Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-09T21:53:36.695Z Has data issue: false hasContentIssue false
  • English
  • Français

A TRANSFORMATION METHOD FOR SOLVING THE HAMILTON–JACOBI–BELLMAN EQUATION FOR A CONSTRAINED DYNAMIC STOCHASTIC OPTIMAL ALLOCATION PROBLEM

Part of: Parabolic equations and systems Hamiltonian and Lagrangian mechanics Asymptotic theory Mathematical economics

Published online by Cambridge University Press:  10 October 2013

S. KILIANOVÁ  and
D. ŠEVČOVIČ
S. KILIANOVÁ*
Affiliation:
Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia email sevcovic@fmph.uniba.sk
D. ŠEVČOVIČ
Affiliation:
Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia email sevcovic@fmph.uniba.sk
*
✉kilianova@fmph.uniba.sk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We propose and analyse a method based on the Riccati transformation for solving the evolutionary Hamilton–Jacobi–Bellman equation arising from the dynamic stochastic optimal allocation problem. We show how the fully nonlinear Hamilton–Jacobi–Bellman equation can be transformed into a quasilinear parabolic equation whose diffusion function is obtained as the value function of a certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence and uniqueness and derive useful bounds of classical Hölder smooth solutions. Furthermore, we construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit travelling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 index as an example of the application of the method.

Keywords

Hamilton–Jacobi–Bellman equationRiccati transformationquasilinear parabolic equationfinite volume approximation schemetravelling wave solution

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 34E05: Asymptotic expansions 70H20: Hamilton-Jacobi equations 91B16: Utility theory 91B70: Stochastic models 90C15: Stochastic programming

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 1 , July 2013 , pp. 14 - 38
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Abe, R. and Ishimura, N., “Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem”, Proc. Japan Acad. Ser. A 84 (2008) 11–14; doi:10.3792/pjaa.84.11.CrossRefGoogle Scholar
Aubin, J.-P., “Lipschitz behavior of solutions to convex minimization problems”, Math. Oper. Res. 9 (1984) 87–111; doi:10.1287/moor.9.1.87.CrossRefGoogle Scholar
Bagagiolo, F., Bardi, M. and Capuzzo Dolcetta, I., “A viscosity solutions approach to some asymptotic problems in optimal control”, in: Partial differential equation methods in control and shape analysis (eds da Prato, G. and Zolézio, J.-P.), (Dekker, New York, 1997), 29–39.Google Scholar
Bank, B., Guddat, J., Klatte, D., Kummer, B. and Tammer, K., Non-linear parametric optimization (Birkhäuser, Basel, 1983).Google Scholar
Benton, S., The Hamilton–Jacobi equation. A global approach (Academic Press, New York, 1977).Google Scholar
Bertsekas, D. P., Dynamic programming and stochastic control (Academic Press, New York, 1976).Google Scholar
Bielecki, T. R., Pliska, S. R. and Sheu, S.-J., “Risk sensitive portfolio management with Cox–Ingersoll–Ross interest rates: the HJB equation”, SIAM J. Control Optim. 44 (2006) 1811–1843; doi:10.1137/S0363012903437952.CrossRefGoogle Scholar
Browne, S., “Risk-constrained dynamic active portfolio management”, Manag. Sci. 46 (2000) 1188–1199; doi:10.1287/mnsc.46.9.1188.12233.CrossRefGoogle Scholar
Crandall, M. G., Ishii, H. and Lions, P.-L., “User’s guide to viscosity solutions of second order partial differential equations”, Bull. Amer. Math. Soc. 27 (1992) 1–67; doi:10.1090/S0273-0979-1992-00266-5.CrossRefGoogle Scholar
Dai, M., Jiang, L., Li, P. and Yi, F., “Finite horizon optimal investment and consumption with transaction costs”, SIAM J. Control Optim. 48 (2009) 1134–1154; doi:10.1137/070703685.CrossRefGoogle Scholar
Dai, M. and Yi, F., “Finite-horizon optimal investment with transaction costs: a parabolic double obstacle problem”, J. Differential Equations 246 (2009) 1445–1469; doi:10.1016/j.jde.2008.11.003.CrossRefGoogle Scholar
Fleming, W. H. and Soner, H. M., Controlled Markov processes and viscosity solutions (Springer, New York, 1993).Google Scholar
Huang, Y., Forsyth, P. A. and Labahn, G., “Combined fixed point and policy iteration for Hamilton–Jacobi–Bellman equations in finance”, SIAM J. Numer. Anal. 50 (2012) 1861–1882; doi:10.1137/100812641.CrossRefGoogle Scholar
Ishimura, N., Koleva, M. N. and Vulkov, L. G., “Numerical solution via transformation methods of nonlinear models in option pricing”, AIP Conf. Proc. 1301 (2010) 387–394; doi:10.1063/1.3526637.CrossRefGoogle Scholar
Ishimura, N., Koleva, M. and Vulkov, L., “Numerical solution of a nonlinear evolution equation for the risk preference”, in: Numerical methods and applications (eds Dimov, E., Dimova, S. and Kolkovska, N.), (Springer, Berlin, 2011), 445–452.CrossRefGoogle Scholar
Ishimura, N. and Maneenop, S., “Traveling wave solutions to the nonlinear evolution equation for the risk preference”, J. SIAM Lett. 3 (2011) 25–28.CrossRefGoogle Scholar
Ishimura, N. and Ševčovič, D., “On traveling wave solutions to a Hamilton-Jacobi-Bellman equation with inequality constraints”, Japan J. Indust. Appl. Math. 30 (2013) 51–67; doi:10.1007/s13160-012-0087-8.CrossRefGoogle Scholar
Jandačka, M. and Ševčovič, D., “On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile”, J. Appl. Math. 2005 (2005) 235–258; doi:10.1155/JAM.2005.235.CrossRefGoogle Scholar
Karatzas, I., Lehoczky, J. P., Sethi, S. P. and Shreve, S., “Explicit solution of a general consumption/investment problem”, Math. Oper. Res. 11 (1986) 261–294; doi:10.1287/moor.11.2.261.CrossRefGoogle Scholar
Klatte, D., “On the Lipschitz behavior of optimal solutions in parametric problems of quadratic optimization and linear complementarity”, Optimization 16 (1985) 819–831; doi:10.1080/02331938508843080.CrossRefGoogle Scholar
Koleva, M. N., “Iterative methods for solving nonlinear parabolic problem in pension saving management”, AIP Conf. Proc. 1404 (2011) 457–463; doi:10.1063/1.3659948.CrossRefGoogle Scholar
Koleva, M. N. and Vulkov, L., “Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance”, Math. Comput. Model. 57 (2013) 2564–2575; doi:10.1016/j.mcm.2013.01.008.CrossRefGoogle Scholar
Kútik, P. and Mikula, K., “Finite volume schemes for solving nonlinear partial differential equations in financial mathematics”, in: Finite volumes for complex applications VI. Problems & perspectives (eds Fořt, J., Fürst, J., Halama, J., Herbin, R. and Hubert, F.), (Springer, Berlin, 2011), 643–651.CrossRefGoogle Scholar
Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N., Linear and quasilinear equations of parabolic type (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar
LeVeque, R. J., Finite volume methods for hyperbolic problems (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
Macová, Z. and Ševčovič, D., “Weakly nonlinear analysis of the Hamilton–Jacobi–Bellman equation arising from pension savings management”, Intl J. Numer. Anal. Model. 7 (2010) 619–638; http://www.math.ualberta.ca/ijnam/volume-7-2010/no-4-10/2010-04-02.pdf.Google Scholar
Merton, R. C., “Lifetime portfolio selection under uncertainty: the continuous-time case”, Rev. Econ. Stat. 51 (1969) 247–257; doi:10.2307/1926560.CrossRefGoogle Scholar
Merton, R. C., “Optimum consumption and portfolio rules in a continuous time model”, J. Econom. Theory 3 (1971) 373–413; doi:10.1016/0022-0531(71)90038-X.CrossRefGoogle Scholar
Milgrom, P. and Segal, I., “Envelope theorems for arbitrary choice sets”, Econometrica 70 (2002) 583–601; doi:10.1111/1468-0262.00296.CrossRefGoogle Scholar
Musiela, M. and Zariphopoulou, T., “An example of indifference prices under exponential preferences”, Finance Stoch. 8 (2004) 229–239; doi:10.1007/s00780-003-0112-5.CrossRefGoogle Scholar
Muthuraman, K. and Kumar, S., “Multidimensional portfolio optimization with proportional transaction costs”, Math. Finance 16 (2006) 301–335; doi:10.1111/j.1467-9965.2006.00273.x.CrossRefGoogle Scholar
Nayak, S. and Papanicolaou, G., “Market influence of portfolio optimizers”, Appl. Math. Finance 15 (2008) 21–40; doi:10.1080/13504860701269285.CrossRefGoogle Scholar
Peyrl, H., Herzog, F. and Geering, H., “Numerical solution of the Hamilton–Jacobi–Bellman equation for stochastic optimal control problems”, Proc. 2005 WSEAS International Conference on Dynamical Systems and Control 489–497.Google Scholar
Pratt, J. W., “Risk aversion in the small and in the large”, Econometrica 32 (1964) 122–136; doi:10.2307/1913738.CrossRefGoogle Scholar
Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations (Springer, New York, 1984).CrossRefGoogle Scholar
Reisinger, C. and Witte, J. H., “On the use of policy iteration as an easy way of pricing American options”, SIAM J. Financial Math. 3 (2012) 459–478; doi:10.1137/110823328.CrossRefGoogle Scholar
Ševčovič, D., Stehlíková, B. and Mikula, K., Analytical and numerical methods for pricing financial derivatives (Nova Science Publishers, Hauppauge, NY, 2011).Google Scholar
Songzhe, L., “Existence of solutions to initial value problem for a parabolic Monge–Ampère equation and application”, Nonlinear Anal. Theory Meth. Appl. 65 (2006) 59–78; doi:10.1016/j.na.2005.05.047.CrossRefGoogle Scholar
Tourin, A. and Zariphopoulou, T., “Numerical schemes for investment models with singular transactions”, Comput. Econ. 7 (1994) 287–307; doi:10.1007/BF01299457.CrossRefGoogle Scholar
Xia, J., “Risk aversion and portfolio selection in a continuous-time model”, SIAM J. Control Optim. 49 (2011) 1916–1937; doi:10.1137/10080871X.CrossRefGoogle Scholar
Zariphopoulou, T., “Consumption–investment models with constraints”, SIAM J. Control Optim. 32 (1994) 59–85; doi:10.1137/S0363012991218827.CrossRefGoogle Scholar