Friday, January 19, 2018

correlation - What is the covariance of two correlated Ornstein-Uhlenbeck processes?


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)eait


and


XitE[Xit]=eaitt0σieaiudBiu


and thus :


Cov(X1t,X2t)=E[ea1tt0σ1ea1udB1uea2tt0σ2ea2udB2u]


and if dB1t,B2t=ρ12dt


Cov(X1t,X2t)=E[e(a1+a2)tt0σ1σ2e(a1+a2)uρ12du]=e(a1+a2)tt0σ1σ2e(a1+a2)uρ12du=σ1σ2ρ12a1+a2(1e(a1+a2)t)


If you want to prove the last formula, you will need :



  • the fact B2t=ρ12B1t+1ρ212Bt


  • 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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