## Girsanov transformation

Hello,

I would like to use the Girsanov transformation in the Black-Scholes model with infinite time horizon in order to find an equivalent martingale measure. The problem is not the Girsanov transformation itself but the existence of the probability measure.

Here is a more detailed explaination of my problem.
I used LaTex for all kinds of mathematical expressions.

Girsanov transformation:
Let $\left( W_{t} \right)_{t\in[0,\infty)}$ be a standard brownian motion on the filtrated probability space $\left( \Omega,\mathcal{F}, \left( \mathcal{F}_{t} \right)_{t \in [0,\infty)}, \mathbf{P} \right)$. For any $\theta \in \mathbb{R}$ and $\mathcal{F}_{\infty} := \sigma\left(\cup_{t \in [0,\infty)}{\mathcal{F}_{t}}\right)$ let $\mathbf{P}^{\theta}$ be a probability measure on $\left( \Omega, \mathcal{F}_{\infty} \right)$ with ${\frac{\mathrm{d}\mathbf{P}^{\theta}}{\mathrm{d} \mathbf{P}}} \big|_{\mathcal{F}_{t}} = e^{\theta W_{t}-\frac{1}{2}\theta^{2}t}$ for all $t\in[0,\infty)$.
Then, the stochastic process $\left(W_{t}\right)_{t\in[0,\infty}$, which is given by $W_{t} := W_{t}-\theta t$ for $t \in [0,\infty)$, is a standard brownian motion under the probability measure $\mathbf{P}^{\theta}$.

If we use this transformation to find an equivalent martingale measure, we need to show that the measure $\mathbf{P}^{\theta}$ exists.

In the case of a limited time horizon $T>0$, the existence is easy to show: We choose the measure with density $e^{\theta W_{T}-\frac{1}{2}\theta^{2}T}$ relative to $\mathbf{P}$.

But how can we show the existence of the measure $\mathbf{P}^{\theta}$ for an infinite time horizon?

Do we need the density on $\left( \Omega, \mathcal{F}_{\infty} \right)$? Or is there another argument that I don't know?

I would be very greatful if you could give me a hint how to go on.