Monday, 16 November 2020 09:02

Optimization in Hilbert Spaces

Written by B. Okelo Nyaare
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B. Okelo

School of Mathematics and Actuarial Science,

Jaramogi Oginga Odinga University of Science and Technology,

P. O. Box 210-40601, Bondo-Kenya.

Corresponding Author: bnyaare@yahoo.com

Studies on optimization has attracted the attention of many mathematicians and researchers over a long period of time. In this paper, we are concerned with the classical results on optimization of convex functions in infinite-dimensional real Hilbert spaces. The methodology involves the use of Ito’s formula and Black-Scholes Model. The results show that a function f has a first order optimality condition. Moreover, if f is differentiable at a point x*in Rn and ifx* is a local minimum of f, then the delof f(x*) = 0. A simple application involving the Dirichlet problem is also given. In conclusion, with regard to Portfolio Optimization, this study is geared towards applications to particularly stochastic optimization with consideration to: Cox-Ross-Rubinstein model and Hamilton-Jacobi-Bellman Equation.

Keywords: Optimization, Hilbert space, Stochastic analysis, Applications

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