In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
. It gives a bound on the probability that a
submartingale
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all ...
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
martingale, but the result is also valid for submartingales.
The inequality is due to the American mathematician
Joseph L. Doob
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, ...
.
Statement of the inequality
The setting of Doob's inequality is a
submartingale
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all ...
relative to a
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
of the underlying probability space. The
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
on the
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
of the martingale will be denoted by . The corresponding
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of a random variable , as defined by
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
, will be denoted by .
Informally, Doob's inequality states that the expected value of the process at some final time controls the probability that a
sample path
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of so ...
will reach above any particular value beforehand. As the proof uses very direct reasoning, it does not require any restrictive assumptions on the underlying filtration or on the process itself, unlike for many other theorems about stochastic processes. In the continuous-time setting, right-continuity (or left-continuity) of the sample paths is required, but only for the sake of knowing that the supremal value of a sample path equals the supremum over an arbitrary countable dense subset of times.
Discrete time
Let be a discrete-time submartingale relative to a filtration
of the underlying probability space, which is to say:
:
The submartingale inequality says that
:
for any positive number . The proof relies on the set-theoretic fact that the event defined by may be decomposed as the disjoint union of the events defined by and for all . Then
:
having made use of the submartingale property for the last inequality and the fact that
for the last equality. Summing this result as ranges from 1 to results in the conclusion
:
which is sharper than the stated result. By using the elementary fact that , the given submartingale inequality follows.
In this proof, the submartingale property is used once, together with the definition of
conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
. The proof can also be phrased in the language of stochastic processes so as to become a corollary of the powerful theorem that a
stopped submartingale is itself a submartingale. In this setup, the minimal index appearing in the above proof is interpreted as a
stopping time
In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of “random time”: a random variable whose value is inter ...
.
Continuous time
Now let be a submartingale indexed by an interval of real numbers, relative to a filtration of the underlying probability space, which is to say:
:
for all The submartingale inequality says that if the sample paths of the martingale are almost-surely right-continuous, then
:
for any positive number . This is a corollary of the above discrete-time result, obtained by writing
:
in which is any sequence of finite sets whose union is the set of all rational numbers. The first equality is a consequence of the right-continuity assumption, while the second equality is only set-theoretic. The discrete-time inequality applies to say that
:
for each , and this passes to the limit to yield the submartingale inequality. This passage from discrete time to continuous time is very flexible, as it only required having a countable dense subset of , which can then automatically be built out of an increasing sequence of finite sets. As such, the submartingale inequality holds even for more general index sets, which are not required to be intervals or
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s.
Further inequalities
There are further submartingale inequalities also due to Doob. Now let be a martingale or a positive submartingale; if the index set is uncountable, then (as above) assume that the sample paths are right-continuous. In these scenarios,
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
implies that is a submartingale for any number , provided that these new random variables all have finite integral. The submartingale inequality is then applicable to say that
:
for any positive number . Here is the ''final time'', i.e. the largest value of the index set. Furthermore one has
: