Doob's martingale inequality

Template:Short description In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the case that the process is a martingale, but the result is also valid for submartingales.

The inequality is due to the American mathematician Joseph L. Doob.

Statement of the inequality

The setting of Doob's inequality is a submartingale relative to a filtration of the underlying probability space. The probability measure on the sample space of the martingale will be denoted by Template:Math. The corresponding expected value of a random variable Template:Mvar, as defined by Lebesgue integration, will be denoted by Template:Math.

Informally, Doob's inequality states that the expected value of the process at some final time controls the probability that a sample path will reach above any particular value beforehand. As the proof uses very direct reasoning, it does not require any restrictive assumptions on the underlying filtration or on the process itself, unlike for many other theorems about stochastic processes. In the continuous-time setting, right-continuity (or left-continuity) of the sample paths is required, but only for the sake of knowing that the supremal value of a sample path equals the supremum over an arbitrary countable dense subset of times.

Discrete time

Let Template:Math be a discrete-time submartingale relative to a filtration ${ℱ}_1,\dots ,{ℱ}_n$ of the underlying probability space, which is to say:

$X_i\le \mathrm{E}[X_{i+1}\mid {ℱ}_i].$

The submartingale inequalityTemplate:What says that

$P[\underset{1\le i\le n}{\max }{X}_i\ge C]\le \frac{\mathrm{E}[\text{max}(X_n,0)]}{C}$

for any positive number Template:Mvar. The proof relies on the set-theoretic fact that the event defined by Template:Math may be decomposed as the disjoint union of the events Template:Math defined by Template:Math and Template:Math for all Template:Math. Then

$CP(E_i)=\underset{E_i}{\int }CdP\le \underset{E_i}{\int }{X}_{i}dP\le \underset{E_i}{\int }\text{E}[X_n\mid {ℱ}_i]dP=\underset{E_i}{\int }{X}_{n}dP,$

having made use of the submartingale property for the last inequality and the fact that $E_i\in {ℱ}_i$ for the last equality. Summing this result as Template:Mvar ranges from 1 to Template:Mvar results in the conclusion

$CP(E)\le \underset{E}{\int }{X}_{n}dP,$

which is sharper than the stated result. By using the elementary fact that Template:Math, the given submartingale inequality follows.

In this proof, the submartingale property is used once, together with the definition of conditional expectation.Template:Sfnm The proof can also be phrased in the language of stochastic processes so as to become a corollary of the powerful theorem that a stopped submartingale is itself a submartingale.Template:Sfnm In this setup, the minimal index Template:Mvar appearing in the above proof is interpreted as a stopping time.

Continuous time

Now let Template:Math be a submartingale indexed by an interval Template:Math of real numbers, relative to a filtration Template:Math of the underlying probability space, which is to say:

$X_s\le \mathrm{E}[X_t\mid {ℱ}_s].$

for all Template:Math The submartingale inequalityTemplate:What says that if the sample paths of the martingale are almost-surely right-continuous, then

$P[\underset{0\le t\le T}{\sup }{X}_t\ge C]\le \frac{\mathrm{E}[\text{max}(X_T,0)]}{C}$

for any positive number Template:Mvar. This is a corollary of the above discrete-time result, obtained by writing

$\underset{0\le t\le T}{\sup }{X}_t=\sup \{X_t:t\in [0,T]\cap \mathbb{Q}\}=\underset{i\to \mathrm{\infty }}{\lim }\sup \{X_t:t\in [0,T]\cap Q_i\}$

in which Template:Math is any sequence of finite sets whose union is the set of all rational numbers. The first equality is a consequence of the right-continuity assumption, while the second equality is purely set-theoretic. The discrete-time inequality applies to say that

$P[\underset{t\in [0,T]\cap Q_i}{\sup }{X}_t\ge C]\le \frac{\mathrm{E}[\text{max}(X_T,0)]}{C}$

for each Template:Mvar, and this passes to the limit to yield the submartingale inequality.Template:Sfnm This passage from discrete time to continuous time is very flexible, as it only required having a countable dense subset of Template:Math, which can then automatically be built out of an increasing sequence of finite sets. As such, the submartingale inequality holds even for more general index sets, which are not required to be intervals or natural numbers.Template:Sfnm

Further inequalities

There are further submartingale inequalities also due to Doob. Now let Template:Math be a martingale or a positive submartingale; if the index set is uncountable, then (as above) assume that the sample paths are right-continuous. In these scenarios, Jensen's inequality implies that Template:Math is a submartingale for any number Template:Math, provided that these new random variables all have finite integral. The submartingale inequality is then applicable to say thatTemplate:Sfnm

$\text{P}[\underset{t}{\sup }|X_t|\ge C]\le \frac{\text{E}[|X_T{|}^p]}{C^p}.$

for any positive number Template:Mvar. Here Template:Mvar is the final time, i.e. the largest value of the index set. Furthermore one has

$\text{E}[|X_T{|}^p]\le \text{E}[\underset{0\le s\le T}{\sup }|X_s{|}^p]\le {\left(\frac{p}{p-1}\right)}^p\text{E}[|X_T{|}^p]$

if Template:Mvar is larger than one. This, sometimes known as Doob's maximal inequality, is a direct result of combining the layer cake representation with the submartingale inequality and the Hölder inequality.Template:Sfnm

In addition to the above inequality, there holdsTemplate:Sfnm

$\text{E}\left|\underset{0\le s\le T}{\sup }{X}_s\right|\le \frac{e}{e-1}(1+\text{E}[\max \{|X_T|\log |X_T|,0\}])$

Related inequalities

Doob's inequality for discrete-time martingales implies Kolmogorov's inequality: if X₁, X₂, ... is a sequence of real-valued independent random variables, each with mean zero, it is clear that

$\begin{array}{cc}\mathrm{E}[X_1+\cdots +X_n+X_{n+1}\mid X_1,\dots ,X_n]& =X_1+\cdots +X_n+\mathrm{E}[X_{n+1}\mid X_1,\dots ,X_n]\\ & =X_1+\cdots +X_n,\end{array}$

so S_n = X₁ + ... + X_n is a martingale. Note that Jensen's inequality implies that |S_n| is a nonnegative submartingale if S_n is a martingale. Hence, taking p = 2 in Doob's martingale inequality,

$P[\underset{1\le i\le n}{\max }\left|S_i\right|\ge \lambda ]\le \frac{\mathrm{E}\left[S_n^2\right]}{{\lambda }^2},$

which is precisely the statement of Kolmogorov's inequality.Template:Sfnm

Application: Brownian motion

Let B denote canonical one-dimensional Brownian motion. ThenTemplate:Sfnm

$P[\underset{0\le t\le T}{\sup }{B}_t\ge C]\le \exp (-\frac{C^2}{2T}).$

The proof is just as follows: since the exponential function is monotonically increasing, for any non-negative λ,

$\{\underset{0\le t\le T}{\sup }{B}_t\ge C\}=\{\underset{0\le t\le T}{\sup }\exp (\lambda B_t)\ge \exp (\lambda C)\}.$

By Doob's inequality, and since the exponential of Brownian motion is a positive submartingale,

$\begin{array}{cc}P[\underset{0\le t\le T}{\sup }{B}_t\ge C]& =P[\underset{0\le t\le T}{\sup }\exp (\lambda B_t)\ge \exp (\lambda C)]\\ & \le \frac{\mathrm{E}[\exp (\lambda B_T)]}{\exp (\lambda C)}\\ & =\exp (\frac{1}{2}{\lambda }^{2}T-\lambda C)& & \mathrm{E}[\exp (\lambda B_t)]=\exp \left(\frac{1}{2}{\lambda }^{2}t\right)\end{array}$

Since the left-hand side does not depend on λ, choose λ to minimize the right-hand side: λ = C/T gives the desired inequality.

References

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External links

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