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 $\hat{P}$.

  • $\begingroup$ Does your numéraire have any specific dynamics? Or is it just generic? $\endgroup$ Commented Apr 6, 2020 at 20:49
  • $\begingroup$ @DaneelOlivaw Does the equation hold for both generic and specific dynamic numeraire? If yes, I would like to see both. $\endgroup$
    – Idonknow
    Commented Apr 7, 2020 at 0:05

1 Answer 1


Both equations you have provided are incorrect.

The first equation should read:

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

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

The second equation should read

$ 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.

  • $\begingroup$ I understand that there is a typo in my first equation. I edited it. But I do not understand your second equation. $\endgroup$
    – Idonknow
    Commented Apr 7, 2020 at 2:33
  • $\begingroup$ 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 $d \hat{P} / d P$). $\endgroup$
    – AXH
    Commented Apr 7, 2020 at 8:11
  • $\begingroup$ 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? $ $\endgroup$
    – Idonknow
    Commented Apr 7, 2020 at 8:16

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