Worked Applications of Doob's Maximal Inequalities

From Maximal Inequalities to Useful Bounds

Doob’s maximal inequalities turn information about a process at a fixed terminal time into control of its entire path up to that time. This is useful when the quantity of interest is a crossing event, such as whether a martingale ever exceeds a prescribed level, or a moment of its running maximum. The resulting estimates need not be sharp for a particular process: comparing them with an exact calculation helps show what the inequalities do and do not tell us.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\). The arguments below do not use Brownian motion or stochastic integrals. In particular, no integrand class is needed. The example filtrations may contain information independent of \(B_t\); adjoining the Brownian filtration then leaves the conditional expectations in the examples unchanged.

Two inequalities will be used. The weak-type maximal inequality says that if \(X\) is a nonnegative, right-continuous submartingale and \(\mathbb{E}[X_T]<\infty\), then for every \(\lambda>0\), \(\mathbb{P}(X_T^*\ge\lambda)\le\mathbb{E}[X_T]/\lambda\). The \(L^p\) maximal inequality says that if \(M\) is a right-continuous martingale and \(\mathbb{E}[|M_T|^p]<\infty\) for some \(p>1\), then \(\mathbb{E}[(M_T^*)^p]\le (p/(p-1))^p\mathbb{E}[|M_T|^p]\). These are the standard Doob maximal inequalities established in the earlier material on continuous-time martingales. In particular, \(|M_t|\) is a nonnegative submartingale, by conditional Jensen’s inequality, so the weak-type bound applies to it.

Tail Bounds from Terminal-Time Information

Proof. The \(L^p\) maximal inequality gives \(\mathbb{E}[(M_T^*)^p]\le (p/(p-1))^p\mathbb{E}[|M_T|^p]\), so \(M_T^*\) has finite \(p\)-th moment. On the event \(\{M_T^*\ge K\}\), the nonnegative random variable \((M_T^*)^p\) is at least \(K^p\). Consequently \(\mathbb{E}[(M_T^*)^p]\ge K^p\mathbb{P}(M_T^*\ge K)\). Combining these inequalities and dividing by \(K^p>0\) proves the claim. A probability is at most \(1\), which gives the stated improvement. \(\square\)

The weak-type inequality often gives a more direct crossing estimate when a first moment is available. For a martingale \(M\) with \(\mathbb{E}[|M_T|]<\infty\), applying it to \(|M|\) yields \(\mathbb{P}(M_T^*\ge K)\le\mathbb{E}[|M_T|]/K\). The threshold event here is inclusive: it counts a path that reaches \(K\) exactly. The bounds also hold for the strict event \(\{M_T^*>K\}\), since that event is contained in \(\{M_T^*\ge K\}\).

Moment Bounds and Lower-Order Moments

Proof. The \(L^p\) maximal inequality first gives \(\mathbb{E}[(M_T^*)^p]\le (p/(p-1))^p\mathbb{E}[|M_T|^p]<\infty\). Apply the monotonicity of probability-space \(L^r\) norms, which follows from Hölder’s inequality: for a nonnegative random variable \(Y\) with finite \(p\)-th moment and \(q<p\), \(\mathbb{E}[Y^q]\le(\mathbb{E}[Y^p])^{q/p}\). Taking \(Y=M_T^*\), then substituting the \(L^p\) maximal bound, proves the displayed inequality. This argument also covers \(M_T^*=0\); no division by that random variable is used. \(\square\)

The \(L^p\) inequality also yields a tail bound by the proposition. This can be useful when the terminal \(p\)-th moment is known but the first absolute moment is not the most convenient quantity. The dependence on \(K^{-p}\) shows how a higher terminal moment controls the probability of very large excursions. It does not imply that the bound is close to the true probability for moderate \(K\).

Choosing and Interpreting a Bound

For a nonnegative submartingale, the weak-type estimate uses the terminal first moment and is often the natural choice for a single crossing event. For a martingale with a known \(p\)-th terminal moment, the \(L^p\) inequality controls a moment of the whole path, and the proposition converts that moment into a tail estimate. If the resulting numerical bound exceeds \(1\), it remains mathematically correct but gives no information beyond the fact that a probability is at most \(1\).

A common mistake is to apply the terminal-time estimate directly to the running maximum. For example, in the two-step example the calculation \(\mathbb{P}(M_2^*\ge3/2)\le\mathbb{E}[|M_2|]/(3/2)\) is justified by the weak-type inequality applied to the submartingale \(|M_t|\), not by treating \(M_2^*\) as though it were \(M_2\). A different incorrect calculation would replace \(\mathbb{E}[|M_2|]\) by \(\mathbb{E}[M_2]=0\), giving a zero upper bound. The correction is essential: the weak-type inequality here concerns the nonnegative process \(|M_t|\), whose terminal expectation in the example is \(1\), not the signed mean \(\mathbb{E}[M_2]=0\).

These inequalities concern true martingales or submartingales with the stated integrability; no local-martingale version is being asserted here. Checking the hypotheses first, then comparing the bound with an exact distribution when one is available, keeps the estimate both valid and informative.