Tricky Proof in Stochastic Processes/ Probability Theory

Question

Say we have a probability space $(\Omega, \mathscr{F}, \mathbb{P}$), and $\mathbb{F} = (\mathscr{F}_t)_{t \geq 0}$ is a filtration of sub-$\sigma$-algebras of $\mathscr{F}$ satisfying the standard conditions. I've been told to consider a one-dimensional $\mathbb{F}$-Wiener process $(W_t)_{t \geq 0}$ and also the process $(X_t)_{t \geq 0}$ given by: $$X_t = f(W_t) - \frac{1}{2} \int_{0}^{t} f''(W_s) ds$$ for every $t > 0$, where $f: \mathbb{R} \mapsto \mathbb{R}$ is a $C^2$ function.

Furthermore, I've been told to assume that there exists a constant $C > 0$ such that $|f''(x)| \leq C$ for all $x \in \mathbb{R}$.

Show that $(X_t)_{t \geq 0}$ is a martingale (with respect to $\mathbb{F}$).

I've been quite stumped by this proof for a while. How could I demonstrate this idea?

Answer 1

Applying Ito's formula to $f$, we have \begin{align*} f(W_t) &= f(W_0) + \int_0^t f'(W_s)dW_s + \frac 12 \int_0^t f''(W_s)ds \end{align*} so \begin{align*} X_t = f(W_t)-\frac 12 \int_0^t f''(W_s)ds &= f(W_0) + \int_0^t f'(W_s)dW_s \end{align*} is a (local) martingale. To prove it is a true martingale, use the fact that $|f''(x)| \le C$ to show that \begin{align*} \mathbb{E}\left[\int_0^t |f'(W_s)|^2 ds \right] < \infty \end{align*} for all $t \ge 0$.