A Homogeneous Distribution Problem with Applications to Finance

We consider the problem of determining the cumulative distribution function and/or moments of the optimal solution value of a nonlinear program dependent upon a single random variable. This problem is difficult computationally because one must in effect determine the optimal solution to an infinite number of nonlinear programs. Bereanu [Bereanu, B., G. Peeters. 1970. A `Wait-and-See' problem in stochastic linear programming. An experimental computer code. Cashiers Centre Etudes Rech. Oper. 12 (3) 133-148.] has provided an algorithm to solve the distribution problem in the linear case based on extensions of the methods of parametric linear programming. (See also [Bereanu, B. 1967. On stochastic linear programming, distribution problems: stochastic technology matrix. Z. f. Wahrscheinlichkeitstheorie u. oerw. Gerbieter 8 148-152; Bereanu, B. 1971. The distribution problem in stochastic linear programming: the Cartesian integration method. Center of Mathematical Statistics of the Academy of RSR, Bucharest, 71-103 (mimeographed); Bereanu, B. 1970. Renewal processes and some stochastic programming problems in economics. SIAM J. Appl. Math. 19 308-322; Bereanu, B. 1973. The Cartesian integration method in stochastic linear programming. L. Collatz, W. Wetterlink, eds. Numerische Methoden bei Optimierungsaufgaben. Springer-Verlag Publishing Co., Inc., Basel; Prekopa, A. 1966. On the probability distribution of the optimum of a random linear program. SIAM J. Control 4 211-222.] for the analysis of more general linear programs.) This paper presents an extremely simple algorithm to solve the problem in the special case when all functions in the nonlinear program are homogeneous. In this instance the infinite class of optimal solutions are known linear homogeneous transformations of the optimal solution to a single nonlinear program. The distribution function may then be determined by substitution of an easily calculated variable into the distribution function of the random variable. The results are useful in the solution and analysis of a number of financial optimization problems. Problems from the analysis of optimal capital accumulation and portfolio separation are treated in some detail.