Doob's martingale inequality

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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 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 martingale, but the result is also valid for submartingales.

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

Statement of the inequality[]

Let X be a submartingale taking 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 a probability measure on the sample space Ω of the stochastic process

and 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 submartingale 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 :

and, for p > 1,

The last of these is sometimes known as Doob's Maximal inequality.

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 Sn = X1 + ... + Xn is a martingale. Note that Jensen's inequality implies that |Sn| is a nonnegative submartingale if Sn 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[]

  • Revuz, Daniel; Yor, Marc (1999). Continuous martingales and Brownian motion (Third ed.). Berlin: Springer. ISBN 3-540-64325-7. (Theorem II.1.7)
  • Shiryaev, Albert N. (2001) [1994], "Martingale", Encyclopedia of Mathematics, EMS Press
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