2326 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION

Here Bt is just one dimensional Wiener process. You would want it to equal

2∫ 1

0

∫ t

0dBsdBt = 2

∫ 1

0BtdBt

So what should this equal? Let F (x) = x2 so F ′ (x) = 2x,F ′′ (x) = 2. Consider F (Bt) .Then using the formalism for the Ito formula,

dF (Bt) = d(B2

t)= 2BtdBt +

12(2)(dBt)

2

= 2BtdBt +dt

Therefore,

B2t = 2

∫ t

0BsdBs + t

and letting t = 1,12

B21−

12=∫ 1

0BsdBs =

∫ 1

0

∫ s

0dBrdBs

and so we would want to have

B21−1 = 2

∫ 1

0

∫ s

0dBrdBs

and we want this to equal∫ 1

0∫ 1

0 dBsdBt so we need to be defining this in a way such that thiswill result. Of course, this is just the simplest example of an iterated integral with respectto these one dimensional Wiener processes.

Now partition [0,1) as 0= t0 < t1 < · · · , tn = 1. Then sum over all [ti−1, ti)× [t j−1, t j) butleave out those which are on the “diagonal”. These would be of the form [ti−1, ti)× [ti−1, ti).

Here you would have in the sum products of the form(Bti −Bti−1

)(Bt j −Bt j−1

). Thus you

would have

∑i, j

(Bti −Bti−1

)(Bt j −Bt j−1

)−

n

∑i=1

(Bti −Bti−1

)2

=

(∑

iBti −Bti−1

)2

−n

∑i=1

(Bti −Bti−1

)2

= (B1−B0)2−

n

∑i=1

(Bti −Bti−1

)2

Then of course you take a limit as the norm of the partition goes to 0. This yields in thelimit

B21−1

which is the thing which is wanted. Thus the idea is to consider only functions which areequal to 0 on the “diagonal” and define an integral for these. Then hopefully these will bedense in L2 ([0,T ]n) and the multiple integral can then be defined as some sort of limit.