Deriving Forward-Measure Brownian Motion with Girsanov’s Theorem

# Proof standard Brownian Motion under change of measure


# Proof standard Brownian Motion under change of measure












Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ 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?

## Answer by Sesame (score 1, accepted)

https://quant.stackexchange.com/a/46265

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}