# How to use Girsanov theorem to prove $\hat{W_t}$ is $\hat{\mathbb P}$-Brownian motion?

Assumptions:

Let $$T > 0$$, and let $$(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$$ be a filtered probability space where $$\mathbb P = \tilde{\mathbb P}$$ (risk-neutral measure) and $$\mathscr F_t = \mathscr F_t^{{W}} = \mathscr F_t^{\tilde{W}}$$ where $$W = \tilde{W} = (\tilde{W_t})_{t \in [0,T]} = ({W_t})_{t \in [0,T]}$$ is standard $$\mathbb P=\tilde{\mathbb P}$$-Brownian motion.

Define forward measure $$\hat{\mathbb P}$$:

$$A_T := \frac{d \hat{\mathbb P}}{d \mathbb P} = \frac{\exp(-\int_0^T r_s ds)}{P(0,T)}$$

It can be shown that $$\exp(-\int_0^t r_s ds)P(t,T)$$ is a $$(\mathscr F_t, \mathbb P)-$$martingale where $$r_t$$ is short rate process and $$P(t,T)$$ is bond price.

We are given that

$$\frac{dP(t,T)}{P(t,T)} = r_t dt + \zeta_t dW_t$$

where $$r_t$$ and $$\zeta_t$$ are $$\mathscr F_t$$-adapted and $$\zeta_t$$ satisfies Novikov's condition. I don't think $$\zeta_t$$ is supposed to represent anything in particular.

Define the stochastic process $$\hat{W} = (\hat{W_t})_{t\in[0,T]}$$ s.t.

$$\hat{W_t} := W_t + \int_0^t -\zeta_s ds$$

Problem:

Use Girsanov Theorem to prove $$\hat{W_t}$$ is standard $$\hat{\mathbb P}$$-Brownian motion.

What I tried:

Since $$\zeta_t$$ satisfies Novikov's condition, $$\int_0^T -\zeta_t dt < \infty$$ a.s. and

$$L_t := \exp(-\int_0^t (-\zeta_s dW_s) - \frac{1}{2} \int_0^t \zeta_s^2 ds)$$

is a $$(\mathscr F_t, \mathbb P)-$$martingale.

By Girsanov Theorem, $$\hat{W_t}$$ is standard $$\mathbb P^{*}$$-Brownian Motion where

$$\frac{d \mathbb P^{*}}{d \mathbb P} = L_T$$

I guess we have that $$\hat{W_t}$$ is standard $$\hat{\mathbb P}$$-Brownian Motion if we can show that

$$L_T = \frac{d \hat{\mathbb P}}{d \mathbb P}$$

I think I was able to show (lost my notes) that $$dL_t = L_t \zeta_t dW_t$$, $$dA_t = A_t \zeta_t dW_t$$ and then $$d(\ln L_t) = d(\ln A_t)$$

From $$d(\ln L_t) = d(\ln A_t)$$, I infer that $$L_t = A_t$$ and hence $$L_T = A_T$$ QED.

Is that right?