# 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.

## 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 and *p* ≥ 1,

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*is a martingale. Note that Jensen's inequality implies that |

_{n}*M*| is a nonnegative submartingale if

_{n}*M*is a martingale. Hence, taking

_{n}*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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