Prove that $d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t$ gives a Brownian motion under forward measure

Let $$N_t$$ be a numeraire and $$(W_t)$$ be the standard Brownian motion under the risk-neutral probability measure $$P$$.

Recall that forward measure $$\hat{P}$$ is defined as the Radon-Nikodym derivative: $$\frac{d\hat{P}}{d P} = e^{-\int_0^t r_s \,ds}\frac{N_t}{N_0}$$ where $$r_s$$ is risk-free interest rate.

Whenever I want to change the underlying measure to forward measure (take bond as numeraire), I always uses the equation $$d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t.$$ However, I am not able to prove that equation above implies that $$(\hat{W}_t)$$ is a Brownian motion under the forward measure $$\hat{P}$$.

• Does your numéraire have any specific dynamics? Or is it just generic? Apr 6 '20 at 20:49
• @DaneelOlivaw Does the equation hold for both generic and specific dynamic numeraire? If yes, I would like to see both. Apr 7 '20 at 0:05

Both equations you have provided are incorrect.

$$\frac{d \hat{P} } {d P}(t) = \frac{N_t}{N_0} \frac{\beta_0}{\beta_t}$$

where $$\beta_t = \exp \int_{0}^{t} r_s ds$$.

$$d \hat{W}_t = d W_t - \sqrt{\frac{ d \langle N \rangle_t }{N^2_t}} dt$$

Choosing $$N_t = P_{tT}$$ means that we have

$$d \hat{W}_t = d W_t - \sigma_{tT} dt$$

where $$\sigma_{tT}$$ is the volatility coefficient of the zero coupon bond.

Now one can use Levy's characterization theorem to show that $$\hat{W}$$ is a $$\hat{P}$$-Brownian motion.

• I understand that there is a typo in my first equation. I edited it. But I do not understand your second equation. Apr 7 '20 at 2:33
• The Girsanov theorem explains the second equation. The new Brownian motion, under the $\hat{P}$ measure, is given as follows $d \hat{W}_t = d W_t - \gamma d t$ where $\gamma$ is the volatility of the stochastic exponential (equivalently the Radon-Nikodym derivative $d \hat{P} / d P$).
– AXH
Apr 7 '20 at 8:11
• How to deduce that $d\hat{W}_t = dW_t - \gamma dt$ from Girsanov Theorem? Does the equation hold for all SDE with volatility $\gamma?$ Apr 7 '20 at 8:16