Closed-Form Covariance for a Multidimensional Ornstein–Uhlenbeck Process
Summary
The discussion concerns the unconditional covariance of a multidimensional Ornstein–Uhlenbeck process, expressed as an integral involving the drift matrix and the diffusion covariance. It asks how to evaluate that integral in a dynamic Nelson–Siegel or arbitrage-free term-structure setting, and whether the quantities denoted by λ are eigenvalues of the drift matrix.
The responses say that the relevant λ values are eigenvalues of the matrix A, while noting that the same symbol can mean market price of diffusion risk elsewhere in the cited term-structure literature. They point to a matrix eigen-decomposition result and report numerical verification, but do not show the derivation itself. The notes therefore clarify the notation and likely interpretation, but provide limited guidance on implementation details, such as handling repeated eigenvalues or non-diagonalizable matrices.
Key ideas
- The covariance integral combines mean reversion through the drift matrix with constant diffusion covariance.
- In the cited covariance formula, λ denotes eigenvalues of the drift matrix A.
- The same symbol can denote market price of diffusion risk in another part of the term-structure literature.
- The responses refer to numerical verification and related references but do not derive the result.
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# Variance of Multi-Dimensional OU process
# Variance of Multi-Dimensional OU process
I'm trying to implement this model shown here:
http://www.sciencedirect.com/science/article/pii/S0304407611000388
As part of the modelling process I have to calculate the unconditional variance of X see page 10).
$\sigma_R^2=\int_u^t \exp^{-A s}\Sigma \Sigma^T \exp^{-A^T s}ds$
They say they use a closed form result from here
http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDEQFjAA&url=http%3A%2F%2Fwww.markfisher.net%2F~mefisher%2Fpapers%2Fterm_prem.pdf&ei=-6yAU6K_M4ziO4m9gPgK&usg=AFQjCNEeJcmAiEzZbfcrWfTGP2uCP5GMFg&bvm=bv.67720277,d.ZWU
see the eqn directly below eqn (3.10) on p5 sadly I can not understand this result much less transform it in to the DNS/AFDNS framework
However the solution is presented here in the DNS/AFNS fraemwork
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1974033
as
$\int_u^t \exp^{-A s}\Sigma \Sigma^T \exp^{-A^T s}ds=\Lambda \Gamma \Lambda^{-1}$
where the (i,j)th element of $\Gamma$ is $=\frac{\sigma_{ij}}{\lambda_i+ \lambda_j}(1-\exp^{-(\lambda_i + \lambda_j)\delta T})$ where $\sigma_{i,j}$ is the element (i,j) of the covariance matrix ($\Sigma \Sigma^T $) assumed constant, and $\Lambda$ is the eigenvector of $\kappa(s_t):=A$.
Unfortunately the solution I found does not say what the $\lambda$ are. I assume they must be the eigenvalues of A?
Q1.) Can someone please confirm my hunch about the $\lambda$ being the eigenvalues
Q2.) Can someone either show me or point me to a reference where I can see how this derivation is done.
## Answer by user12348 (score 2, accepted)
https://quant.stackexchange.com/a/11429
This interesting question provides excellent links to Dynamic Nelson-Siegel Term Structure Models for interest rates for No Arbitrage and exposes key formulation in an interesting way.
Appendix in p37 of ssrn link says $\lambda$ is market price of diffusion risk. However, in the DNS model the $\lambda$ is eigenvalues of $\kappa$, which then part of covariance matrix elements as you described. See section 5.2 of the ssrn reference for an example. Actually you can the data from Bloomberg for the period and be able to validate their results.
## Answer by ArtificiallyIntelligent (score 0)
https://quant.stackexchange.com/a/45613
- I verified numerically, $\lambda$ is the eigenvalue of $A$.
- Also, if you looked at this paper: http://liu.diva-portal.org/smash/get/diva2:1140151/FULLTEXT02.pdf, equation 11 is a special case of the above general formula in your question. It is clearly showing that $\lambda$ is the eigenvalue.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.