# Proving a True Martingale

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 quote here my old question (Tricky Proof in Stochastic Processes/ Probability Theory). This question was answered by @user6247850 who uses Ito's formula to show that it is a local martingale. His answer below I will paste here:

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 $$t \ge 0$$.

How can I go on to prove that it is a True martingale? I am new to this topic and these kind of questions so any demonstration of a solution would really help me. Many thanks.

By mean value theorem

$$f'(x)=f'(0)+f''(c)x$$

for some $$0\leq c \leq x$$ when $$x>0$$and hence

$$-Cx+f'(0)\leq f'(x)\leq f'(0)+C x$$

$$f'(0)$$ exists because $$f$$ is $$C^2$$. When $$x<0$$

$$Cx+f'(0)\leq f'(x)\leq f'(0)-C x$$

Then $$|f'(x)|^2 \leq \max((f'(0)+Cx)^2,(f'(0)-Cx)^2)$$. Hence $$|f'(x)|^2$$ is bounded by a max of two integrable polynomials and is hence integrable. A result presented in any stochastic calculus textbook is that a stochastic integral has mean zero if the integrand function is square integrable. Hence

$$\mathbb{E}[X_t]=f(W_0)$$

and $$X_t$$ is a true martingale.