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.
Contents
- Statement of the inequality
- Further inequalities
- Related inequalities
- Application Brownian motion
- References
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.
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.