Covariance of the Vasicek Process at Different Times

Summary

The question asks how to derive the covariance between values of a mean reverting Vasicek process at two times. The answer first expresses the process as the solution to its stochastic differential equation: its initial level and long run mean terms are deterministic, while the random component is an integral of an exponentially decaying kernel against Brownian motion.

Because deterministic terms do not affect covariance, the calculation reduces to the covariance of the stochastic integral at the two times. The answer suggests finding that covariance for arbitrary time points and then applying covariance properties to the requested pair. It provides a route to the derivation rather than the final formula, and the question’s displayed process appears to need correction to the standard integral form. No numerical example or empirical evidence is given.

Key ideas

• The Vasicek process separates into deterministic terms and a stochastic integral driven by Brownian motion.
• Deterministic components do not contribute to covariance.
• The covariance calculation reduces to evaluating the covariance of the stochastic integral at two time points.
• Once the general two time covariance is known, it can be applied to the requested shifted times.
• The response outlines the derivation but does not state the resulting covariance formula.

Tags

# Covariance of mean-reverting Vasicek process?


# Covariance of mean-reverting Vasicek process?












I am dealing with a mean-reverting Vasicek process defined as:

\begin{equation} S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t \end{equation}

I want to determine the following covariance:

\begin{equation} Cov[(S_{t+i}),(S_{t})] \end{equation}

Could someone help me with the analytical derivation? Thanks in advance!

## Answer by ir7 (score 0, accepted)

https://quant.stackexchange.com/a/58462

Hint: We need to start with SDE:

\begin{equation} S_t = S_0 e^{-at} + b(1-{\rm e}^{-at}) + \sigma \int_0^t {\rm e}^{-a(t-u)}\; dW_u \end{equation}

As first two terms are deterministic, using standard properties of covariance, computation of $$ {\rm cov} (S_{t_1}, S_{t_2}) $$ can be reduced to the computation of

$$ {\rm cov} (Y_{t_1}, Y_{t_2}) $$ where

$$ Y_t = \int_0^t {\rm e}^{au}\; dW_u. $$

Last covariance calculation can be found here.

Edit: Once ${\rm cov} (S_{t_1}, S_{t_2})$ is available for all $t_1$ and $t_2$, covariance properties (on linear combinations) can then be used again to answer the original question.

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.