Martingale

2020 Mathematics Subject Classification: Primary: 60G42 Secondary: 60G44 [MSN][ZBL]

A stochastic process $ X = ( X _ {t} , {\mathcal F} _ {t} ) $, $ t \in T \subseteq [ 0 , \infty ) $, defined on a probability space $ ( \Omega , {\mathcal F} , {\mathsf P} ) $ with a non-decreasing family of $ \sigma $- algebras $ ( {\mathcal F} _ {t} ) _ {t \in T } $, $ {\mathcal F} _ {s} \subseteq {\mathcal F} _ {t} \subseteq {\mathcal F} $, $ s \leq t $, such that $ {\mathsf E} | X _ {t} | < \infty $, $ X _ {t} $ is $ {\mathcal F} _ {t} $- measurable and

$$ \tag{1 } {\mathsf E} ( X _ {t} \mid {\mathcal F} _ {s} ) = X _ {s} $$

(with probability 1). In the case of discrete time $ T = \{ 1 , 2 ,\dots \} $; in the case of continuous time $ T = [ 0 , \infty ) $. Related notions are stochastic processes which form a submartingale, if

$$ {\mathsf E} ( X _ {t} \mid {\mathcal F} _ {s} ) \geq X _ {s} , $$

or a supermartingale, if

$$ {\mathsf E} ( X _ {t} \mid {\mathcal F} _ {s} ) \leq X _ {s} . $$

Example 1. If $ \xi _ {1} , \xi _ {2} \dots $ is a sequence of independent random variables with $ {\mathsf E} \xi _ {j} = 0 $, then $ X = ( X _ {n} , {\mathcal F} _ {n} ) $, $ n \geq 1 $, with $ X _ {n} = \xi _ {1} + \dots + \xi _ {n} $ and $ {\mathcal F} _ {n} = \sigma \{ \xi _ {1} \dots \xi _ {n} \} $ the $ \sigma $- algebra generated by $ \xi _ {1} \dots \xi _ {n} $, is a martingale.

Example 2. Let $ Y = ( Y _ {n} , {\mathcal F} _ {n} ) $ be a martingale (submartingale), $ V = ( V _ {n} , {\mathcal F} _ {n} ) $ a predictable sequence (that is, $ V _ {n} $ is not only $ {\mathcal F} _ {n} $- measurable but also $ {\mathcal F} _ {n-1} $- measurable, $ n \geq 1 $), $ {\mathcal F} _ {0} = \{ \emptyset , \Omega \} $, and let

$$ ( V \cdot Y ) _ {n} = \ V _ {1} Y _ {1} + \sum _ { k=2}^ { n } V _ {k} \Delta Y _ {k} ,\ \ \Delta Y _ {k} = Y _ {k} - Y _ {k-1} . $$

Then, if the variables $ ( V \cdot Y ) _ {n} $ are integrable, the stochastic process $ ( ( V \cdot Y ) _ {n} , {\mathcal F} _ {n} ) $ forms a martingale (submartingale). In particular, if $ \xi _ {1} , \xi _ {2} \dots $ is a sequence of independent random variables corresponding to a Bernoulli scheme

$$ {\mathsf P} \{ \xi _ {i} = \pm 1 \} = \frac{1}{2} ,\ \ Y _ {k} = \xi _ {1} + \dots + \xi _ {k} , $$

$$ {\mathcal F} _ {k} = \sigma \{ \xi _ {1} \dots \xi _ {k} \} , $$

and

$$ \tag{2 } V _ {k} = \ \left \{ \begin{array}{ll} 2 &\textrm{ if } \xi _ {1} = \dots = \xi _ {k-1} = 1 , \\ 0 &\textrm{ otherwise } , \\ \end{array} \right .$$

then $ ( ( V \cdot Y ) _ {n} , {\mathcal F} _ {n} ) $ is a martingale. This stochastic process is a mathematical model of a game in which a player wins one unit of capital if $ \xi _ {k} = + 1 $ and loses one unit of capital if $ \xi _ {k} = - 1 $, and $ V _ {k} $ is the stake at the $ k $- th game. The game-theoretic sense of the function $ V _ {k} $ defined by (2) is that the player doubles his stake when he loses and stops the game on his first win. In the gambling world such a system is called a martingale, which explains the origin of the mathematical term "martingale" .

One of the basic facts of the theory of martingales is that the structure of a martingale (submartingale) $ X = ( X _ {t} , {\mathcal F} _ {t} ) $ is preserved under a random change of time. A precise statement of this (called the optimal sampling theorem) is the following: If $ \tau _ {1} $ and $ \tau _ {2} $ are two finite stopping times (cf. Markov moment), if $ {\mathsf P} \{ \tau _ {1} \leq \tau _ {2} \} = 1 $ and if

$$ \tag{3 } {\mathsf E} | X _ {\tau _ {i} } | \ < \infty ,\ \lim\limits _ { t } \ \inf \int\limits _ {\{ \tau _ {i} > t \} } | X _ {t} | d {\mathsf P} = 0 , $$

then $ {\mathsf E} ( X _ {\tau _ {2} } \mid {\mathcal F} _ {\tau _ {1} } ) ( \geq ) = X _ {\tau _ {1} } $( with probability 1), where

$$ {\mathcal F} _ {\tau _ {1} } = \ \{ {A \in {\mathcal F} } : {A \cap \{ \tau _ {1} \leq t \} \in {\mathcal F} _ {t} \textrm{ for all } t \in T } \} . $$

As a particular case of this the Wald identity follows:

$$ {\mathsf E} ( \xi _ {1} + \dots + \xi _ \tau ) = {\mathsf E} \xi _ {1} {\mathsf E} \tau . $$

Among the basic results of the theory of martingales is Doob's inequality: If $ X = ( X _ {n} , {\mathcal F} _ {n} ) $ is a non-negative submartingale,

$$ X _ {n} ^ {*} = \max _ {1 \leq j \leq n } X _ {j} , $$

$$ \| X _ {n} \| _ {p} = ( {\mathsf E} | X _ {n} | ^ {p} ) ^ {1/p} ,\ p \geq 1 ,\ n \geq 1 , $$

then

$$ \tag{4 } {\mathsf P} \{ X _ {n} ^ {*} \geq \epsilon \} \leq \ \frac{ {\mathsf E} X _ {n} } \epsilon , $$

$$ \tag{5 } \| X _ {n} \| _ {p} \leq \| X _ {n} ^ {*} \| _ {p} \leq \frac{p}{p-1} \| X _ {n} \| _ {p} ,\ p > 1 , $$

$$ \tag{6 } \| X _ {n} ^ {*} \| _ {p} \leq \frac{e}{e-1} [ 1 + \| X _ {n} \mathop{\rm ln} ^ {+} X _ {n} \| _ {p} ] ,\ p = 1 . $$

If $ X = ( X _ {n} , {\mathcal F} _ {n} ) $ is a martingale, then for $ p > 1 $ the Burkholder inequalities hold (generalizations of the inequalities of Khinchin and Marcinkiewicz–Zygmund for sums of independent random variables):

$$ \tag{7 } A _ {p} \| \sqrt {[ X ] _ {n} } \| _ {p} \leq \| X _ {n} \| _ {p} \leq B _ {p} \| \sqrt {[ X ] _ {n} } \| _ {p} , $$

where $ A _ {p} $ and $ B _ {p} $ are certain universal constants (not depending on $ X $ or $ n $), for which one can take

$$ A _ {p} = \ \left ( \frac{18 p ^ {3/2} }{p - 1 } \right ) ^ {-1} ,\ \ B _ {p} = \ \frac{18 p ^ {3/2} }{( p - 1 ) ^ {1/2} } , $$

and

$$ [ X ] _ {n} = \ \sum _ { i=1} ^ { n } ( \Delta X _ {i} ) ^ {2} ,\ \ X _ {0} = 0 . $$

Taking (5) and (7) into account, it follows that

$$ \tag{8 } A _ {p} \| \sqrt {[ X ] _ {n} } \| _ {p} \ \leq \| X _ {n} ^ {*} \| _ {p} \ \leq \widetilde{B} _ {p} \| \sqrt {[ X ] _ {n} } \| _ {p} , $$

where

$$ \widetilde{B} _ {p} = \ \frac{18 p ^ {5/2} }{( p - 1 ) ^ {3/2} } . $$

When $ p = 1 $ inequality (8) can be generalized. Namely, Davis' inequality holds: There are universal constants $ A $ and $ B $ such that

$$ A \| \sqrt {[ X ] _ {n} } \| _ {1} \ \leq \| X _ {n} ^ {*} \| _ {1} \ \leq B \| \sqrt {[ X ] _ {n} } \| _ {1} . $$

In the proof of a different kind of theorem on the convergence of submartingales with probability 1, a key role is played by Doob's inequality for the mathematical expectation $ {\mathsf E} \beta _ {n} ( a , b) $ of the number of upcrossings, $ \beta _ {n} ( a , b ) $, of the interval $ [ a , b ] $ by the submartingale $ X = ( X _ {n} , {\mathcal F} _ {n} ) $ in $ n $ steps; namely

$$ \tag{9 } {\mathsf E} \beta _ {n} ( a , b ) \leq \ \frac{ {\mathsf E} | X _ {n} | + | a | }{b - a } . $$

The basic result on the convergence of submartingales is Doob's theorem: If $ X = ( X _ {n} , {\mathcal F} _ {n} ) $ is a submartingale and $ \sup {\mathsf E} | X _ {n} | < \infty $, then with probability 1, $ \lim\limits _ {n \rightarrow \infty } X _ {n} $( $ = X _ \infty $) exists and $ {\mathsf E} | X _ \infty | < \infty $. If the submartingale $ X $ is uniformly integrable, then, in addition to convergence with probability $ 1 $, convergence in $ L _ {1} $ holds, that is,

$$ {\mathsf E} | X _ {n} - X _ \infty | \rightarrow 0 ,\ n \rightarrow \infty . $$

A corollary of this result is Lévy's theorem on the continuity of conditional mathematical expectations: If $ {\mathsf E} | \xi | < \infty $, then

$$ {\mathsf E} ( \xi | {\mathcal F} _ {n} ) \rightarrow {\mathsf E} ( \xi | {\mathcal F} _ \infty ) , $$

where $ {\mathcal F} _ {1} \subseteq {\mathcal F} _ {2} \subseteq \dots $ and $ {\mathcal F} _ \infty = \sigma ( \cup _ {n} {\mathcal F} _ {n} ) $.

A natural generalization of a martingale is the concept of a local martingale, that is, a stochastic process $ X = ( X _ {t} , {\mathcal F} _ {t} ) $ for which there is a sequence $ ( \tau _ {m} ) _ {m \geq 1 } $ of finite stopping times $ \tau _ {m} \uparrow \infty $( with probability 1), $ m \geq 1 $, such that for each $ m \geq 1 $ the "stopped" processes

$$ X ^ {\tau _ {m} } = \ ( X _ {t \wedge \tau _ {m} } I ( \tau _ {m} > 0 ) , {\mathcal F} _ {t} ) $$

are martingales. In the case of discrete time each local martingale $ X = ( X _ {n} , {\mathcal F} _ {n} ) $ is a martingale transform, that is, can be represented in the form $ X _ {n} = ( V \cdot Y ) _ {n} $, where $ V $ is a predictable process and $ Y $ is a martingale.

Each submartingale $ X = ( X _ {t} , {\mathcal F} _ {t} ) $ has, moreover, a unique Doob–Meyer decomposition $ X _ {t} = M _ {t} + A _ {t} $, where $ M = ( M _ {t} , {\mathcal F} _ {t} ) $ is a local martingale and $ A = ( A _ {t} , {\mathcal F} _ {t} ) $ is a predictable non-decreasing process. In particular, if $ m = ( m _ {t} , {\mathcal F} _ {t} ) $ is a square-integrable martingale, then its square $ m ^ {2} = ( m _ {t} ^ {2} , {\mathcal F} _ {t} ) $ is a submartingale in whose Doob–Meyer decomposition $ m _ {t} ^ {2} = M _ {t} + \langle m \rangle _ {t} $ the process $ \langle m \rangle = ( \langle m \rangle _ {t} , {\mathcal F} _ {t} ) $ is called the quadratic characteristic of the martingale $ m $. For each square-integrable martingale $ m $ and predictable process $ V = ( V _ {t} , {\mathcal F} _ {t} ) $ such that $ \int _ {0} ^ {t} V _ {s} ^ {2} d \langle m \rangle _ {s} < \infty $( with probability 1), $ t > 0 $, it is possible to define the stochastic integral

$$ ( V \cdot m ) _ {t} = \int\limits _ { 0 } ^ { t } V _ {s} d m _ {s} , $$

which is a local martingale. In the case of a Wiener process $ W = ( W _ {t} , {\mathcal F} _ {t} ) $, which is a square-integrable martingale, $ \langle m \rangle _ {t} = t $ and the stochastic integral $ ( V \cdot W ) _ {t} $ is none other than the Itô stochastic integral with respect to the Wiener process.

In the case of continuous time the Doob, Burkholder and Davis inequalities are still true (for right-continuous processes having left limits).

References

[D]	J.L. Doob, "Stochastic processes" , Chapman & Hall (1953) MR1570654 MR0058896 Zbl 0053.26802
[GS]	I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) (Translated from Russian) MR0346882 Zbl 0291.60019

Comments

Stopping times are also called optimal times, or, in the older literature, Markov times or Markov moments, cf. Markov moment. The optimal sampling theorem is also called the stopping theorem or Doob's stopping theorem.

The notion of a martingale is one of the most important concepts in modern probability theory. It is basic in the theories of Markov processes and stochastic integrals, and is useful in many parts of analysis (convergence theorems in ergodic theory, derivatives and lifting in measure theory, inequalities in the theory of singular integrals, etc.). More generally one can define martingales with values in $ \mathbf C $, $ \mathbf R ^ {n} $, a Hilbert or a Banach space; Banach-valued martingales are used in the study of Banach spaces (Radon–Nikodým property, etc.).

References

[DM]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. V-VIII. Theory of martingales (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001
[D2]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001
[N]	J. Neveu, "Discrete-parameter martingales" , North-Holland (1975) (Translated from French) MR0402915 Zbl 0345.60026
[V]	J. Ville, "Etude critique de la notion de collectif" , Gauthier-Villars (1939) Zbl 0021.14601 Zbl 0021.14505 Zbl 65.0547.05
[WH]	P. Wall, C.C. Heyde, "Martingale limit theory and its application" , Acad. Press (1980) MR624435