The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 224 Wednesday, February 4, 2004 12:50 PM

224 The Mathematics of Financial Modeling and Investment Management

the Brownian motion. In fact, in the case of continuous paths, we wrote

the approximating functions as follows:

φn( t, ω)= ∑gt( i, ω)[Bi + 1 ()ω – Bi ()ω]

i

taking the function g in the left extreme of each subinterval.

However, the definition of stochastic integrals in the sense of Stra-

tonovich admits anticipation. In fact, the stochastic integral in the sense

of Stratonovich, written as follows:

T

∫ ft(, ω)°dBt()ω

S

uses the following approximation under the assumption of continuous

paths:

φ (n t, ω)= ∑gt(i * , ω)[Bi + 1 ()ω – Bi ()ω]

i

where

ti + 1 – ti

t*i = --------------------

2

is the midpoint of the i-th subinterval.

Whose definition—Itô’s or Stratonovich’s—is preferable? Note that

neither can be said to be correct or incorrect. The choice of the one over

the other is a question of which one best represents the phenomena

under study. The lack of anticipation is one reason why the Itô integral

is generally preferred in finance theory.

We have just outlined the definition of stochastic integrals leaving

aside mathematical details and rigor. The following two sections will

make the above process mathematically rigorous and will discuss the

question of anticipation of information. While these sections are a bit

technical and might be skipped by those not interested in the mathemat-

ical details of stochastic calculus, they explain a number of concepts

that are key to the modern development of finance theory.