Martingale (probability theory)
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probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, a martingale is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given the past, is equal to the present value. Martingales are used to model fair games, where future expected winnings are equal to the current amount regardless of past outcomes.


History

Originally, '' martingale'' referred to a class of betting strategies that was popular in 18th-century
France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan ...
. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the
exponential growth Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
of the bets eventually bankrupts its users due to finite bankrolls. Stopped Brownian motion, which is a martingale process, can be used to model the trajectory of such games. The concept of martingale in probability theory was introduced by Paul Lévy in 1934, though he did not name it. The term "martingale" was introduced later by , who also extended the definition to continuous martingales. Much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance.


Definitions

A basic definition of a discrete-time martingale is a discrete-time
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
(i.e., a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s) ''X''1, ''X''2, ''X''3, ... that satisfies for any time ''n'', :\mathbf ( \vert X_n \vert )< \infty :\mathbf (X_\mid X_1,\ldots,X_n)=X_n. That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.


Martingale sequences with respect to another sequence

More generally, a sequence ''Y''1, ''Y''2, ''Y''3 ... is said to be a martingale with respect to another sequence ''X''1, ''X''2, ''X''3 ... if for all ''n'' :\mathbf ( \vert Y_n \vert )< \infty :\mathbf (Y_\mid X_1,\ldots,X_n)=Y_n. Similarly, a continuous-time martingale with respect to the
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
''Xt'' is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
''Yt'' such that for all ''t'' :\mathbf ( \vert Y_t \vert )<\infty :\mathbf ( Y_ \mid \ ) = Y_s\quad \forall s \le t. This expresses the property that the conditional expectation of an observation at time ''t'', given all the observations up to time s , is equal to the observation at time ''s'' (of course, provided that ''s'' ≤ ''t''). The second property implies that Y_n is measurable with respect to X_1 \dots X_n.


General definition

In full generality, a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
Y:T\times\Omega\to S taking values in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
S with norm \lVert \cdot \rVert_ is a martingale with respect to a filtration \Sigma_* and
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
\mathbb P if * Σ is 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 filte ...
of the underlying
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
(Ω, Σ, \mathbb P); * ''Y'' is adapted to the filtration Σ, i.e., for each ''t'' in the
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
''T'', the random variable ''Yt'' is a Σ''t''-
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
; * for each ''t'', ''Yt'' lies in the ''Lp'' space ''L''1(Ω, Σ''t''\mathbb P; ''S''), i.e. ::\mathbf_ (\lVert Y_ \rVert_) < + \infty; * for all ''s'' and ''t'' with ''s'' < ''t'' and all ''F'' ∈ Σ''s'', ::\mathbf_ \left( _t-Y_schi_F\right) =0, :where ''χF'' denotes the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of the event ''F''. In Grimmett and Stirzaker's ''Probability and Random Processes'', this last condition is denoted as ::Y_s = \mathbf_ ( Y_t \mid \Sigma_s ), :which is a general form of conditional expectation. It is important to note that the property of being a martingale involves both the filtration ''and'' the probability measure (with respect to which the expectations are taken). It is possible that ''Y'' could be a martingale with respect to one measure but not another one; the Girsanov theorem offers a way to find a measure with respect to which an Itō process is a martingale. In the Banach space setting the conditional expectation is also denoted in operator notation as \mathbf^ Y_t.


Examples of martingales

* An unbiased
random walk In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a rand ...
, in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where at each time step a move to the right or left is equally likely. * A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair. The gambler is playing a game of coin flipping. Suppose ''Xn'' is the gambler's fortune after ''n'' tosses of a fair coin, such that the gambler wins $1 if the coin toss outcome is heads and loses $1 if the coin toss outcome is tails. The gambler's conditional expected fortune after the next game, given the history, is equal to his present fortune. This sequence is thus a martingale. * Let ''Yn'' = ''Xn''2 − ''n'' where ''Xn'' is the gambler's fortune from the prior example. Then the sequence is a martingale. This can be used to show that the gambler's total gain or loss varies roughly between plus or minus the
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the number of games of coin flipping played. * de Moivre's martingale: Suppose the coin toss outcomes are unfair, i.e., biased, with probability ''p'' of coming up heads and probability ''q'' = 1 − ''p'' of tails. Let ::X_=X_n\pm 1 :with "+" in case of "heads" and "−" in case of "tails". Let ::Y_n=(q/p)^ :Then is a martingale with respect to . To show this :: \begin E _ \mid X_1,\dots,X_n& = p (q/p)^ + q (q/p)^ \\ pt& = p (q/p) (q/p)^ + q (p/q) (q/p)^ \\ pt& = q (q/p)^ + p (q/p)^ = (q/p)^=Y_n. \end * Pólya's urn contains a number of different-coloured marbles; at each
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
a marble is randomly selected from the urn and replaced with several more of that same colour. For any given colour, the fraction of marbles in the urn with that colour is a martingale. For example, if currently 95% of the marbles are red then, though the next iteration is more likely to add red marbles than another color, this bias is exactly balanced out by the fact that adding more red marbles alters the fraction much less significantly than adding the same number of non-red marbles would. * Likelihood-ratio testing in
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
: A random variable ''X'' is thought to be distributed according either to probability density ''f'' or to a different probability density ''g''. A random sample ''X''1, ..., ''X''''n'' is taken. Let ''Y''''n'' be the "likelihood ratio" ::Y_n=\prod_^n\frac : If X is actually distributed according to the density ''f'' rather than according to ''g'', then is a martingale with respect to * In an ecological community, i.e. a group of species that are in a particular trophic level, competing for similar resources in a local area, the number of individuals of any particular species of fixed size is a function of (discrete) time, and may be viewed as a sequence of random variables. This sequence is a martingale under the unified neutral theory of biodiversity and biogeography. * If is a
Poisson process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points ...
with intensity ''λ'', then the compensated Poisson process is a continuous-time martingale with right-continuous/left-limit sample paths. * Wald's martingale * A d-dimensional process M=(M^,\dots,M^) in some space S^d is a martingale in S^d if each component T_i(M)=M^ is a one-dimensional martingale in S.


Submartingales, supermartingales, and relationship to harmonic functions

There are two generalizations of a martingale that also include cases when the current observation ''Xn'' is not necessarily equal to the future conditional expectation ''E''  ''X''1,...,''Xn''but instead an upper or lower bound on the conditional expectation. These generalizations reflect the relationship between martingale theory and
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...
, that is, the study of
harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...
s. Just as a continuous-time martingale satisfies E  nbsp;− ''X''''s'' = 0 ∀''s'' ≤ ''t'', a harmonic function ''f'' satisfies the
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
Δ''f'' = 0 where Δ is the Laplacian operator. Given a
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
process ''W''''t'' and a harmonic function ''f'', the resulting process ''f''(''W''''t'') is also a martingale. * A discrete-time submartingale is a sequence X_1,X_2,X_3,\ldots of integrable random variables satisfying ::\operatorname E _\mid X_1,\ldots,X_n\ge X_n. : Likewise, a continuous-time submartingale satisfies ::\operatorname E _t\mid\\ge X_s \quad \forall s \le t. :In potential theory, a subharmonic function ''f'' satisfies Δ''f'' ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary of a ball is bounded above by the harmonic function for all points inside the ball. Similarly, if a submartingale and a martingale have equivalent expectations for a given time, the history of the submartingale tends to be bounded above by the history of the martingale. Roughly speaking, the
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"sub-" is consistent because the current observation ''Xn'' is ''less than'' (or equal to) the conditional expectation ''E''  ''X''1,...,''Xn'' Consequently, the current observation provides support ''from below'' the future conditional expectation, and the process tends to increase in future time. * Analogously, a discrete-time supermartingale satisfies ::\operatorname E _\mid X_1,\ldots,X_n\le X_n. : Likewise, a continuous-time supermartingale satisfies ::\operatorname E _t\mid\\le X_s \quad \forall s \le t. :In potential theory, a superharmonic function ''f'' satisfies Δ''f'' ≤ 0. Any superharmonic function that is bounded below by a harmonic function for all points on the boundary of a ball is bounded below by the harmonic function for all points inside the ball. Similarly, if a supermartingale and a martingale have equivalent expectations for a given time, the history of the supermartingale tends to be bounded below by the history of the martingale. Roughly speaking, the prefix "super-" is consistent because the current observation ''Xn'' is ''greater than'' (or equal to) the conditional expectation ''E''  ''X''1,...,''Xn'' Consequently, the current observation provides support ''from above'' the future conditional expectation, and the process tends to decrease in future time.


Examples of submartingales and supermartingales

* Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that is ''both'' a submartingale and a supermartingale is a martingale. * Consider again the gambler who wins $1 when a coin comes up heads and loses $1 when the coin comes up tails. Suppose now that the coin may be biased, so that it comes up heads with probability ''p''. ** If ''p'' is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale. ** If ''p'' is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale. ** If ''p'' is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale. * A
convex function In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that ''Xn''2 − ''n'' is a martingale). Similarly, a
concave function In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any funct ...
of a martingale is a supermartingale.


Martingales and stopping times

A stopping time with respect to a sequence of random variables ''X''1, ''X''2, ''X''3, ... is a random variable τ with the property that for each ''t'', the occurrence or non-occurrence of the event ''τ'' = ''t'' depends only on the values of ''X''1, ''X''2, ''X''3, ..., ''X''''t''. The intuition behind the definition is that at any particular time ''t'', you can look at the sequence so far and tell if it is time to stop. An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings (for example, he might leave only when he goes broke), but he can't choose to go or stay based on the outcome of games that haven't been played yet. In some contexts the concept of ''stopping time'' is defined by requiring only that the occurrence or non-occurrence of the event ''τ'' = ''t'' is probabilistically independent of ''X''''t'' + 1, ''X''''t'' + 2, ... but not that it is completely determined by the history of the process up to time ''t''. That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used. One of the basic properties of martingales is that, if (X_t)_ is a (sub-/super-) martingale and \tau is a stopping time, then the corresponding stopped process (X_t^\tau)_ defined by X_t^\tau:=X_ is also a (sub-/super-) martingale. The concept of a stopped martingale leads to a series of important theorems, including, for example, the optional stopping theorem which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.


Martingale problem

The martingale problem is a framework in stochastic analysis for characterizing solutions to stochastic differential equations (SDEs) through martingale conditions.


General Martingale Problem (A, μ)

Let E be a Polish space with Borel \sigma-algebra \mathcal, and let \mathcal(E) be the set of probability measures on E. Suppose A : \mathcal(A) \to C(E) is a Markov pregenerator, where \mathcal(A) is a dense subspace of C(E). A probability measure \mathbb on the Skorokhod space D_E[0,\infty) solves the martingale problem (A, \mu) for \mu \in \mathcal(E) if: For every \Gamma \in \mathcal, \mathbb = \mu(\Gamma). For every f \in \mathcal(A), the process f(\zeta_t) - \int_0^t A f(\zeta_s),ds is a local martingale under \mathbb with respect to its natural filtration. If \mu = \delta_\eta (the Dirac measure at \eta), then \mathbb is said to solve the martingale problem for A with initial point \eta.


Martingale Problem for Diffusions M(a, b)

A process X = (X_t)_ on a filtered probability space (\Omega, \mathcal, (\mathcalt), \mathbb) solves the martingale problem M(a, b) for measurable functions a : \mathbb^d \to \mathbb+^d and b : \mathbb^d \to \mathbb^d if: For each 1 \le i \le d, M^i_t = X^i_t - \int_0^t b_i(X_s),ds is a local martingale. For each 1 \le i,j \le d, M^i_t,M^j_t - \int_0^t a_(X_s),ds is a local martingale.


Connection to Stochastic Differential Equations

Solutions to M(a, b) correspond (in a weak sense) to solutions of the SDE dX_t = b(X_t),dt + \sigma(X_t),dB_t, where \sigma\sigma^\top = a. One sees this by applying the generator A to simple functions such as x_i or x_i,x_j, thereby recovering the drift b and the diffusion matrix a.


See also

* Azuma's inequality *
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
* Doob martingale * Doob's martingale convergence theorems * Doob's martingale inequality *
Doob–Meyer decomposition theorem The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a Martingale (probability theory)#Submartingales and supermartingales, submartingale may be decomposed in a unique way as the sum of a ...
* Local martingale *
Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
* Markov property * Martingale (betting system) * Martingale central limit theorem * Martingale difference sequence * Martingale representation theorem * Normal number * Semimartingale


Notes


References

* * Entire issue dedicated to Martingale probability theory (Laurent Mazliak and Glenn Shafer, Editors). * * * * * * * Stroock, D. W. and Varadhan, S. R. S. (1979). ''Multidimensional Diffusion Processes''. Springer. * Ethier, S. N. and Kurtz, T. G. (1986). ''Markov Processes: Characterization and Convergence''. Wiley. {{Authority control Stochastic processes Martingale theory Game theory Paul Lévy (mathematician)