The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 226 Wednesday, February 4, 2004 12:50 PM

226 The Mathematics of Financial Modeling and Investment Management

assumed that it is possible to associate to each moment t a σ-algebra of

events ℑt ⊂ ℑ formed by all events that are known prior to or at time t. It

is assumed that events are never “forgotten,” i.e., that ℑt ⊂ ℑs, if t < s.

An increasing sequence of σ-algebras, each associated to the time at

which all its events are known, represents the propagation of informa-

tion. This sequence (called a filtration) is typically indicated as ℑt.

The economy is therefore represented as a probability space (Ω,ℑ,P)

equipped with a filtration {ℑt}. The key point is that every process Xt(ω)

that represents economic or financial quantities must be adapted to the

filtration {ℑt}, that is, the random variable Xt(ω) must be measurable

with respect to the σ-algebras ℑt. In simple terms, this means that each

event of the type Xt(ω) ≤ x belongs to ℑt while each event of the type

Xs(ω) ≤ y for t ≤ s belongs to ℑs. For instance, consider a process Pt(ω)

which might represent the price of a stock. Any coherent representation

of the economy must ensure that events such as {ω: Ps(ω) ≤ c} are not

known at any time t < s. The filtration {ℑt} prescribes all events admissi-

ble at time t.

Why do we have to use the complex concept of filtration? Why can’t

we simply identify information at time t with the values of all the vari-

ables known at time t as opposed to identifying a set of events? The

principal reason is that in a continuous-time continuous-state environ-

ment any individual value has probability zero; we cannot condition on

single values as the standard definition of conditional probability would

become meaningless. In fact, in the standard definition of conditional

probability (see Chapter 6) the probability of the conditioning event

appears in the denominator and cannot be zero.

It is possible, however, to reverse this reasoning and construct a fil-

tration starting from a process. Suppose that a process Xt(ω) does not

admit any anticipation of information, for instance because the Xt(ω)

are all mutually independent. We can therefore construct a filtration ℑt

as the strictly increasing sequence of σ-algebras generated by the process

Xt(ω). Any other process must be adapted to ℑt.

Let’s now go back to the definition of the Brownian motion. Sup-

pose that a probability space (Ω,ℑ,P) equipped with a filtration ℑt is

given. A one-dimensional standard Brownian motion is a stochastic

process Bt(ω) with the following properties:

■ Bt(ω) is defined over the probability space (Ω,ℑ,P).

■ Bt(ω) is continuous for 0 ≤ t < ∞.

■ B 0 (ω) = 0.

■ Bt(ω) is adapted to the filtration ℑt.

■ The increments Bt(ω) –Bs(ω) are independent and normally distributed

with variance (t–s) and zero mean.