Using Girsanov’s Theorem to Turn a Gaussian Process into Brownian Motion

Summary

The document formulates a Gaussian process as a combination of two independent Brownian motions plus a drift involving a Vasicek short-rate process and a deterministic volatility-related function. It asks how to choose a new probability measure under which the resulting process is Brownian motion. The proposed route is Girsanov’s theorem: shift the drift through a change of measure and verify a sufficient integrability condition, here expressed as Novikov’s condition over a finite horizon.

The material supplies the process and the condition to be checked, but no derivation establishing that the condition holds for the stated Vasicek dynamics. Its usefulness is therefore mainly as a setup for measure-change analysis. Whether Novikov’s condition is satisfied depends on the behavior of the drift integral and the model parameters; stating the condition alone does not establish existence of the desired measure. The question also sits in mathematical finance rather than presenting a trading strategy or empirical result.

Key ideas

• The process combines a drift term with a weighted sum of independent Brownian motions.
• Girsanov’s theorem can remove an appropriate drift under a changed probability measure.
• Novikov’s condition is presented as a sufficient check for the proposed measure change.
• The document does not prove that the condition holds for the specified Vasicek-based drift.
• The setup concerns stochastic modeling rather than empirical trading performance.

Tags

# On the construction of a Brownian motion from a Gaussian process


# On the construction of a Brownian motion from a Gaussian process












Let $X$ a Gaussian process defined by $$ X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T] $$ Where $\sigma>0$, $T>0$, $\rho\in]-1,+1[$, $Z_1$ and $Z_2$ two independent Brownian motion defined on the same probability space, $\sigma_P(s,T)$ a deterministic function and $r_t$ a process whose dynamics are described by the Vasicek model.

My problem is to define a new probability measure under which the process $X$ is a Brownian motion. The theorem Girsanov allows this construction when the Novikov condition is checked, namely: $$ E\left(\exp\left(\frac{1}{2}\int_{0}^{T}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)^2\mathrm{d}s) \right)\right)<\infty $$ I would like your opinion on this issue, thank you in advance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.