Martingale (probability theory)

Definitions

A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) ${\displaystyle X_1,X_2,X_3,\dots }$ that satisfies for any time ${\displaystyle n}$:

${\displaystyle \mathbf{E}[|X_n|]<\mathrm{\infty }}$

${\displaystyle \mathbf{E}[X_{n+1}\mid X_1,\dots ,X_n]=X_n.}$

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 ${\displaystyle Y_1,Y_2,Y_3,\dots }$ is said to be a martingale with respect to another sequence ${\displaystyle X_1,X_2,X_3,\dots }$ if for all ${\displaystyle n}$:

${\displaystyle \mathbf{E}[|Y_n|]<\mathrm{\infty }}$

${\displaystyle \mathbf{E}[Y_{n+1}\mid X_1,\dots ,X_n]=Y_n.}$

Similarly, a continuous-time martingale with respect to the stochastic process ${\displaystyle X_t}$ is a stochastic process ${\displaystyle Y_t}$ such that for all ${\displaystyle t}$:

${\displaystyle \mathbf{E}[|Y_t|]<\mathrm{\infty }}$

${\displaystyle \mathbf{E}[Y_t\mid \{X_{\tau },\tau \le s\}]=Y_s\mathrm{\forall }s\le t.}$

This expresses the property that the conditional expectation of an observation at time ${\displaystyle t}$, given all the observations up to time ${\displaystyle s}$, is equal to the observation at time ${\displaystyle s}$ (provided that ${\displaystyle s\le t}$). The second property implies that ${\displaystyle Y_n}$ is measurable with respect to ${\displaystyle X_1,\dots ,X_n}$.

General definition

In full generality, a stochastic process ${\displaystyle Y:T\times \mathrm{\Omega }\to S}$ taking values in a Banach space ${\displaystyle S}$ with norm ${\displaystyle \Vert \cdot {\Vert }_S}$ is a martingale with respect to a filtration ${\displaystyle {\mathrm{\Sigma }}_{\ast }}$ and probability measure ${\displaystyle \mathbb{P}}$ if:

• ${\displaystyle {\mathrm{\Sigma }}_{\ast }}$ is a filtration of the underlying probability space ${\displaystyle (\mathrm{\Omega },\mathrm{\Sigma },\mathbb{P})}$;
• ${\displaystyle Y}$ is adapted to the filtration ${\displaystyle {\mathrm{\Sigma }}_{\ast }}$, i.e., for each ${\displaystyle t}$ in the index set ${\displaystyle T}$, the random variable ${\displaystyle Y_t}$ is a ${\displaystyle {\mathrm{\Sigma }}_t}$-measurable function;
• for each ${\displaystyle t}$, ${\displaystyle Y_t}$ lies in the L^p space ${\displaystyle L^1(\mathrm{\Omega },{\mathrm{\Sigma }}_t,\mathbb{P};S)}$, i.e.,

${\displaystyle {\mathbf{E}}_{\mathbb{P}}[\Vert Y_t{\Vert }_S]<+\mathrm{\infty };}$

• for all ${\displaystyle s}$ and ${\displaystyle t}$ with ${\displaystyle s<t}$ and all ${\displaystyle F\in {\mathrm{\Sigma }}_s}$,

${\displaystyle {\mathbf{E}}_{\mathbb{P}}\left[(Y_t-Y_s){\chi }_F\right]=0,}$

where ${\displaystyle {\chi }_F}$ denotes the indicator function of the event ${\displaystyle F}$. In Grimmett and Stirzaker's Probability and Random Processes, this last condition is denoted as:

${\displaystyle Y_s={\mathbf{E}}_{\mathbb{P}}[Y_t\mid {\mathrm{\Sigma }}_s],}$

which is a general form of conditional expectation. This is also called the martingale property.

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 ${\displaystyle Y}$ 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 ${\displaystyle {\mathbf{E}}^{{\mathrm{\Sigma }}_s}{Y}_t}$.

Examples of martingales

• An unbiased random walk, 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. Suppose ${\displaystyle X_n}$ is the gambler's fortune after ${\displaystyle n}$ tosses of a fair coin, such that the gambler wins $1 if the coin toss outcome is heads and loses $1 if the outcome is tails. The gambler's conditional expected fortune after the next game, given the history, is equal to his present fortune.
• Let ${\displaystyle Y_n=X_n^2-n}$ where ${\displaystyle X_n}$ is the gambler's fortune from the prior example. Then the sequence ${\displaystyle \{Y_n:n=1,2,3,\dots \}}$ 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 of the number of games played.
• de Moivre's martingale: Suppose the coin toss outcomes are unfair (biased), with probability ${\displaystyle p}$ of coming up heads and probability ${\displaystyle q=1-p}$ of tails. Let

${\displaystyle X_{n+1}=X_n\pm 1}$

with "+" in case of "heads" and "−" in case of "tails". Let

${\displaystyle Y_n=(q/p{)}^{X_n}}$

Then ${\displaystyle \{Y_n:n=1,2,3,\dots \}}$ is a martingale with respect to ${\displaystyle \{X_n:n=1,2,3,\dots \}}$. To show this:

${\displaystyle \begin{array}{cc}\mathbf{E}[Y_{n+1}\mid X_1,\dots ,X_n]& =p(q/p{)}^{X_n+1}+q(q/p{)}^{X_n-1}\\ & =p(q/p)(q/p{)}^{X_n}+q(p/q)(q/p{)}^{X_n}\\ & =q(q/p{)}^{X_n}+p(q/p{)}^{X_n}=(q/p{)}^{X_n}=Y_n.\end{array}}$

• Pólya's urn contains a number of different-coloured marbles; at each 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.
• Likelihood-ratio testing in statistics: A random variable ${\displaystyle X}$ is thought to be distributed according either to probability density ${\displaystyle f}$ or to a different probability density ${\displaystyle g}$. A random sample ${\displaystyle X_1,\dots ,X_n}$ is taken. Let ${\displaystyle Y_n}$ be the "likelihood ratio":

${\displaystyle Y_n=\prod _{i=1}^n\frac{g(X_i)}{f(X_i)}}$

If ${\displaystyle X}$ is actually distributed according to the density ${\displaystyle f}$ rather than ${\displaystyle g}$, then ${\displaystyle \{Y_n\}}$ is a martingale with respect to ${\displaystyle \{X_n\}}$.

• Doob martingale: Let ${\displaystyle Y}$ be any random variable with a finite expected value, and let ${\displaystyle \{{\mathcal{F}}_n\}}$ be a filtration. The sequence defined by ${\displaystyle X_n=\mathbf{E}[Y\mid {\mathcal{F}}_n]}$ is a martingale. This represents the sequentially updated best estimate of ${\displaystyle Y}$ as more information becomes available.

• In an ecological community, the number of individuals of any particular species of fixed size is a function of 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.
• In population genetics, the frequency of an allele in a population of fixed size evolving under genetic drift (with no mutation or selection) follows a martingale. This is a defining feature of the Wright–Fisher model.
• If ${\displaystyle \{N_t:t\ge 0\}}$ is a Poisson process with intensity ${\displaystyle \lambda }$, then the compensated Poisson process ${\displaystyle \{N_t-\lambda t:t\ge 0\}}$ is a continuous-time martingale with right-continuous/left-limit sample paths.
• Exponential martingale: If ${\displaystyle W_t}$ is a standard one-dimensional Brownian motion, then the process ${\displaystyle Z_t=\exp \left(\theta W_t-\frac{1}{2}{\theta }^{2}t\right)}$ is a continuous-time martingale for any real parameter ${\displaystyle \theta }$. This process plays a critical role in stochastic calculus and the Girsanov theorem for changing probability measures.
• Wald's martingale: If ${\displaystyle X_1,X_2,\dots }$ are independent and identically distributed random variables with moment-generating function ${\displaystyle M(\theta )=\mathbf{E}[\exp (\theta X_1)]}$, and their partial sums are denoted by ${\displaystyle S_n=X_1+\cdots +X_n}$, then the sequence ${\displaystyle W_n=\exp (\theta S_n)/M(\theta {)}^n}$ is a martingale.
• Asset pricing in mathematical finance: By the fundamental theorem of asset pricing, in an arbitrage-free market, the discounted price of any tradable asset is a martingale under the risk-neutral measure. For example, if ${\displaystyle S_t}$ is a stock price and ${\displaystyle r}$ is the risk-free interest rate, then ${\displaystyle e^{-rt}{S}_t}$ is a martingale under the equivalent martingale measure.
• A ${\displaystyle d}$-dimensional process ${\displaystyle M=(M^{(1)},\dots ,M^{(d)})}$ in some space ${\displaystyle S^d}$ is a martingale if each component ${\displaystyle T_i(M)=M^{(i)}}$ is a one-dimensional martingale in ${\displaystyle S}$.

Submartingales, supermartingales, and relationship to harmonic functions

There are two generalizations of a martingale that include cases when the current observation ${\displaystyle X_n}$ is not necessarily equal to the future conditional expectation, but instead acts as an upper or lower bound on the conditional expectation.

These generalizations reflect the relationship between martingale theory and potential theory (the study of harmonic functions):

• Just as a continuous-time martingale satisfies ${\displaystyle \mathbf{E}[X_t\mid \{X_{\tau }:\tau \le s\}]-X_s=0}$ for all ${\displaystyle s\le t}$, a harmonic function ${\displaystyle f}$ satisfies the partial differential equation ${\displaystyle \mathrm{\Delta }f=0}$ where ${\displaystyle \mathrm{\Delta }}$ is the Laplacian operator.
• Given a Brownian motion process ${\displaystyle W_t}$ and a harmonic function ${\displaystyle f}$, the resulting process ${\displaystyle f(W_t)}$ is also a martingale.

Submartingales:

• A discrete-time submartingale is a sequence ${\displaystyle X_1,X_2,X_3,\dots }$ of integrable random variables satisfying:

${\displaystyle \mathbf{E}[X_{n+1}\mid X_1,\dots ,X_n]\ge X_n.}$

• Likewise, a continuous-time submartingale satisfies:

${\displaystyle \mathbf{E}[X_t\mid \{X_{\tau }:\tau \le s\}]\ge X_s\mathrm{\forall }s\le t.}$

• In potential theory, a subharmonic function ${\displaystyle f}$ satisfies ${\displaystyle \mathrm{\Delta }f\ge 0}$. Any subharmonic function bounded above by a harmonic function for all points on the boundary of a ball is bounded above by the harmonic function 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 prefix "sub-" is consistent because the current observation ${\displaystyle X_n}$ is less than (or equal to) the conditional expectation. Consequently, the process tends to increase in future time.

Supermartingales:

• Analogously, a discrete-time supermartingale satisfies:

${\displaystyle \mathbf{E}[X_{n+1}\mid X_1,\dots ,X_n]\le X_n.}$

• Likewise, a continuous-time supermartingale satisfies:

${\displaystyle \mathbf{E}[X_t\mid \{X_{\tau }:\tau \le s\}]\le X_s\mathrm{\forall }s\le t.}$

• In potential theory, a superharmonic function ${\displaystyle f}$ satisfies ${\displaystyle \mathrm{\Delta }f\le 0}$.
• Roughly speaking, the prefix "super-" is consistent because the current observation ${\displaystyle X_n}$ is greater than (or equal to) the conditional expectation. Consequently, 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 it comes up tails. Suppose now that the coin may be biased, coming up heads with probability ${\displaystyle p}$.
  • If ${\displaystyle p=1/2}$, the gambler on average neither wins nor loses money, and the fortune over time is a martingale.
  • If ${\displaystyle p<1/2}$, the gambler loses money on average, and the fortune over time is a supermartingale.
  • If ${\displaystyle p>1/2}$, the gambler wins money on average, and the fortune over time is a submartingale.
• A convex function 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 ${\displaystyle X_n^2-n}$ is a martingale). Similarly, a concave function of a martingale is a supermartingale.

Martingales and stopping times

A stopping time with respect to a sequence of random variables ${\displaystyle X_1,X_2,X_3,\dots }$ is a random variable ${\displaystyle \tau }$ with the property that for each ${\displaystyle t}$, the occurrence or non-occurrence of the event ${\displaystyle \tau =t}$ depends only on the values of ${\displaystyle X_1,X_2,\dots ,X_t}$. The intuition behind the definition is that at any particular time ${\displaystyle t}$, you can look at the sequence so far and tell if it is time to stop. An example might be the time at which a gambler leaves the gambling table, which might depend on previous winnings, but not on 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 ${\displaystyle \tau =t}$ is probabilistically independent of ${\displaystyle X_{t+1},X_{t+2},\dots }$ but not completely determined by the history up to time ${\displaystyle t}$.

One of the basic properties of martingales is that, if ${\displaystyle (X_t{)}_{t>0}}$ is a (sub-/super-) martingale and ${\displaystyle \tau }$ is a stopping time, then the corresponding stopped process ${\displaystyle (X_t^{\tau }{)}_{t>0}}$ defined by ${\displaystyle X_t^{\tau }:=X_{min\{\tau ,t\}}}$ is also a (sub-/super-) martingale.

The concept of a stopped martingale leads to a series of important theorems, including 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 convergence

One of the central results of martingale theory is Doob's martingale convergence theorem. In one standard discrete-time form, if ${\displaystyle (X_n,{\mathcal{F}}_n{)}_{n\ge 0}}$ is a supermartingale satisfying

${\displaystyle \underset{n\ge 0}{sup}\mathrm{E}[|X_n|]<\mathrm{\infty },}$

then there exists an integrable random variable ${\displaystyle X_{\mathrm{\infty }}}$ such that

${\displaystyle X_n⟶X_{\mathrm{\infty }}}$ almost surely.

In particular, every nonnegative supermartingale converges almost surely, and its limiting expectation satisfies

${\displaystyle \mathrm{E}[X_{\mathrm{\infty }}]\le \mathrm{E}[X_0].}$

For martingales, stronger convergence is characterized by uniform integrability. A martingale ${\displaystyle (X_n,{\mathcal{F}}_n)}$ is uniformly integrable if and only if it converges almost surely and in ${\displaystyle L^1}$ to an integrable random variable ${\displaystyle X_{\mathrm{\infty }}}$. In that case,

${\displaystyle X_n=\mathrm{E}[X_{\mathrm{\infty }}\mid {\mathcal{F}}_n]}$

for every ${\displaystyle n}$. Thus each value of the martingale is the conditional expectation of its terminal value given the information available at that time. Almost-sure convergence alone does not generally imply ${\displaystyle L^1}$ convergence or preservation of expectations.

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 ${\displaystyle E}$ be a Polish space with Borel ${\displaystyle \sigma }$-algebra ${\displaystyle \mathcal{E}}$, and let ${\displaystyle \mathcal{P}(E)}$ be the set of probability measures on ${\displaystyle E}$. Suppose ${\displaystyle A:\mathcal{D}(A)\to C(E)}$ is a Markov pregenerator, where ${\displaystyle \mathcal{D}(A)}$ is a dense subspace of ${\displaystyle C(E)}$. A probability measure ${\displaystyle \mathbb{P}}$ on the Skorokhod space ${\displaystyle D_E[0,\mathrm{\infty })}$ solves the martingale problem ${\displaystyle (A,\mu )}$ for ${\displaystyle \mu \in \mathcal{P}(E)}$ if:

• For every ${\displaystyle \mathrm{\Gamma }\in \mathcal{E}}$, ${\displaystyle \mathbb{P}({\zeta }_0\in \mathrm{\Gamma })=\mu (\mathrm{\Gamma }).}$
• For every ${\displaystyle f\in \mathcal{D}(A)}$, the process ${\displaystyle f({\zeta }_t)-{\int }_0^{t}Af({\zeta }_s)ds}$ is a local martingale under ${\displaystyle \mathbb{P}}$ with respect to its natural filtration.

If ${\displaystyle \mu ={\delta }_{\eta }}$ (the Dirac measure at ${\displaystyle \eta }$), then ${\displaystyle \mathbb{P}}$ is said to solve the martingale problem for ${\displaystyle A}$ with initial point ${\displaystyle \eta }$.

Martingale Problem for Diffusions M(a, b)

A process ${\displaystyle X=(X_t{)}_{t\ge 0}}$ on a filtered probability space ${\displaystyle (\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t),\mathbb{P})}$ solves the martingale problem ${\displaystyle M(a,b)}$ for measurable functions ${\displaystyle a:{\mathbb{R}}^d\to {\mathbb{S}}_{+}^d}$ and ${\displaystyle b:{\mathbb{R}}^d\to {\mathbb{R}}^d}$ if:

• For each ${\displaystyle 1\le i\le d}$, ${\displaystyle M_t^i=X_t^i-{\int }_0^t{b}_i(X_s)ds}$ is a local martingale.
• For each ${\displaystyle 1\le i,j\le d}$, ${\displaystyle M_t^i{M}_t^j-{\int }_0^t{a}_{ij}(X_s)ds}$ is a local martingale.

Connection to Stochastic Differential Equations

Solutions to ${\displaystyle M(a,b)}$ correspond (in a weak sense) to solutions of the SDE ${\displaystyle d{X}_t=b(X_t)dt+\sigma (X_t)d{B}_t}$, where ${\displaystyle \sigma {\sigma }^{\mathrm{\top }}=a}$. One sees this by applying the generator ${\displaystyle A}$ to simple functions such as ${\displaystyle x_i}$ or ${\displaystyle x_i{x}_j}$, thereby recovering the drift ${\displaystyle b}$ and the diffusion matrix ${\displaystyle a}$.

Applications in Mathematical Finance

Martingales form the foundational mathematics behind modern quantitative finance and asset pricing theory.

• The Fundamental Theorem of Asset Pricing: This theorem states that a financial market is free of arbitrage if and only if there exists a risk-neutral probability measure (also known as an equivalent martingale measure).
• Risk-Neutral Pricing: Under this equivalent martingale measure, the discounted price process of any tradable asset is a martingale. This implies that the current price of an asset is exactly equal to the expected value of its future discounted payoffs.
• Derivatives Pricing: Models such as the Black–Scholes model rely heavily on martingale representations. By finding the correct martingale measure via the Girsanov theorem, complex financial derivatives and options can be priced accurately without needing to predict the actual real-world direction of the stock market.