# Submartingale example: proof

I am trying to prove if the process $M_t = e^{W_t^2-t}$ is a submartingale ($W_t$ is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward.

Let me show you what I have done so far:

\begin{align*} \mathbb E\left[e^{W_t^2-t} \middle| \mathscr{F}_s\right] = & \mathbb E\left[e^{(W_t-W_s+W_s)^2-t} \middle| \mathscr{F}_s\right] = &\\ & \mathbb E\left[e^{(W_t-W_s)^2 + 2W_s(W_t-W_s)+W_s^2-t} \middle| \mathscr{F}_s\right] = \\ & e^{W_s^2-t}\mathbb E(e^{(W_t-W_s)^2})\mathbb E\left[e^{2W_s(W_t-W_s)} \middle| \mathscr{F}_s\right] \end{align*}

And here is where I am unsure about how to solve the different expectations in order to prove that this is indeed a submartingale: I know I am nearly there, but I have tried different things from this point on without succeding. Can somebody perhaps give me a hint?

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In the last line, you need $$e^{W_s^2-s}\mathbb E(e^{(W_t-W_s)^2 - (t-s)})\mathbb E\left[e^{2W_s(W_t-W_s)} \middle| \mathscr{F}_s\right]$$
which is equal to $$e^{W_s^2-s}\mathbb E(e^{W_h^2 - h})\mathbb E\left[e^{k W_h} \right]$$ where $k = 2 W_s$ and $h=t-s$. Now you can use Jensen to show that each of the expectations is $\ge 1$.
Let $(M_t)_{t \geq 0}$ be a martingale and $f: \mathbb{R} \to \mathbb{R}$ a convex function. Then it follows directly from Jensen's inequality that $(f(M_t))_{t \geq 0}$ is a submartingale. Apply this to the martingale $$M_t := W_t^2-t$$ and $$f(x) := \exp(x).$$