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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.

Contents

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,

X s ≤ E [ X t | F s ] .

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

P [ sup 0 ≤ t ≤ T X t ≥ C ] ≤ E [ | X T | ] C .

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

X : [ 0 , T ] × Ω → [ 0 , + ∞ )

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

E [ X T ] = ∫ Ω X T ( ω ) d P ( ω )

in the sense of Lebesgue integration. F s 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

S t = sup 0 ≤ s ≤ t X s ,

and for p ≥ 1 let

∥ X t ∥ p = ∥ X t ∥ L p ( Ω , F , P ) = ( E [ | X t | p ] ) 1 p .

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

P [ S T ≥ C ] ≤ ∥ X T ∥ 1 C .

The following inequalities also hold: for p = 1,

∥ S T ∥ p ≤ e e − 1 ( 1 + ∥ X T log + ⁡ X T ∥ p )

and, for p > 1,

∥ X T ∥ p ≤ ∥ S T ∥ p ≤ p p − 1 ∥ X T ∥ p .

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

E [ X 1 + ⋯ + X n + X n + 1 | X 1 , … , X n ] = X 1 + ⋯ + X n + E [ X n + 1 | X 1 , … , X n ] = X 1 + ⋯ + X n ,

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,

P [ max 1 ≤ i ≤ n | M i | ≥ λ ] ≤ E [ M n 2 ] λ 2 ,

which is precisely the statement of Kolmogorov's inequality.

Application: Brownian motion

Let B denote canonical one-dimensional Brownian motion. Then

P [ sup 0 ≤ t ≤ T B t ≥ C ] ≤ exp ⁡ ( − C 2 2 T ) .

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

{ sup 0 ≤ t ≤ T B t ≥ C } = { sup 0 ≤ t ≤ T exp ⁡ ( λ B t ) ≥ exp ⁡ ( λ C ) } .

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

P [ sup 0 ≤ t ≤ T B t ≥ C ] = P [ sup 0 ≤ t ≤ T exp ⁡ ( λ B t ) ≥ exp ⁡ ( λ C ) ] ≤ E [ exp ⁡ ( λ B T ) ] exp ⁡ ( λ C ) = exp ⁡ ( 1 2 λ 2 T − λ C ) E [ exp ⁡ ( λ B t ) ] = exp ⁡ ( 1 2 λ 2 t )

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

Doob's martingale inequality Wikipedia
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