Maximal inequalities convert control at a terminal time into control of an entire path. Upcrossing inequalities convert one-sided expectation bounds into almost-sure convergence. These two mechanisms are the martingale counterparts of Kolmogorov's maximal inequality and compactness by oscillation control.
Weak maximal inequality
Definition. For a process \(X\), write \[ X_n^*=\max_{0\le k\le n}|X_k|. \] For a one-sided submartingale inequality, write \(\overline X_n=\max_{0\le k\le n}X_k\).
Theorem (Doob weak maximal inequality). If \(X_0,\ldots,X_n\) is a nonnegative submartingale, then, for every \(\lambda>0\), \[ \lambda P(\overline X_n\ge\lambda) \le \mathbb E[X_n1_{\{\overline X_n\ge\lambda\}}] \le\mathbb EX_n. \]
Proof. Partition the first-crossing event into \[ A_k=\{X_0<\lambda,\ldots,X_{k-1}<\lambda,\ X_k\ge\lambda\}, \qquad0\le k\le n. \] The \(A_k\) are disjoint, \(A_k\in\mathcal F_k\), and their union is \(\{\overline X_n\ge\lambda\}\). Since a submartingale satisfies \(X_k\le\mathbb E[X_n\mid\mathcal F_k]\), \[ \lambda P(A_k) \le\mathbb E[X_k1_{A_k}] \le\mathbb E[X_n1_{A_k}]. \] Sum over \(k\). Nonnegativity gives the final inequality.
Corollary. If \(M\) is a martingale, then \[ \lambda P(M_n^*\ge\lambda) \le\mathbb E[|M_n|1_{\{M_n^*\ge\lambda\}}], \] because \(|M_k|\) is a nonnegative submartingale.
The \(L^p\) maximal inequality
Theorem (Doob \(L^p\) inequality). If \(p>1\), \(q=p/(p-1)\), and \(M_n\in L^p\), then \[ \|M_n^*\|_p\le q\|M_n\|_p. \] More generally, the same bound holds for the positive maximum of a nonnegative submartingale.
Proof. For \(X=\max_{k\le n}X_k\) and nonnegative submartingale \(X_k\), the layer-cake formula and the weak inequality give \[ \begin{aligned} \mathbb EX^p &=p\int_0^\infty\lambda^{p-1}P(X\ge\lambda)\,\mathrm{d}\lambda\\ &\le p\int_0^\infty\lambda^{p-2} \mathbb E[X_n1_{\{X\ge\lambda\}}]\,\mathrm{d}\lambda\\ &=\frac{p}{p-1}\mathbb E[X_nX^{p-1}]. \end{aligned} \] H?lder's inequality bounds the last expression by \(q\|X_n\|_p\|X\|_p^{p-1}\). Divide by \(\|X\|_p^{p-1}\). Apply the result to \(|M_k|\).
Corollary (\(L^p\) martingale convergence). If \(p>1\) and \(\sup_n\mathbb E|M_n|^p<\infty\), then there is \(M_\infty\in L^p\) such that \[ M_n\to M_\infty \quad\text{almost surely and in }L^p. \]
Proof. The submartingale convergence theorem below gives almost-sure convergence because \(\sup_n\mathbb E|M_n|<\infty\). Fatou gives \(M_\infty\in L^p\). For fixed \(m\), the process \(M_{m+k}-M_m\) is a martingale, so Doob's inequality and conditional expectation representation yield uniform \(L^p\) control of tails. Equivalently, \(L^p\)-boundedness with \(p>1\) implies uniform integrability of \(|M_n|^p\); almost-sure convergence plus Vitali then gives \(\mathbb E|M_n-M_\infty|^p\to0\).
Upcrossings
Definition. For a real sequence \(x_0,\ldots,x_n\) and \(a<b\), the number \(U_n[a,b]\) of completed upcrossings is the largest \(r\) for which there are indices \[ s_1<t_1<s_2<t_2<\cdots<s_r<t_r\le n \] with \(x_{s_j}\le a\) and \(x_{t_j}\ge b\).
Theorem (Doob upcrossing inequality). If \(X_0,\ldots,X_n\) is a submartingale, then \[ (b-a)\mathbb EU_n[a,b] \le \mathbb E\!\left[(X_n-a)^+-(X_n-b)^+\right] \le \mathbb E(X_n-a)^+. \]
Proof. Construct the predictable \(0\)-\(1\) strategy that alternates between waiting for \(X\le a\) and waiting for the subsequent value \(X\ge b\). Apply the associated pathwise trading identity not to unrestricted terminal wealth, but to the capped payoff \[ h(x)=(x-a)^+-(x-b)^+ = \begin{cases} 0,&x\le a,\\ x-a,&a<x<b,\\ b-a,&x\ge b. \end{cases} \] At every completed upcrossing the capped account increases by \(b-a\); an unfinished crossing contributes only the terminal value \(h(X_n)\). The remaining increment terms form the negative of a predictable nonnegative transform of the submartingale. Their expectation is nonpositive because \[ \mathbb E[H_k\Delta X_k] =\mathbb E\!\left[ H_k\mathbb E(\Delta X_k\mid\mathcal F_{k-1})\right]\ge0. \] Taking expectations in the pathwise account therefore gives \((b-a)\mathbb EU_n[a,b]\le\mathbb Eh(X_n)\). Since \(0\le h(x)\le(x-a)^+\), the second inequality follows.
Almost-sure convergence
Theorem (submartingale convergence). If \(X_n\) is a submartingale and \[ \sup_n\mathbb EX_n^+<\infty, \] then \(X_n\) converges almost surely to a finite integrable random variable \(X_\infty\).
Proof. Since \(\mathbb EX_n\ge\mathbb EX_0\), \[ \mathbb EX_n^- =\mathbb EX_n^+-\mathbb EX_n \le\sup_m\mathbb EX_m^+-\mathbb EX_0, \] so \(\sup_n\mathbb E|X_n|<\infty\). For rational \(a<b\), the upcrossing inequality makes \(\sup_n\mathbb EU_n[a,b]<\infty\), hence \(U_\infty[a,b]<\infty\) almost surely. Outside one null set this holds for every rational interval. If a real sequence has \(\liminf x_n<\limsup x_n\), some rational interval between them is crossed infinitely often. Therefore \(X_n\) has an extended-real limit almost surely. Fatou's lemma gives \[ \mathbb E|X_\infty| \le\liminf_n\mathbb E|X_n|<\infty, \] so the limit is finite almost surely.
Remark. Almost-sure convergence plus bounded \(L^1\) norms does not imply \(L^1\) convergence. Uniform integrability is the missing condition and is treated after optional stopping.