Incomplete Cholesky factorization

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,

$\mathbf {E} \left[X_{t}{\big |}{\mathcal {F}}_{s}\right]\geq X_{s}.$

(For a continuous-time submartingale, assume further that the process is càdlàg.) Then, for any constant C > 0 and p ≥ 1,

$\mathbf {P} \left[\sup _{0\leq t\leq T}X_{t}\geq C\right]\leq {\frac {\mathbf {E} \left[X_{T}^{p}\right]}{C^{p}}}.$

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

$X:[0,T]\times \Omega \to [0,+\infty )$

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

$\mathbf {E} [X_{T}]=\int _{\Omega }X_{T}(\omega )\,\mathrm {d} \mathbf {P} (\omega )$

in the sense of Lebesgue integration. ${\mathcal {F}}_{s}$ 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

$S_{t}=\sup _{0\leq s\leq t}X_{s},$

and for p ≥ 1 let

$\|X_{t}\|_{p}=\|X_{t}\|_{L^{p}(\Omega ,{\mathcal {F}},\mathbf {P} )}=\left(\mathbf {E} \left[|X_{t}|^{p}\right]\right)^{\frac {1}{p}}.$

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

$\mathbf {P} \left[S_{T}\geq C\right]\leq {\frac {\|X_{T}\|_{p}^{p}}{C^{p}}}.$

The following inequalities also hold: for p = 1,

$\|S_{T}\|_{p}\leq {\frac {e}{e-1}}\left(1+\|X_{T}\log X_{T}\|_{p}\right)$

and, for p > 1,

$\|X_{T}\|_{p}\leq \|S_{T}\|_{p}\leq {\frac {p}{p-1}}\|X_{T}\|_{p}.$

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

${\begin{aligned}\mathbf {E} \left[X_{1}+\dots +X_{n}+X_{n+1}{\big |}X_{1},\dots ,X_{n}\right]&=X_{1}+\dots +X_{n}+\mathbf {E} \left[X_{n+1}{\big |}X_{1},\dots ,X_{n}\right]\\&=X_{1}+\cdots +X_{n},\end{aligned}}$

so M_n = X₁ + ... + 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,

$\mathbf {P} \left[\max _{1\leq i\leq n}\left|M_{i}\right|\geq \lambda \right]\leq {\frac {\mathbf {E} \left[M_{n}^{2}\right]}{\lambda ^{2}}},$

which is precisely the statement of Kolmogorov's inequality.

Application: Brownian motion

Let B denote canonical one-dimensional Brownian motion. Then

$\mathbf {P} \left[\sup _{0\leq t\leq T}B_{t}\geq C\right]\leq \exp \left(-{\frac {C^{2}}{2T}}\right).$

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

$\left\{\sup _{0\leq t\leq T}B_{t}\geq C\right\}=\left\{\sup _{0\leq t\leq T}\exp(\lambda B_{t})\geq \exp(\lambda C)\right\}.$

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

${\begin{aligned}\mathbf {P} \left[\sup _{0\leq t\leq T}B_{t}\geq C\right]&=\mathbf {P} \left[\sup _{0\leq t\leq T}\exp(\lambda B_{t})\geq \exp(\lambda C)\right]\\&\leq {\frac {\mathbf {E} \left[\exp(\lambda B_{T})\right]}{\exp(\lambda C)}}\\&=\exp \left({\tfrac {1}{2}}\lambda ^{2}T-\lambda C\right)&&\mathbf {E} \left[\exp(\lambda B_{t})\right]=\exp \left({\tfrac {1}{2}}\lambda ^{2}t\right)\end{aligned}}$

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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