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For my thesis on a Bayesian sampling routine for a modification on arbitrage-free Nelson-Siegel I came across an equation that involves a matrix exponential within an integral, i.e.

$\int_{0}^{\Delta t} e^{-K*s}\Sigma \Sigma' e^{-(K)'*s}ds = Q$

where I need to extract $\Sigma$ $(3 \times 3)$ from the integral. Here, $s$ is a scalar and $K$ $(3 \times 3)$ and $Q$ $(3 \times 3)$ are known.

In one model the off-diagonal elements of all are set to zero and algebraic expressions may easily be obtained. I however seek to derive the elements of $\Sigma$ for a more general case, where $\Sigma$ is a lower triangular matrix, K is a non-symmetric matrix of state dynamics with positive real eigenvalues and Q is a non-diagonal covariance matrix. Additionally, $\Sigma$ and $e^{-K*s}$ do not commute. Numerical methods would also be more than welcome.

Kind regards,

Gert

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  • $\begingroup$ What is the connection between $K$ and $\Sigma$? $\endgroup$ – Taylor May 10 at 16:00
  • $\begingroup$ @Taylor Hi Taylor, thank you for your reply. To be specific, I am looking at a three factor latent state Ornstein-Uhlenbeck process under the P-measure, where $\text{d}X_{t} = K( \theta^{P} - X_{t} ) \text{d}t + \Sigma \text{d}W_{t}^{P}$. $\endgroup$ – Gert van Dasler May 10 at 20:11
  • $\begingroup$ Do your latent states in this case represent things like dynamic slope and curvature of the yield curve? I'm more familiar with these kinds of models in discrete time and using things like Kalman filtering. Instead of make the state dynamics correlated you could probably just change $K$ and rotate the states, and then adjust the "observation" equation. By the way, where does this integral equation pop up? $\endgroup$ – Taylor May 11 at 1:34
  • $\begingroup$ @Taylor Yes, in traditional Dynamic Nelson-Siegel they do correspond to level, slope and curvature. I am however using the adjusted base from Nyholm (2018), such that the factor interpretation of [Short Rate, Positive Slope, Curvature] emerges in Rotated Nelson-Siegel (AFRNS). For this model I derived an arbitrage-free specification, (AFRNS), but the exclusion of arbitrage is maintained by this relation between the risk-neutral covariance matrix, $\Sigma$, and the real-world covariance matrix, $Q$. Moreover, I am using KF as well, but this condition is required (Christensen (2011)). $\endgroup$ – Gert van Dasler May 11 at 7:23
  • $\begingroup$ @Taylor I think that the problem is only explicitly solvable by somehow making the matrices commute or, stronger, diagonalization. For the diagonalization-case the method works, as I could easily write the diagonal elements of $\Sigma$ as function of the diagonal elements of $K$ and $Q$. $\endgroup$ – Gert van Dasler May 11 at 7:27

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