Efficient portfolio dependent on Cox–Ingersoll–Ross interest rate

G. S. Kambarbaeva   ; O. S. Rozanova

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2013), pp. 3-10

Résumé

We solve a certain problem of portfolio optimization in the case of the asset prices trends depending on the bank interest rate governed by the Cox–Ingersoll–Ross dynamics. This work continues a series of papers where the interest rate is modeled by a linear stochastic differential equation with a constant volatility.

Bibliographie

[1] Oksendal B., Stokhasticheskie differentsialnye uravneniya. Vvedenie v teoriyu i prilozheniya, Mir, M., 2003

[2] Albeverio S., Rozanova O., “The non-viscous Burgers equation associated with random positions in coordinate space: a threshold for blow up behavior”, Math. Models and Methods in Appl. Sci., 19:5 (2009), 1–19 | DOI | MR

[3] Albeverio S., Rozanova O., “Suppression of unbounded gradients in a SDE associated with the Burgers equation”, Proc. Amer. Math. Soc., 138:1 (2010), 241–251 | DOI | MR

[4] Korshunova A., Rozanova O., “On effects of stochastic regularization for the pressureless gas dynamics”, Proc. 13th Int. Conf. “Hyperbolic Problems: Theory, Numerics and Applications”, Ser. Contemp. Appl. Math., 18, eds. S. Jiang, T. Li, World Sci. Publ., 2012, 486–493 | DOI | MR

[5] Martynov M.A., Rozanova O.S., “On dependence of the implied volatility on returns for stochastic volatility models”, Stochastics: Int. J. Probab. and Stochast. Proc., 83 (2012), 541–552 | MR

[6] Bielecki T., Pliska S., “Risk sensitive dynamic asset management”, J. Appl. Math. and Optim., 37 (1999), 337–360 | DOI | MR

[7] Bielecki T., Pliska S., Sherris M., “Risk sensitive asset allocation”, J. Economic Dynamics and Control, 24 (2000), 1145–1177 | DOI | MR

[8] Bielecki T., Pliska S., Sheu S.-J., “Risk sensitive portfolio management with Cox–Ingersoll–Ross interest rates: the HJB equation”, SIAM J. Control and Optim., 44 (2005), 1811–1843 | DOI | MR

[9] Kambarbaeva G.S., “O nekotorykh yavnykh formulakh dlya vychisleniya uslovnykh matematicheskikh ozhidanii sluchainykh velichin i ikh primeneniyakh”, Vestn. Mosk. un-ta. Matem. Mekhan., 2010, no. 5, 10–15 | MR

[10] Kambarbaeva G.S., “Zadacha sostavleniya effektivnogo portfelya v modeli rynka soglasno Beletskomu i Pliske”, Vestn. Mosk. un-ta. Matem. Mekhan., 2011, no. 5, 14–20 | MR

[11] Risken H., The Fokker–Planck equation. Methods of solution and applications, 2nd ed., Springer, N.Y., 1989 | MR

[12] Shiryaev A.N., Veroyatnost-1, 3-e izd., pererab. i dop., MTsNMO, M., 2004 | MR

[13] Cox J.C., Ingersoll J.E., Ross S.A., “A theory of term structure of interest rates”, Econometrica, 53 (1985), 385–407 | DOI | MR

[14] Feller W., “Two singular diffusion problems”, Ann. Math., 54 (1951), 173–182 | DOI | MR

[15] Vasicek O., “On equilibrium characterization of the term structure”, J. Finan. Economics, 5 (1977), 177–188 | DOI

[16] Brennan M.J., Schwartz E.S., Lagnado R., “Strategic asset allocation”, J. Economic Dynamics and Control, 21 (1997), 1377–1403 | DOI | MR

[17] Rey D., Current research topics — stock market predictability: is it there?, Finan. Markets and Portfolio Management, 17 (2003), 379–387

[18] Heston S.L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Finan. Stud., 6 (1993), 327–343 | DOI

[19] Hata N., Sekine J., “Solving long term optimal investment problems with Cox–Ingersoll–Ross interest rates”, Adv. Math. Economics, 8:1 (2006), 231–255 | DOI | MR

[20] Hata N., ““Down-Side Risk” probability minimization problem with Cox–Ingersoll–Ross's interest rates”, Asia-Pacific Financial Markets, 2011 | DOI

[21] Duffie D., Filipović D., Schachermayer W., “Affine processes and applications in finance”, Ann. Apl. Probab., 13 (2003), 984–1053 | DOI | MR