Proof standard Brownian Motion under change of measure

Let's split the usual time horizon $$[0,T]$$ like $$0=T_{0} and consider the bond price $$P(t,T_{i})$$ for $$i=1,...,n$$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{i}(t)dB_{t}$$ By Ito we can recall $$P(t,T_{i})=P(0,T)\exp(\int_{0}^{t}r_{s}ds+\int^{t}_{0}\xi_{i}(s)dB_{s}-\frac{1}{2}\int^{t}_{0}|\xi_{i}(s)|^{2}ds)$$ Now, I am supposed to proof using Girsanov I theorem, that the process $$W_{t}^{i}=B_{t}-\int^{t}_{0}\xi_{i}(s)ds$$ is a standard Brownian motion under the forward measure $$Q_{T_{i}}$$ using $$P(t,T)$$ as a numeraire for $$i=1,...,n$$. The question states $$dW_{t}^{i}=dB_{t}-\frac{1}{N_{t}}dN_{t}dB_{t}=...=dB_{t}-\xi_{i}(t)dt$$ "Complete the ... part and look at the $$\frac{1}{N_{t}}dN_{t}$$, what is it and why do we use it here?" I cannot find how Girsanov is used for nond pricing and this expression with the ... part is derived from this?

• It appears that you are referring to a look. However, how can one know which book are you talking about? – Gordon Jun 17 at 13:19
• My professor based his own lecture notes on this document and the problem is on page 494: ntu.edu.sg/home/nprivault/MA5182/… – rs4rs35 Jun 20 at 12:49
• You may edit your question to make it more readable. – Gordon Jun 20 at 19:21

Thanks to the Girsanov theorem, we have the following relationship between the forward measure $$\mathbb{Q}^{T_i}$$ and the historical measure $$\mathbb{P}$$. \begin{align} \left.\frac{d\mathbb{Q}^{T_i}}{d\mathbb{P}}\right|_{\mathcal{F}_t} &= e^{-\int_t^Tr_sds}\frac{P_t(T_i)}{P_0(T_i)} \\ &= \exp\left(\int^{T_i}_{t}\xi_{i}(s)dB_{s}-\frac{1}{2}\int^{T_i}_{0}|\xi_{i}(s)|^{2}ds\right)\\ &=\frac{N_{T_i}}{N_t} \end{align} where $$N_t = \exp\left(\int^{t}_{0}\xi_{i}(s)dB_{s}-\frac{1}{2}\int^{t}_{0}|\xi_{i}(s)|^{2}ds\right)$$. This process is an exponential martingale widely known as the Doléans-Dade exponential. By the Ito formula, the dynamic of $$N_t$$ is
$$\begin{equation} dN_t = N_t\xi_i(t)dB_t \end{equation}$$
Giranov tells us as well that it exists a Brownian motion under $$\mathbb{Q}^{T_i}$$ given by : \begin{align} dW_t &= dB_t - \frac{1}{N_t}\langle N_., W_.\rangle_t\\ &=dB_t -\frac{1}{N_t}N_t\xi_i(t)\langle W_., W_.\rangle_t\\ &= dB_t - \xi_i(t)dt \end{align}