Chapter 65

Stochastic Integration65.1 Integrals Of Elementary Processes

Stochastic integration starts with a Q Wiener process having values in a separable Hilbertspace U . Thus it satisfies the following definition.

Definition 65.1.1 Let W (t) be a stochastic process with values in U, a real separableHilbert space which has the properties that t →W (t,ω) is continuous. Whenever t1 <t2 < · · ·< tm, the increments {W (ti)−W (ti−1)} are independent, W (0) = 0, and whenevers < t,

L (W (t)−W (s)) = N (0,(t− s)Q)

which means that whenever h ∈ H,

L ((h,W (t)−W (s))) = N (0,(t− s)(Qh,h))

AlsoE ((h1,W (t)−W (s))(h2,W (t)−W (s))) = (Qh1,h2)(t− s) .

Here Q is a nonnegative trace class operator. Recall this means

Q =∞

∑i=1

λ iei⊗ ei

where {ei} is a complete orthonormal basis, λ i ≥ 0, and

∞

∑i=1

λ i < ∞

Such a stochastic process is called a Q Wiener process.

Recall that such Wiener processes are always of the form

∞

∑k=1

ψk (t)Jgk

where J is a Hilbert Schmidt operator from a suitable space U0 to U and the ψk are realindependent Wiener processes described earlier. This follows from Theorem 64.5.4 whereyou let U0 ⊆U be such that for J the inclusion map, Jek =

√λ kek for Q = ∑k λ kek⊗ ek,

the ek an orthonormal set in U . Thus

(Qx,y) =

(∑k

λ kek (x,ek) ,y

)= ∑

k

(x,√

λ kek

)(y,√

λ kek

)= ∑

k(x,Jek)(y,Jek) = ∑

k(J∗x,ek)(J∗y,ek) = (J∗x,J∗y) = (JJ∗x,y)

so it follows that Q = JJ∗. Of course in finite dimensions, there is no issue because theidentity map is Hilbert Schmidt.

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