 239-284 (2012) · Zbl 1247.91172
[19] Guasoni, P.; Wang, G., Consumption and investment with interest rate risk, J. Math. Anal. Appl., 476, 215-239 (2019) · Zbl 1411.91504
[20] Hata, H.; Sheu, S.-J., On the Hamilton-Jacobi-Bellman equation for an optimal consumption problem: I. Existence of solution, SIAM J. Control Optim., 50, 2373-2400 (2012) · Zbl 1252.49026
[21] Hata, H.; Sheu, S.-J., On the Hamilton-Jacobi-Bellman equation for an optimal consumption problem: II. Verification theorem, SIAM J. Control Optim., 50, 2401-2430 (2012) · Zbl 1252.49041
[22] Haugh, M.; Kogan, L.; Wang, J., Evaluating portfolio policies: a duality approach, Oper. Res., 54, 405-418 (2006) · Zbl 1167.91370
[23] He, H.; Pearson, N. D., Consumption and portfolio policies with incomplete markets and short-sale constraints: the infinite dimensional case, J. Econ. Theory, 54, 259-304 (1991) · Zbl 0736.90017
[24] Heath, D.; Schweizer, M., Martingales versus PDEs in finance: an equivalence result with examples, J. Appl. Probab., 37, 947-957 (2000) · Zbl 0996.91069
[25] Horn, R. A.; Johnson, C. R., Matrix Analysis (2013), Cambridge: Cambridge University Press, Cambridge
[26] Jin, X., Consumption and portfolio turnpike theorems in a continuous-time finance model, J. Econ. Dyn. Control, 22, 1001-1026 (1998) · Zbl 0899.90030
[27] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E., Optimal portfolio and consumption decisions for a small investor on a finite horizon, SIAM J. Control Optim., 25, 1557-1586 (1987) · Zbl 0644.93066
[28] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.; Xu, G.-L., Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim., 29, 702-730 (1991) · Zbl 0733.93085
[29] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), New York: Springer, New York · Zbl 0734.60060
[30] Kim, T. S.; Omberg, E., Dynamic nonmyopic portfolio behavior, Rev. Financ. Stud., 9, 141-161 (1996)
[31] Liu, J., Portfolio selection in stochastic environment, Rev. Financ. Stud., 20, 1-39 (2007)
[32] Merton, R. C., Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat., 51, 247-257 (1969)
[33] Merton, R. C., An intertemporal capital asset pricing model, Econometrica, 41, 867-887 (1973) · Zbl 0283.90003
[34] Pohl, W.; Schmedders, K.; Wilms, O., Higher order effects in asset pricing models with long-run risks, J. Finance, 73, 1061-1111 (2018)
[35] Pusz, W.; Woronowicz, S. L., Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys., 8, 159-170 (1975) · Zbl 0327.46032
[36] Revuz, D.; Yor, M., Continuous Martingales and Brownian Motion (2001), New York: Springer, New York
[37] Rogers, L. C.G., Optimal Investment. Springer Briefs in Quantitative Finance (2013), Heidelberg: Springer, Heidelberg
[38] Stroock, D. W.; Varadhan, S. R., Multidimensional Diffusion Processes (2006), Berlin: Springer, Berlin · Zbl 1103.60005
[39] Wachter, J., Portfolio and consumption decision under mean-reverting returns: an exact solution for complete markets, J. Financ. Quant. Anal., 37, 63-91 (2002)
[40] Zariphoupoulou, T., A solution approach to valuation with unhedgeable risks, Finance Stoch., 5, 61-82 (2001) · Zbl 0977.93081
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