# Apply Novikov's criterion

I want to apply Girsanov's theorem to

$$B_t = X_t - \int_0^t b(X_s)ds$$

where $$X_t$$ is a Brownian motion and $$b$$ is Lipschitz. Therefore I have to show that

$$Z_t = \exp(\int\limits_0^t b(X_s)dX_s - \frac{1}{2} \int\limits_0^t|b(X_s)|^2ds)$$

is a martingale. In order to do so I want to apply Novikov's condition, which gives me the martingale property for $$Z$$, if

$$\mathbb{E}[\exp(\frac{1}{2} \int\limits_0^t|b(X_s)|^2ds)] < +\infty$$

Jensen and the Lipschitz property give me

$$\mathbb{E}[\exp(\frac{1}{2} \int\limits_0^t|b(X_s)|^2ds)] \: \leq \: ...\leq \mathbb{E} [\int\limits_0^t\exp(\frac{C}{2}|X_s|^2)ds]$$

But how can I continue to find the finiteness?

Here is a partial result for small $$t$$:

$$\begin{eqnarray*} E\left[\int_0^t exp\left(\frac{C}{2}X_s^2\right)ds\right] &=& \int_0^t E\left[exp\left(\frac{C}{2}X_s^2\right)\right]ds\\ &=& \int_0^t E\left[exp\left(\frac{Cs}{2}\Big(\frac{X_s}{\sqrt{s}}\Big)^2\right)\right]ds\\ &=& \int_0^t E\left[exp\left(\frac{Cs}{2}\chi_1^2\right)\right]ds\\ &=& \int_0^t M_{\chi_1^2}\Big(\frac{Cs}{2}\Big)ds\\ &=& \int_0^t (1-Cs)^{-\frac{1}{2}} ds\\ \end{eqnarray*}$$ which is finite for $$t\in [0,\frac{1}{C})$$. I have used the following:

(1) Fubini's theorem to switch the order of integration: $$E\left[\int\cdots\right]=\int E\left[\cdots\right]$$

(2) $$X_s\sim N(0,s)$$ so $$\frac{X_s}{\sqrt{s}}\sim N(0,1)$$ and hence $$\Big(\frac{X_s}{\sqrt{s}}\Big)^2\sim \chi_1^2$$

(3) $$M_{\chi_1^2}(t)=E[exp(t\chi_1^2)]=(1-2t)^{-\frac{1}{2}}$$ is the MGF of $$\chi_1^2$$.

I'll try to find a better upper bound :)

• Thank you very much, thats perfect. To apply Girsanov this is already enough: Novikov gives us that $Z_t$ is a martingale on $[0,T]$. This gives us $\mathbb{E}(Z_t)=1$ on that interval, which is also a sufficient condition for $Z_t$ to be a martingale in that setting. We can now prove that $Z_{Kt}$ has expectation 1 for any $K \in \mathbb{N}$ by telescoping the integrals. Apr 15, 2020 at 12:29