2208 CHAPTER 64. WIENER PROCESSES

64.6 Wiener Processes, Another Approach64.6.1 Lots Of Independent Normally Distributed Random VariablesYou can use the Kolmogorov extension theorem to prove the following corollary. It isCorollary 59.20.3 on Page 1936.

Corollary 64.6.1 Let H be a real Hilbert space. Then there exist real valued randomvariables W (h) for h ∈H such that each is normally distributed with mean 0 and for everyh,g,(W ( f ) ,W (g)) is normally distributed and

E (W (h)W (g)) = (h,g)H

Furthermore, if {ei} is an orthogonal set of vectors of H, then {W (ei)} are independentrandom variables. Also for any finite set { f1, f2, · · · , fn} ,

(W ( f1) ,W ( f2) , · · · ,W ( fn))

is normally distributed.

Corollary 64.6.2 The map h→W (h) is linear. Also, {W (h) : h ∈ H} is a closed subspaceof L2 (Ω,F ,P) where F = σ (W (h) : h ∈ H).

Proof: This follows from the above description.

E([W (g+h)− (W (g)+W (h))]2

)= E

(W (g+h)2

)+E((W (g)+W (h))2

)−2E (W (g+h)(W (g)+W (h)))

= |g+h|2 + |g|2 + |h|2 +2(g,h)−2(g+h,g)−2(g+h,h)

= |g|2 + |h|2 +2(g,h)++2(g,h)+ |g|2

+ |h|2−2 |g|2−2(g,h)−2(g,h)−2 |h|2 = 0

Hence W (h+g) =W (g)+W (h).

E((W (α f )−αW ( f ))2

)= E

(W (α f )2

)+E

(α

2W ( f )2)−2E (W (α f )αW ( f ))

= α2 | f |2 +α

2 | f |2−2α (α f , f ) = 0.

Why is {W (h) : h ∈ H} a subspace? This is obvious because W is linear. Why is itclosed? Say W (hn)→ f ∈ L2 (Ω) . This requires that {hn} is a Cauchy sequence. Thushn→ h and so

E(| f −W (h)|2

)≤ 2

[limn→∞

E(| f −W (hn)|2

)+E

(|W (hn)−W (h)|2

)]= 2 lim

n→∞E(|W (hn)−W (h)|2

)= 2 lim

n→∞|hn−h|2H = 0

and so f =W (h) showing that this is indeed a closed subspace.Next is a technical lemma which will be of considerable use.