Doob's martingale inequality

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In mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process 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 non-negative martingale, but the result is also valid for non-negative submartingales.

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

Statement of the inequality

Let X be a submartingale taking non-negative real values, either in discrete or continuous time. That is, for all times s and t with s < t,

(For a continuous-time submartingale, assume further that the process is càdlàg.) Then, for any constant C > 0,

In the above, as is conventional, P denotes the probability measure on the sample space Ω of the stochastic process

and E denotes the expected value with respect to the probability measure P, i.e. the integral

in the sense of Lebesgue integration. denotes the σ-algebra generated by all the random variables Xi with i ≤ s; the collection of such σ-algebras forms a filtration of the probability space.

Further inequalities

There are further (sub)martingale inequalities also due to Doob. With the same assumptions on X as above, let

and for p ≥ 1 let

In this notation, Doob's inequality as stated above reads

The following inequalities also hold, : for p = 1,

and, for p > 1,

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

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

which is precisely the statement of Kolmogorov's inequality.

Application: Brownian motion

Let B denote canonical one-dimensional Brownian motion. Then

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

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

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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