# Characterization of Brownian Motion (Problem Karatzas/Shreve)

In the book "Brownian Motion and Stochastic Calculus" by Karatzas/Shreve, they state the following problem (chapter 5, problem 4.4):

A continuous, adapted process $$W= \{W_t,\mathcal{F}_t;0\leq t < \infty \}$$ is a Brownian motion if and only if $$\begin{equation} f(W_t) - f(W_0) - \frac{1}{2} \int_0^t f^{\prime\prime}(W_s) \mathrm{d}s,\quad \mathcal{F}_t; \quad 0 \leq t <\infty \end{equation}$$ is a continuous local martingale for every $$f\in C^2(\mathbb{R})$$.

For the "only if" part, one applies Ito's formula and gets that $$\begin{equation} f(W_t) - f(W_0) - \frac{1}{2} \int_0^t f^{\prime\prime}(W_s) \mathrm{d}s= \int_0^t f^{\prime}(W_s) \mathrm{d}W_s, \end{equation}$$ which is a continuous local martingale.

But I am struggling with the "if" part. Does anyone have a hint on how to prove that?

Thank you very much!

Best, Luke

• That's a standard result; look up Lévy's characterization of Brownian motion. – saz Jun 13 at 16:03

## 1 Answer

If you take $$f(x) = x$$, you see that $$W_t$$ is a continuous local martingale.

Then, taking $$f(x) = x^2$$ gives us that $$W_t^2 - W_0^2 - t$$ is also a continuous local martingale so that $$W_t$$ has quadratic variation at time $$t$$ equal to $$t$$. Now apply Levy's characterisation of Brownian motion to conclude.

• Thank you very much! – Luke Jun 13 at 16:16