Proof of Fourth-Moment Maximal Inequality for Bounded Right-Continuous Martingales on a Compact Time Interval

Step 1: the square of $M$ is a nonnegative submartingale. Since $|M_t(\omega)|\le K$ for every $t$ and $\omega$, we have $M_t^4\le K^4$ pointwise, so $\mathbb{E}[M_t^4]\le K^4<\infty$ by monotonicity of the integral, that is, $M_t^2$ is square-integrable for every $t\in[0,T]$. By the lemma on the square of a martingale, $X=(M_t^2)_{t\in[0,T]}$ is a square-integrable submartingale with respect to $(\mathcal{F}_t)_{t\in[0,T]}$ with $X_t(\omega)\ge0$ everywhere.

Step 2: extension to all nonnegative times. Define, for $t\ge0$, $\mathcal{F}^{\mathrm{ext}}_t=\mathcal{F}_{\min(t,T)}$ and $X^{\mathrm{ext}}_t=X_{\min(t,T)}$. The family $(\mathcal{F}^{\mathrm{ext}}_t)_{t\ge0}$ is a filtration, since $r\le t$ implies $\min(r,T)\le\min(t,T)$. Each $X^{\mathrm{ext}}_t$ is $\mathcal{F}^{\mathrm{ext}}_t$-measurable, nonnegative and square-integrable, being one of the $X_u$ with $u\in[0,T]$. For $0\le r\le t$ and $A\in\mathcal{F}^{\mathrm{ext}}_r$ the submartingale inequality $\mathbb{E}[X^{\mathrm{ext}}_t\mathbf{1}_{A}]\ge\mathbb{E}[X^{\mathrm{ext}}_r\mathbf{1}_{A}]$ holds: if $t\le T$ it is the inequality for $X$ at the times $r\le t$; if $r\le T\le t$ it is the inequality for $X$ at the times $r\le T$, because $X^{\mathrm{ext}}_t=X_T$ and $\mathcal{F}^{\mathrm{ext}}_r=\mathcal{F}_r$; and if $T\le r$ both sides equal $\mathbb{E}[X_T\mathbf{1}_{A}]$. Hence $X^{\mathrm{ext}}$ is a nonnegative square-integrable submartingale indexed by $t\ge0$.

Step 3: the inequality on finite dyadic grids. For a natural number $n$ let $D_n=\{kT2^{-n}:k\in\{0,1,\dots,2^n\}\}$, whose elements listed in increasing order form sample times $0=t_0<t_1<\dots<t_{2^n}=T$. Claim 1 of Doob's discrete-time $L^2$ maximal inequality, applied to the nonnegative submartingale $X^{\mathrm{ext}}$ and these $2^n+1$ sample times (the index called $n$ in that theorem being $2^n$ here), gives

Since $x\mapsto x^2$ is nondecreasing on $[0,\infty)$, $\big(\max_{d\in D_n}M_d^2\big)^2=\max_{d\in D_n}M_d^4$ pointwise.

Step 4: passage to the supremum. Let $D$ be the set of dyadic partition points of $[0,T]$ from the statement, the points $kT2^{-n}$ with $n$ zero or a natural number; since $D_0=\{0,T\}\subseteq D_1$, one has $D=\bigcup_{n\in\mathbb{N}}D_n$. Let $\overline{M}=\sup_{t\in D}\big(|M_t|\mathbf{1}_{\Omega_0}\big)$, a random variable with $0\le\overline{M}\le K$ by the supremum lemma, whose hypotheses hold by assumption (boundedness of every path by $K$ holding even at every $\omega\in\Omega$). Define

a random variable by claims 1, 3 and 4 of the arithmetic lemma for measurable functions (finitely many maxima of fourth powers of random variables). Since $D_n\subseteq D_{n+1}$ (each $kT2^{-n}$ equals $2kT2^{-(n+1)}$), the sequence $(W_n)$ is nondecreasing; since $(|M_d|\mathbf{1}_{\Omega_0})^4\le M_d^4$, Step 3 gives $\mathbb{E}[W_n]\le4\,\mathbb{E}[M_T^4]$ by monotonicity. The sequence converges pointwise to $\overline{M}^{\,4}$: on one hand $W_n\le\overline{M}^{\,4}$ for every $n$, because $x\mapsto x^4$ is nondecreasing on $[0,\infty)$ and each $|M_d|\mathbf{1}_{\Omega_0}$ with $d\in D_n$ is at most $\overline{M}$; on the other hand, if $\overline{M}(\omega)>0$ and $0<\varepsilon<\overline{M}(\omega)$, claim 3 of the approximation property of the supremum yields $t\in D$ with $|M_t(\omega)|\mathbf{1}_{\Omega_0}(\omega)>\overline{M}(\omega)-\varepsilon$, and this $t$ lies in $D_n$ for all large $n$, so $W_n(\omega)\ge(\overline{M}(\omega)-\varepsilon)^4$ for all large $n$; hence $\sup_nW_n(\omega)\ge(\overline{M}(\omega)-\varepsilon)^4$ for every such $\varepsilon$, and since $(\overline{M}(\omega)-\varepsilon)^4$ can be made larger than any number below $\overline{M}(\omega)^4$ by taking $\varepsilon$ small ($x\mapsto x^4$ being continuous on $[0,\infty)$), $\sup_nW_n(\omega)=\overline{M}(\omega)^4$; the nondecreasing sequence $(W_n(\omega))_n$ therefore converges to $\overline{M}(\omega)^4$; the case $\overline{M}(\omega)=0$ is immediate. By the monotone convergence theorem applied to the nondecreasing sequence of nonnegative random variables $(W_n)$,