What is the covariance of two correlated Ornstein-Uhlenbeck processes? I was trying correlation(1,2)*Var1^(1/2)*Var2^(1/2), but I am not sure! I took Var1=(sigma1^2/(2*speedofmeanreversion1))*(1-exp(-2*speedofmeanreversion1*dt)) and Var2 accordingly. Thank you.
Answer
Using https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Solution
Xit=(Xi0+∫t0σieaiudBiu)e−ait
and
Xit−E[Xit]=e−ait∫t0σieaiudBiu
and thus :
Cov(X1t,X2t)=E[e−a1t∫t0σ1ea1udB1ue−a2t∫t0σ2ea2udB2u]
and if d⟨B1t,B2t⟩=ρ12dt
Cov(X1t,X2t)=E[e−(a1+a2)t∫t0σ1σ2e(a1+a2)uρ12du]=e−(a1+a2)t∫t0σ1σ2e(a1+a2)uρ12du=σ1σ2ρ12a1+a2(1−e−(a1+a2)t)
If you want to prove the last formula, you will need :
- the fact B2t=ρ12B1t+√1−ρ212B⊥t
- https://en.wikipedia.org/wiki/Quadratic_variation#Martingales
- the fact that 2E[MtNt]=E[(M+N)2t]−E[M2t]−E[N2t]with Mt=∫t0σ1ea1udB1u and Nt=∫t0σ2ea2uρ12dB1u
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