# 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.

• do you want to know how to apply Girsanov ? or you are not sure if Novikov condition is verified ? – MJ73550 Apr 19 '16 at 17:24
• Yes, my problem is to check the condition of Novikov. thank you. – M. A. Kacef Apr 19 '16 at 21:44