7
A Primer in Stochastic Integration
7.1 The Set-up .............................................
In the previous chapters we mainly developed the arbitrage theory for mod-
els in finite discrete time. In the setting of the previous chapter, where the
probability space was not finite, several features of infinite dimensional func-
tional analysis played a role. When trading takes place in continuous time
the difficulties increase even more. It is here that we need the full power of
stochastic integration theory. Before giving precise definitions, let us give a
short overview of the different models and of their mutual relation. In financial
problems the following concepts play a dominant role:
(i) assets to be traded
(ii) trading dates
(iii) trade procedures
(iv) uncertainty
We assume for the moment that the set oftrading datesTis a subset ofR+.
The following cases are of particular importance
(i) T={ 0 , 1 ,...,n}finite discrete time
(ii) T=N={ 0 , 1 ,...}infinite discrete time
(iii)T=[0,1] finite horizon continuous time
(iv)T=R+=[0,∞[ infinite horizon continuous time
Theuncertaintyis modelled using a filtered probability space (Ω,(Ft)t∈T,P).
The filtration (Ft)t∈Tis formed by an increasing family of sub-σ-algebras
ofF∞where (Ω,F∞,P) is a probability space. The role of the filtration is
very important since it describes the information available at each timet.
We suppose that there are finitely manyassets, indexed byi=1,...,d.An
asset to be traded is described by a stochastic processSi:T×Ω→R.The
collection of assets is therefore described by a finite dimensional stochastic
processS:T×Ω→Rd. There is no need to suppose that prices are positive.
It is also understood that there is an asset number 0, that describes “cash”.