The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 138 Friday, March 12, 2004 12:39 PM

----------------------------------

138 The Mathematics of Financial Modeling and Investment Management

Fourier transforms are linear operators. The Fourier transform of

the convolutions is the product of Fourier transforms; the Fourier trans-

form of derivatives and integrals have similar properties to the Laplace

transform.

CALCULUS IN MORE THAN ONE VARIABLE

The previous concepts of calculus can be extended in a multivariate envi-

ronment, that is, they can be extended to functions of several variables.

Given a function of nvariables, y= f(x 1 ,...,xn), we can define n partial

derivatives

∂fx( 1 , , ...xn )

∂xi

i= 1,...,nholding constant n– 1 variables and then using the definition

for derivatives of univariate functions:

∂fx( 1 , , ...xn) fx( 1 , , ...xi+ h, , ...xn)– fx( 1 , ,, , ...xi ...xn)

---------------------------------- = lim ----------------------------------------------------------------------------------------------------------------

∂xi h→^0 h

Repeating this process we can define partial derivatives of any order.

Consider, for example, the following function of two variables:

2

fxy ) = e

• (x^2 + σxy y )
  ( ,

Its partial derivatives up to order 2 are given by the following formulas

∂f +^2

------ = –( 2 x+ σy)e

• (x^2 + σxy y )
  ∂x

∂f +^2

------= –( 2 y+ σx)e

• (x^2 + σxy y )
  ∂y

∂^2 f –(x^2 + σxy y^2 )^2

+ ( 2 x+ σy)^2 e –(x + σxy y

+^2 )

---------= – 2 e +^

∂x

2