##
**Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem.**
*(English)*
Zbl 1139.35059

Summary: We are concerned with the solvablity of certain nonlinear partial differential equation, which is derived from the optimal investment problem under the random risk process. The equation describes the evolution of the Arrow-Pratt coefficient of absolute risk aversion with respect to the optimal value function. Employing the fixed point approach combined with the convergence argument we show the existence of solutions.

### MSC:

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

91B28 | Finance etc. (MSC2000) |

91B30 | Risk theory, insurance (MSC2010) |

PDF
BibTeX
XML
Cite

\textit{R. Abe} and \textit{N. Ishimura}, Proc. Japan Acad., Ser. A 84, No. 1, 11--14 (2008; Zbl 1139.35059)

### References:

[1] | R. Abe, Dynamic optimal investment problem in ALM via the theory of partial differential equations, Thesis for the Master-course degree, Graduate School of Economics, Hitotsubashi University, (2006). (in Japanese). |

[2] | S. Browne, Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Math. Operations Research 20 (1995), no. 4, 937-958. · Zbl 0846.90012 |

[3] | F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973), 637-659. · Zbl 1092.91524 |

[4] | L. C. Evans, Partial differential equations , Amer. Math. Soc., Providence, RI, 1998. |

[5] | A. Friedman, Partial differential equations of parabolic type , Krieger Publishing Company, Florida, 1983. · Zbl 0524.14011 |

[6] | D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , Springer Classics in Math., Springer-Verlag, 2001. · Zbl 1042.35002 |

[7] | C. Hipp, Stochastic control with application in insurance, in Stochastic methods in finance , 127-164, Lecture Notes in Math., 1856, Springer, Berlin. · Zbl 1134.91024 |

[8] | H. Imai, N. Ishimura and M. Kushida, Numerical treatment of a singular nonlinear partial differential equation arising in the optimal investment. (Preprint). · Zbl 1213.91172 |

[9] | H. Imai, N. Ishimura, I. Mottate and M. A. Nakamura, On the Hoggard-Whalley-Wilmott equation for the pricing of options with transaction costs, Asia-Pacific Financial Markets 13 (2007), 315-326. · Zbl 1283.91176 |

[10] | R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci. 4 (1973), 141-183. · Zbl 1257.91043 |

[11] | J. W. Pratt, Risk aversion in the small and in the large, Econometrica 32 (1964), 122-136. · Zbl 0132.13906 |

[12] | P. Wilmott, S. Howison and J. Dewynne, The mathematics of financial derivatives , Cambridge Univ. Press, Cambridge, 1995. · Zbl 0842.90008 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.