Doob's martingale inequality
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.
Contents
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 X_{i} 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 X_{1}, X_{2}, ... is a sequence of real-valued independent random variables, each with mean zero, it is clear that
so M_{n} = X_{1} + ... + X_{n} is a martingale. Note that Jensen's inequality implies that |M_{n}| is a nonnegative submartingale if M_{n} 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", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4