112 7 A Primer in Stochastic Integration
Cash is convenient to transform money from one date to another. We assume
that there are no costs in carrying cash and that interest rate is zero. This
results in a constant price process whose value is 1. If interest rate is not
zero then we introduce a process that describes the cumulative value of an
account earning the instantaneous riskless interest rate and we renormalise
all the prices by dividing them by this process. This is exactly what we did
in Chap. 2. The choice of a convenient num ́eraire depends on the application,
and the change of num ́eraire is an important technique in finance. In Chap. 2
we introduced the reader to this technique. We will come back to this later
and we will show in Chap. 11 in what cases the change of num ́eraire can be
performed without distorting the model. For the moment we suppose that a
traded asset has been chosen as num ́eraire and that all prices are expressed
in units of this num ́eraire. We will therefore not need the process indexed by
the number 0 which is simply identically equal to one. The priceStiat time
t∈Tis part of the information available at timet.Inmathematicalterms
we translate this by the statement that the processesSiare adapted, i.e.,
StiisFt-measurable, for eacht∈T. The filtration (Ft)t∈Tis not necessarily
generated by the processS. This means that other sources of information
than prices can be observed (e.g. balance sheets, weather conditions, ...). All
agents have access to the same filtration, i.e. information. Agents can buy and
sell assets, short selling is allowed. There arenotransaction costs. In buying
and selling assets, only information available from the past is to be used. We
cannot buy 100.000 shares of some stock conditionally on the event that the
price next year will be doubled.
The spaceRdwill be endowed with the usual Euclidean structure. The
inner product of two vectorsxandyinRd, written as (x, y), is to be inter-
preted as (x, y)=x^1 y^1 +···+xdyd. In some cases we will simply writexy.We
do not put a dot since this is reserved for stochastic integration. The norm
of a vectorxinRdis written as|x|, reserving the notation‖.‖for norms in
Lp-spaces etc.
7.2 Introductory on Stochastic Processes
The following notation, coming from probability theory, will be used. We write
T=T∪+∞.AmapT:Ω→Tis called astopping timeif for allt∈Tthe
set{T≤t}∈Ft.ForT 1 ≤T 2 , two stopping times, we denote by [[T 1 ,T 2 ]] t h e
set{(t, ω)|t∈T,ω∈ΩandT 1 (ω)≤t≤T 2 (ω)}. Other stochastic intervals
are denoted in an analogous way: ]]T 1 ,T 2 ]] , [[T 1 ,T 2 [[ , ]]T 1 ,T 2 [[. Remark that for
T=R+the interval [[0,∞]] denotesR+×Ω and not [0,∞]×Ω. The symbol
πdenotes the projectionπ:R+×Ω→Ω.
To avoid technical complications we suppose that in continuous time, the
filtration (Ft)tsatisfies theusual conditions:
(i) for alltwe have:Ft=
⋂
s>tFs(right continuity),