The Mathematics of Financial Modelingand Investment Management

7-Optimization Page 213 Wednesday, February 4, 2004 12:50 PM

Optimization 213

such displacements from the optimal ysuch that yε = y), the deriv-

ative of Jwith respect to εmust vanish for ε= 0. ε=^0

If we compute this derivative, we arrive at the following differential

equation that must be satisfied by the optimal solution y

∂Fxyy( ′ ,, ) d∂Fxyy( ′ ,, )

-----------------------------– ------- ----------------------------- = 0

∂y dx ∂y′

First established by Leonard Euler in 1744, this differential equation is

known as the Euler equation or the Euler-Lagrange equation.^6

Though fundamental in the physical sciences, this formulation of

variational principles, is rarely encountered in finance theory. In finance

theory, as in engineering, one is primarily interested in controlling the

evolution of a process. For instance, in investment management, one is

interested in controlling the composition of a portfolio in order to attain

some objective. This is the realm of control theory. Let’s now define con-

trol theory in a deterministic setting. The following section will discuss

stochastic programming—a computational implementation of control

theory in a stochastic setting.

Consider a dynamic process which starts at a given initial time t 0 and

ends at a given terminal time t 1. Let’s suppose that the state of the system is

described by only one variable x(t) called the state variable. The state of the

system is influenced by a set of control variables that we represent as a vec-

tor u(t) = [u 1 (t),...,un(t)]. The control vector must lie inside a given subset of

a Euclidean r-dimensional space, Uwhich is assumed to be closed and time-

invariant. An entire path of the control vector is called a control. A control

is admissible if it stays in Uand satisfies some regularity conditions.

The dynamics of the state variables are specified through the differ-

ential equation

dx

-------= f 1 [xt(), u () t]

dt

where f 1 is assumed to be continuously differentiable with respect to

both arguments. Suppose that the initial state is given but the terminal

state is unrestricted.

The problem to be solved is that of maximizing the objective func-

tional:

(^6) Lagrange himself attributed the equation to Euler.