Martingale (probability theory) - Wikipedia, the free encyclopedia. Stopped Brownian motion is an example of a martingale. It can model an even coin- toss betting game with the possibility of bankruptcy. In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings. In particular, a martingale is a sequence of random variables (i. To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time.
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1 Zetasizer Nano series technical note MRK654-01 Introduction Zeta potential is a physical property which is exhibited by any particle in suspension. The stochastic process defined by. is called a Wiener process with drift μ and infinitesimal variance σ 2. These processes exhaust continuous Lévy processes.
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This website is progressively migrating to http:// The Electronic Journal of Probability (EJP) publishes full-length research articles in. In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings.
However, knowledge of the prior outcomes (e. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games.
History[edit]Originally, martingale referred to a class of betting strategies that was popular in 1. France.[1][2] The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his 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, his probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users, assuming the obvious and realistic finite bankrolls (one of the reasons casinos, though normatively enjoying a mathematical edge in the games offered to their patrons, impose betting limits). 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 LГ©vy in 1.
Ville (1. 93. 9), 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. Definitions[edit]A basic definition of a discrete- timemartingale is a discrete- time stochastic process (i. X1, X2, X3, .. that satisfies for any time n,That is, the conditional expected value of the next observation, given all the past observations, is equal to the last observation. Due to the linearity of expectation, this second requirement is equivalent to: or which states that the average "winnings" from observation to observation are 0. Martingale sequences with respect to another sequence[edit]More generally, a sequence Y1, Y2, Y3 ..
X1, X2, X3 .. if for all n. Similarly, a continuous- time martingale with respect to the stochastic process. Xt is a stochastic process. Yt such that for all t.
This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time , is equal to the observation at time s (of course, provided that s ≤ t). General definition[edit]In full generality, a stochastic process is a martingale with respect to a filtrationand probability measure P if. Σ∗ is a filtration of the underlying probability space (Ω, Σ, P); Y is adapted to the filtration Σ∗, i. T, the random variable Yt is a Σt- measurable function; for each t, Yt lies in the Lp space. L1(Ω, Σt, P; S), i. F ∈ Σs,where χF denotes the indicator function of the event F.
In Grimmett and Stirzaker's Probability and Random Processes, this last condition is denoted as. 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 It. ЕЌ process is a martingale. Examples of martingales[edit]An unbiased random walk (in any number of dimensions) is an example of a martingale.
A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair. Polya's urn contains a number of different coloured marbles, and each iteration a marble is randomly selected out of the urn and replaced with several more of that same colour. For any given colour, the ratio of marbles inside the urn with that colour is a martingale. For example, if currently 9. Suppose Xn is a gambler's fortune after n tosses of a fair coin, where the gambler wins $1 if the coin comes up heads and loses $1 if the coin comes up tails. The gambler's conditional expected fortune after the next trial, given the history, is equal to his present fortune, so this sequence is a martingale.
Let Yn = Xn. 2 в€’ n where Xn is the gambler's fortune from the preceding example. Then the sequence { Yn : n = 1, 2, 3, .. 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 steps.(de Moivre's martingale) Now suppose an "unfair" or "biased" coin, with probability p of "heads" and probability q = 1 в€’ p of "tails". Letwith "+" in case of "heads" and "в€’" in case of "tails".
Let. Then { Yn : n = 1, 2, 3, .. Xn : n = 1, 2, 3, .. To show this. (Likelihood- ratio testing in statistics) A population is thought to be distributed according to either a probability density f or another probability density g.
A random sample is taken, the data being X1, .., Xn. Let Yn be the "likelihood ratio"(which, in applications, would be used as a test statistic). If the population is actually distributed according to the density f rather than according to g, then { Yn : n = 1, 2, 3, .. } is a martingale with respect to { Xn : n = 1, 2, 3, .. }. Suppose each amoeba either splits into two amoebas, with probability p, or eventually dies, with probability 1 в€’ p.
Let Xn be the number of amoebas surviving in the nth generation (in particular Xn = 0 if the population has become extinct by that time). Let r be the probability of eventual extinction. Finding r as function of p is an instructive exercise. Hint: The probability that the descendants of an amoeba eventually die out is equal to the probability that either of its immediate offspring dies out, given that the original amoeba has split.) Thenis a martingale with respect to { Xn: n = 1, 2, 3, .. Software- created martingale series. In an ecological community (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 { Nt : t ≥ 0 } is a Poisson process with intensity λ, then the compensated Poisson process { Nt − λt : t ≥ 0 } is a continuous- time martingale with right- continuous/left- limit sample paths. Wald's martingale. Submartingales, supermartingales, and relationship to harmonic functions[edit]There are two popular generalizations of a martingale that also include cases when the current observation Xn is not necessarily equal to the future conditional expectation E[Xn+1|X1..,Xn] but instead an upper or lower bound on the conditional expectation. These definitions reflect a relationship between martingale theory and potential theory, which is the study of harmonic functions.
Just as a continuous- time martingale satisfies E[Xt|{Xτ : τ≤s}] − Xs = 0 ∀s ≤ t, a harmonic function f satisfies the partial differential equation Δf = 0 where Δ is the Laplacian operator. Given a Brownian motion process Wt and a harmonic function f, the resulting process f(Wt) is also a martingale. Likewise, a continuous- time submartingale satisfies. In potential theory, a subharmonic functionf satisfies Δf ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary of a ball are 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 prefix "sub- " is consistent because the current observation Xn is less than (or equal to) the conditional expectation E[Xn+1|X1..,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. Likewise, a continuous- time supermartingale satisfies. In potential theory, a superharmonic functionf satisfies Δf ≤ 0. Any superharmonic function that is bounded below by a harmonic function for all points on the boundary of a ball are 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[Xn+1|X1..,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[edit]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.
Wiener process - Wikipedia, the free encyclopedia. A single realization of a one- dimensional Wiener process.
A single realization of a three- dimensional Wiener process. In mathematics, the Wiener process is a continuous- time stochastic process named in honor of Norbert Wiener.
It is often called standard Brownian motion, after Robert Brown. It is one of the best known LГ©vy processes (c. Г dl. Г g stochastic processes with stationaryindependent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, and physics. The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described.
As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution.
In applied mathematics, the Wiener process is used to represent the integral of a white noise. Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and unknown forces in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations.
It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schr. ödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model. Characterisations of the Wiener process[edit]The Wiener process Wt is characterised by the following properties: [1]W0 = 0 a. W has independent increments: Wt+u - Wt is independent of σ(Ws : s ≤ t) for u ≥ 0. W has Gaussian increments: Wt+u - Wt is normally distributed with mean 0 and variance u, Wt+u−Wt ~ N(0, u)W has continuous paths: With probability 1, Wt is continuous in t. The independent increments means that if 0 ≤ s.
Wt. 1в€’Ws. 1 and Wt. Ws. 2 are independent random variables, and the similar condition holds for n increments.
An alternative characterisation of the Wiener process is the so- called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with W0 = 0 and quadratic variation [Wt, Wt] = t (which means that Wt. A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent N(0, 1) random variables. This representation can be obtained using the Karhunen–Lo. ève theorem. Another characterisation of a Wiener process is the Definite integral (from zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.[citation needed]The Wiener process can be constructed as the scaling limit of a random walk, or other discrete- time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher[citation needed].
Unlike the random walk, it is scale invariant, meaning thatis a Wiener process for any nonzero constant О±. The Wiener measure is the probability law on the space of continuous functionsg, with g(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral. Wiener process as a limit of random walk[edit]Let be i. For each n, define a continuous time stochastic process.
It is a random step function. Increment of is independent because are independent. For large n, is close to by the central limit theorem. It's tempting to believe that as , will approach Wiener process. The proof is provided by Donsker's theorem. This formulation explained why Brownian motion is ubiquitous.[2]Properties of a one- dimensional Wiener process[edit]Basic properties[edit]The unconditional probability density function, which follows Normal Distribution with mean = 0 and variance = t, at a fixed time t: The expectation is zero: The variance, using the computational formula, is t: Covariance and correlation[edit]The covariance and correlation: The results for the expectation and variance follow immediately from the definition that increments have a normal distribution, centered at zero. Thus. The results for the covariance and correlation follow from the definition that non- overlapping increments are independent, of which only the property that they are uncorrelated is used.
Suppose that t. 1 < t. Substitutingwe arrive at: Since W(t.
W(t. 1)−W(t. 0) and W(t. W(t. 1), are independent,Thus. Wiener representation[edit]Wiener (1. Brownian path in terms of a random Fourier series. If are independent Gaussian variables with mean zero and variance one, thenandrepresent a Brownian motion on . The scaled processis a Brownian motion on (cf. Karhunen–Lo. ève theorem).
Running maximum[edit]The joint distribution of the running maximumand Wt is. To get the unconditional distribution of , integrate over −∞ < w ≤ m : And the expectation[3]Self- similarity[edit]. A demonstration of Brownian scaling, showing for decreasing c. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically. Brownian scaling[edit]For every c > 0 the process is another Wiener process. Time reversal[edit]The process for 0 ≤ t ≤ 1 is distributed like Wt for 0 ≤ t ≤ 1.
Time inversion[edit]The process is another Wiener process. A class of Brownian martingales[edit]If a polynomialp(x, t) satisfies the PDEthen the stochastic processis a martingale. Example: is a martingale, which shows that the quadratic variation of W on [0, t] is equal to t. It follows that the expected time of first exit of W from (в€’c, c) is equal to c.
More generally, for every polynomial p(x, t) the following stochastic process is a martingale: where a is the polynomial. Example: the processis a martingale, which shows that the quadratic variation of the martingale on [0, t] is equal to. About functions p(xa, t) more general than polynomials, see local martingales.
Some properties of sample paths[edit]The set of all functions w with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely. Qualitative properties[edit]For every Оµ > 0, the function w takes both (strictly) positive and (strictly) negative values on (0, Оµ). The function w is continuous everywhere but differentiable nowhere (like the Weierstrass function).
Points of local maximum of the function w are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if w has a local maximum at t then. The same holds for local minima. The function w has no points of local increase, that is, no t > 0 satisfies the following for some ε in (0, t): first, w(s) ≤ w(t) for all s in (t − ε, t), and second, w(s) ≥ w(t) for all s in (t, t + ε). Local increase is a weaker condition than that w is increasing on (t − ε, t + ε).) The same holds for local decrease.
The function w is of unbounded variation on every interval. The quadratic variation of w over [0,t] is t. Zeros of the function w are a nowhere denseperfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).
Quantitative properties[edit]Local modulus of continuity: Global modulus of continuity (LГ©vy): Local time[edit]The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density Lt(В·). Thus,for a wide class of functions f (namely: all continuous functions; all locally integrable functions; all non- negative measurable functions). The density Lt is (more exactly, can and will be chosen to be) continuous. The number Lt(x) is called the local time at x of w on [0, t].
It is strictly positive for all x of the interval (a, b) where a and b are the least and the greatest value of w on [0, t], respectively. For x outside this interval the local time evidently vanishes.) Treated as a function of two variables x and t, the local time is still continuous.
Treated as a function of t (while x is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of w. These continuity properties are fairly non- trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone.
In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non- smoothness of the trajectory. Related processes[edit]. The generator of a Brownian motion is ½ times the Laplace–Beltrami operator. The image above is of the Brownian motion on a special manifold: the surface of a sphere. The stochastic process defined byis called a Wiener process with drift μ and infinitesimal variance σ2.
These processes exhaust continuous LГ©vy processes. Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge.