# Portfolio optimization

The problem of optimally choosing a distribution of available wealth over various investment opportunities may be studied in several settings. One may consider static problems, in which a decision is made once and for all, or dynamic problems, in which rearrangements are possible in the course of time. The latter may be formulated either in discrete time or in continuous time, on a finite interval or on an infinite interval. The characteristics of the available investment opportunities are typically described in stochastic terms, and here many options are open with respect to the distributions that are used and the behaviour in time; in continuous time one may use models in which asset prices follow continuous paths or models in which prices can be subject to jumps. One may or may not include transaction costs, position limits, and other frictions and side constraints in the problem formulation. Finally, a specification has to be given of the criterion that will be used to compare investment strategies. A frequently used criterion is expected utility, that is, the expected value of a utility function of portfolio value, summed or integrated over time as appropriate. Other criteria may be applied as well however, for instance relating to asymptotic properties of portfolio value, or to worst-case behaviour with respect to a given class of probability measures.

The work of H. Markowitz [a8] is usually viewed as the starting point of modern portfolio theory. Markowitz considered static problems and called a portfolio mean-variance efficient if it achieves a given mean with minimal variance (cf. also Dispersion; Average). Such portfolios may be found by solving a quadratic programming problem. Early results on portfolio problems in continuous time with criteria of the expected utility type were obtained by R.C. Merton [a9], [a10], who used the method of dynamic programming. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. This martingale approach was developed in [a11], [a4], [a1]. An important technical achievement, needed to overcome difficulties associated with limited regularity of value functions, has been the introduction of viscosity solutions [a6], [a7].

#### References

[a1] | J. Cox, C.F. Huang, "Optimal consumption and portfolio policies when asset prices follow a diffusion process." J. Economic Th. , 49 (1989) pp. 33–83 |

[a2] | W.H. Fleming, R.W. Rishel, "Deterministic and stochastic optimal control" , Springer (1975) |

[a3] | W.H. Fleming, H.M. Soner, "Controlled Markov processes and viscosity solutions" , Springer (1993) |

[a4] | I. Karatzas, J.P. Lehoczky, S.E. Shreve, "Optimal portfolio and consumption decisions for a small investor on a finite horizon" SIAM J. Control Optim. , 27 (1987) pp. 1157–1186 |

[a5] | R. Korn, "Optimal portfolios. Stochastic models for optimal investment and risk management in continuous time" , World Sci. (1997) |

[a6] | P.L. Lions, "Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. Part 1: The dynamic programming principle and applications" Commun. Partial Diff. Eqs. , 8 (1983) pp. 1101–1174 |

[a7] | P.L. Lions, "Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. Part 2: Viscosity solutions and uniqueness" Commun. Partial Diff. Eqs. , 8 (1983) pp. 1229–1276 |

[a8] | H. Markowitz, "Portfolio selection" J. Finance , 7 (1952) pp. 77–91 |

[a9] | R.C. Merton, "Lifetime portfolio selection under uncertainty: The continuous case" Rev. Economical Statist. , 51 (1969) pp. 247–257 |

[a10] | R.C. Merton, "Optimum consumption and portfolio rules in a continuous time model" J. Economic Th. , 3 (1971) pp. 373–413 |

[a11] | S.R. Pliska, "A stochastic calculus model of continuous trading: Optimal portfolios" Math. Operat. Res. , 11 (1986) pp. 371–382 |

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Portfolio optimization.

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