Add a compensator to submartingale to create a martingale.

Question

Let $W_t$ be a Brownian motion. By Jensen's inequality, $W_t^2$ is a submartingale. I was wondering if it were possible to add another process to $W_t^2$, say $X_t$, such that $W_t^2 + X_t$ is a martingale. How would I go about doing this?

I know I need to solve for $X_t$ which satisfies the equation: $$ E[W_t^2+X_t|\mathcal{F_s}] = W_s^2+ X_s$$ $ \forall s<t$. How do I go about doing this?

Answer 1

For many stochastic processes (but certainly not all) you can find such a function $X_t$. It turns out that it will be a finite variation process, called the bracket process, or quadratic variation process.

In this particular case, one can use the markov property in place of Ito's formula:

\begin{align} \Bbb E[ W_t^2 | \mathcal{F}_s] &= \Bbb E\left[ (W_t-W_s)^2 + \left. 2W_sW_t - W_s^2 \ \right|\ \mathcal{F}_s\right] \\ &= \Bbb E[(W_t-W_s)^2 | \mathcal{F}_s] + 2W_s\Bbb E[W_t|\mathcal{F}_s] - W^2_s && \text{"taking out what is known"}\\ &= \Bbb E[(W_t-W_s)^2 | \mathcal{F}_s] + 2W^2_s - W_s^2 && W\text{ is a martingale}\\ &= \Bbb E[(W_t-W_s)^2] + W_s^2 &&\text{by independence from the past} \\ &= \Bbb EX_{t-s}^2 + W_s^2 && (\star)\\ &= (t-s) + W_s^2 &&(t>s)\end{align} where in the line marked $(\star)$, $X_{t-s}$ is a Brownian Motion restarted at time $s$. This tells us that $W_t^2 - t$ is a martingale.

Answer 2

By Ito's formula:

$$W_t^2=2 \int_0^t W_s dW_s + \int_0^t ds = 2 \int_0^t W_s dW_s + t.$$

Now the Ito integral (of a square integrable, independent-of-future function) is a martingale, so $W_t^2-t$ is a martingale.

This procedure works for a wide class of diffusion processes.