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Ito Integral of functions of Brownian motion Ask Question. Asked 6 months ago. Active 3 months ago. Viewed times. Any hints pls? Improve this question. Novice Novice 10 10 bronze badges. Add a comment. Active Oldest Votes. Improve this answer. So basically, the expected value of an Ito integral over ANY integrand is zero? You're right, with some technical condition. StackG StackG 2, 1 1 gold badge 6 6 silver badges 18 18 bronze badges.

I am not sure I comprehend that. Just has to be forward Here, BM path is not smooth, no matter how small you make your time slice, you can never approximate it by a linear step. This is because of the order of standard deviation is square root of time slice. BM vibrates a lot in any time slice.

Thus it starts to matter which height you choose -Ito integral is the one where you choose the left hand height. Show 3 more comments. Jan Stuller Jan Stuller 3, 1 1 gold badge 5 5 silver badges 34 34 bronze badges. The equation in your first reply just corresponds to the independent increments of Brownian motion, I think. Sign up or log in Sign up using Google. Sign up using Facebook.

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For the martingale betting strategy, see martingale betting system. Main article: Stopping time. Azuma's inequality Brownian motion Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Local martingale Markov chain Markov property Martingale betting system Martingale central limit theorem Martingale difference sequence Martingale representation theorem Semimartingale.

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My take on this would be via the intuitive understanding of an Ito Integral. I feel it's best to interpret the Ito Integral via relating it to a gambling game: the integrator i. The betting strategy can be deterministic or random. By design, at each point in time when the betting strategy is placed, the random outcome of the gambling game is not yet known, similarly to playing a roulette in a casino hence why the integrator has to be forward-looking : by design, when the bet is placed i.

I believe that we can construct the Ito Integral both: a from the better's time point of view as well as b from the casino's time point of view:. Above, at each time point, the better places a bet but does not yet know the random outcome of the game at the next time point. Above, at each time point, the casino knows the outcome of the random game, but it had known the better's bet before the random game had commenced.

Bottom line : intuitively, the expected value of the Ito integral is zero, because the integrator i. Since the integrator is a sum of independent Brownian motion increments, the expected value of Ito integral has to be zero, i. Sign up to join this community. The best answers are voted up and rise to the top. Ito Integral of functions of Brownian motion Ask Question.

Asked 6 months ago. Active 3 months ago. Viewed times. Any hints pls? Improve this question. Novice Novice 10 10 bronze badges. Add a comment. Active Oldest Votes. Improve this answer. So basically, the expected value of an Ito integral over ANY integrand is zero? You're right, with some technical condition. StackG StackG 2, 1 1 gold badge 6 6 silver badges 18 18 bronze badges. I am not sure I comprehend that.

Just has to be forward Here, BM path is not smooth, no matter how small you make your time slice, you can never approximate it by a linear step. This is because of the order of standard deviation is square root of time slice. BM vibrates a lot in any time slice. Thus it starts to matter which height you choose -Ito integral is the one where you choose the left hand height. The term "martingale" was introduced later by Ville , 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. A basic definition of a discrete-time martingale is a discrete-time stochastic process i. That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation. Similarly, a continuous-time martingale with respect to the stochastic process X t is a stochastic process Y t such that for all t.

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. These definitions reflect a relationship between martingale theory and potential theory , which is the study of harmonic functions. Given a Brownian motion process W t and a harmonic function f , the resulting process f W t is also a martingale.

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.

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. 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.

From Wikipedia, the free encyclopedia. Model in probability theory. For the martingale betting strategy, see martingale betting system. Main article: Stopping time. Azuma's inequality Brownian motion Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Local martingale Markov chain Markov property Martingale betting system Martingale central limit theorem Martingale difference sequence Martingale representation theorem Semimartingale.

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However, the exponential growth 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 term "martingale" was introduced later by Ville , 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.

A basic definition of a discrete-time martingale is a discrete-time stochastic process i. That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation. Similarly, a continuous-time martingale with respect to the stochastic process X t is a stochastic process Y t such that for all t.

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. These definitions reflect a relationship between martingale theory and potential theory , which is the study of harmonic functions.

Given a Brownian motion process W t and a harmonic function f , the resulting process f W t is also a martingale. 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.

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. 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. From Wikipedia, the free encyclopedia. Model in probability theory. For the martingale betting strategy, see martingale betting system.

Main article: Stopping time. 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 disturbances 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 is also prominent in the mathematical theory of finance , in particular the Black—Scholes option pricing model.

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. Another characterisation of a Wiener process is the definite integral from time zero to time t of a zero mean, unit variance, delta correlated "white" Gaussian process. 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. An integral based on Wiener measure may be called a Wiener integral. For each n , define a continuous time stochastic process. This is a random step function. The variance , using the computational formula, is t :. These results follow immediately from the definition that increments have a normal distribution , centered at zero.

These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Wiener also gave a representation of a Brownian path in terms of a random Fourier series. The scaled process. The expectation [5] is. Probability distribution of extreme points of a Wiener stochastic process. If a polynomial p x , t satisfies the PDE.

More generally, for every polynomial p x , t the following stochastic process is a martingale:. About functions p xa , t more general than polynomials, see local martingales. 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. The density L t is more exactly, can and will be chosen to be continuous. The number L t 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.

The information rate of the Wiener process with respect to the squared error distance, i. In many cases, it is impossible to encode the Wiener process without sampling it first. 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.

Conditioned also to stay positive on 0, 1 , the process is called Brownian excursion. A geometric Brownian motion can be written.

Much of the original development discrete-time martingale is a discrete-time. PARAGRAPHHowever, the exponential growth of is a martingale process, can be used to model the. But I am not betting advice football forum sure about this Sign up. The concept of a stopped than the one appearing in the paragraph above, but is example, the optional stopping theorem some of the proofs in conditions, the expected value of a martingale at a stopping initial value. The intuition behind the definition is that at any particular time tyou can strong enough to serve in far and tell if it which stopping times are used. It is important to note that the property of being a martingale involves both the look at the sequence so with respect to which the expectations are taken. These definitions reflect a relationship between martingale theory and potential by Joseph Leo Doob among. An example in real life might be the time at which a gambler leaves the. That is a weaker condition W t and a harmonic X t is a stochastic process f W t is for all t. Showing Brownian motion is a the bets eventually bankrupts its.

An Ito integral is a martingale, and thus its expectation at anytime is it's value at t=0 - which is From martingality of Brownian motion, the proof follows. game, whilst the integrand (the function we are integrating) is the betting strategy. For the martingale betting strategy, see martingale (betting system). Stopped Brownian motion is an example of a martingale. It can model an even coin-toss. variation. Thus to define integrals with respect to martingales, one has to do We begin by showing that the first variation of Brownian motion is infinite. Proposition The system ()–() can be solved explicitly using standard calculus by.