The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 442 Wednesday, February 4, 2004 1:08 PM

442 The Mathematics of Financial Modeling and Investment Management

Each security i is characterized by a payoff-rate process δti and by a

price process Si i

t. In this continuous-state setting, δt

i and S

t are real vari-

ables with a continuous range such that δi t() ω and Si t() ω are, respectively,

the payoff-rate and the price of the i-th asset at time t, 0 ≤t ≤T and in state

ω ∈Ω. Note that δti represents a rate of payoff and not a payoff as was the

case in the discrete-time setting. The payoff-rate process must be inter-

preted in the sense that the cumulative payoff of each individual asset is

t

Dti = ∫δi s sd

0

We assume that the number of assets is finite. We can therefore use

the vector notation to indicate a set of processes. For example, we write

δt and St to indicate the vector process of payoff rates and prices respec-

tively. Following Chapter 6, all payoff-rates and prices are stochastic

processes adapted to the filtration ℑ. One can make assumptions about

the price and the payoff-rate processes. For example, it can be assumed

that price and payoff-rate processes satisfy a set of stochastic differen-

tial equations or that they exhibit finite jumps. Later in this chapter we

will explore a number of these processes.

As explained in Chapter 6, conditional expectations are defined as

partial averaging. In fact, given a variable Xs, s > t, its conditional

expectation Et[Xs] is defined as a variable that is ℑt-measurable and

whose average on each set A ∈ ℑt is the same as that of X:

Yt = Et[Xs] ⇔EY[ t()ω] = EX[ s ()ω]

for ω ∈ A, ∀A ∈ ℑt and Y is ℑt-measurable.

The law of iterated expectations applies as in the finite-state case:

Et[Eu(Xs)] = Et[Xs]

In a continuous-state setting, conditional expectations are variables

that assume constant values on the sets of infinite partitions. Imagine

the evolution of a variable X. At the initial date, X 0 identifies the entire

space Ω. At each subsequent date t, the space Ω is partitioned into an

infinite number of sets, each determined by one of the infinite values of

Xt.^1 However, these sets have measure zero. In fact, they are sets of the

(^1) One can visualize this process as a tree structure with an infinite number of branch-
es and an infinite number of branching points. However, as the number of branches
and of branching points is a continuum, intuition might be misleading.