Convex functions of martingales and submartingales

Theorem 9.7.4 states: assume that $h$ is a convex function from $\mathbb {R}$ to $\mathbb {R}$, that $\left\{ Z_{n}; n \geq 1\right\}$ is a martingale, and that $\mathbb {E}\left[\left|h(Z_{n})\right|\right] < \infty$ for all $n$. Then $\left\{ h(Z_{n}); n \geq 1\right\}$ is a submartingale.

Show that Theorem 9.7.4 is also valid for martingales relative to a joint process. That is, show that if $h$ is a convex function of a real variable and if $\left\{ Z_{n}; n \geq 1\right\}$ is a martingale relative to a joint process $\left\{ Z_{n}, X_{n}; n \geq 1\right\}$, then $\left\{ h(Z_{n}); n \geq 1\right\}$ is a submartingale relative to $\left\{ h(Z_{n}), X_{n}; n \geq 1\right\}$.

Let $Y$ be a martingale and let $u$ be a convex function mapping $\mathbb {R}$ to $\mathbb {R}$. Show that $\left\{ u\left(Y_{n}\right): n \geq 0\right\}$ is a submartingale provided that $\mathbb {E}\left[u\left(Y_{n}\right)^{+}\right] < \infty$ for all $n$.

Show that $\left|Y_{n}\right|, Y_{n}^{2}$, and $Y_{n}^{+}$constitute submartingales whenever the appropriate moment conditions are satisfied.

Let $Y$ be a submartingale and let $u$ be a convex non-decreasing function mapping $\mathbb {R}$ to $\mathbb {R}$. Show that $\left\{ u\left(Y_{n}\right): n \geq 0\right\}$ is a submartingale provided that $\mathbb {E}\left[u\left(Y_{n}\right)^{+}\right] < \infty$ for all $n$.

Show that (subject to a moment condition) $Y_{n}^{+}$constitutes a submartingale, but that $\left|Y_{n}\right|$ and $Y_{n}^{2}$ need not constitute submartingales.

Let $\left\{ \xi_{n}\right\}$ be a martingale, $\mathbb {E}\left[\xi_{n}^{2}\right] < \infty$. Prove that $\left\{ \xi_{n}^{2}\right\}$ is a submartingale.

Let $\left\{ X_{n}, \mathscr {F}_{n}\right\}$ be a martingale, and $g(x)$ a convex function such that $\mathbb {E}\left[\left|g\left(X_{n}\right)\right|\right] < \infty , n = 0,1, \ldots$ Prove that the sequence $\left\{ g\left(X_{n}\right), \mathscr {F}_{n}\right\}$ forms a submartingale.

Let $\left\{ X_{n}, \mathscr {F}_{n}\right\}$ be a submartingale, and $g(x)$ a convex nondecreasing function such that $\mathbb {E}\left[\left|g\left(X_{n}\right)\right|\right] < \infty , n = 0,1, \ldots$ Prove that the sequence $\left\{ g\left(X_{n}\right), \mathscr {F}_{n}\right\}$ also forms a submartingale.