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• Stochastic ProcessesSOLO HERMELINUpdated: 10.05.11 15.06.14http://www.solohermelin.com

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• SOLOStochastic Processes Table of Content Langevin EquationLvy Process

• SOLOStochastic Processes Table of Content (continue)

• *Random ProcessesSOLORandom Variable: A variable x determined by the outcome of a random experiment. Random Process or Stochastic Process: A function of time x determined by the outcome of a random experiment. This is a family or an ensemble of functions of time, in general different for each outcome . Mean or Ensemble Average of the Random Process: Autocorrelation of the Random Process: Autocovariance of the Random Process:

• *SOLOStationarity of a Random Process1. Wide Sense Stationarity of a Random Process: Mean Average of the Random Process is time invariant: Autocorrelation of the Random Process is of the form: since: We have: Power Spectrum or Power Spectral Density of a Stationary Random Process: 2. Strict Sense Stationarity of a Random Process: All probability density functions are time invariant:Ergodicity:A Stationary Random Process for which Time Average = Assembly AverageRandom Processes

• *SOLOTime Autocorrelation: Ergodicity:For a Ergodic Random Process defineFinite Signal Energy Assumption:Define:Let compute:therefore:Random Processes

• *SOLOErgodicity (continue):Let compute:Define:Since the Random Process is Ergodic we can use the Wide Stationarity Assumption:Random Processes

• *SOLOErgodicity (continue):We obtained the Wiener-Khinchine Theorem (Wiener 1930): The Power Spectrum or Power Spectral Density of a Stationary Random Process S () is the Fourier Transform of the Autocorrelation Function R ().Random Processes

• *SOLOWhite Noise A (not necessary stationary) Random Process whose Autocorrelation is zero for any two different times is called white noise in the wide sense.- instantaneous varianceWide Sense WhitenessStrict Sense Whiteness A (not necessary stationary) Random Process in which the outcome for any two different times is independent is called white noise in the strict sense.A Stationary White Noise Random has the Autocorrelation:Note In general whiteness requires Strict Sense Whiteness. In practice we have only moments (typically up to second order) and thus only Wide Sense Whiteness.Random Processes

• *SOLOWhite NoiseA Stationary White Noise Random has the Autocorrelation: The Power Spectral Density is given by performing the Fourier Transform of the Autocorrelation: We can see that the Power Spectrum Density contains all frequencies at the same amplitude. This is the reason that is called White Noise. The Power of the Noise is defined as:Random Processes

• *SOLOMarkov Processes A Markov Process is defined by: i.e. the Random Process, the past up to any time t1 is fully defined by the process at t1. Examples of Markov Processes: 1. Continuous Dynamic System 2. Discrete Dynamic System - state space vector (n x 1) - input vector (m x 1) - white input noise vector (n x 1) - measurement vector (p x 1) - white measurement noise vector (p x 1)Random Processes

• SOLOStochastic Processes The earliest work on SDEs was done to describe Brownian motion in Einstein's famous paper, and at the same time by Smoluchowski. However, one of the earlier works related to Brownian motion is credited to Bachelier (1900) in his thesis 'Theory of Speculation'. This work was followed upon by Langevin. Later It and Stratonovich put SDEs on more solid mathematical footing. In physical science, SDEs are usually written as Langevin Equations. These are sometimes confusingly called "the Langevin Equation" even though there are many possible forms. These consist of an ordinary differential equation containing a deterministic part and an additional random white noise term. A second form is the Smoluchowski Equation and, more generally, the Fokker-Planck Equation. These are partial differential equations that describe the time evolution of probability distribution functions. The third form is the stochastic differential equation that is used most frequently in mathematics and quantitative finance (see below). This is similar to the Langevin form, but it is usually written in differential form. SDEs come in two varieties, corresponding to two versions of stochastic calculus.BackgroundTerminologyA stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. SDE are used to model diverse phenomena such as fluctuating stock prices or physical system subject to thermal fluctuations. Typically, SDEs incorporate white noise which can be thought of as the derivative of Brownian motion (or the Wiener process); however, it should be mentioned that other types of random fluctuations are possible, such as jump processes. Stochastic Differential Equation (SDE)

• SOLOStochastic Processes Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is non-differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the Ito Stochastic Calculus and the Stratonovich Stochastic Calculus. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist and conveniently, one can readily convert an Ito SDE to an equivalent Stratonovich SDE and back again. Still, one must be careful which calculus to use when the SDE is initially written down.Stochastic Calculus

• Stochastic ProcessesSOLOBrownian MotionIn 1827 Brown, a botanist, discovered the motion of pollen particles in water. At the beginning of the twentieth century, Brownian motion was studied by Einstein, Perrin and other physicists. In 1923, against this scientific background, Wiener defined probability measures in path spaces, and used the concept of Lebesgue integrals to lay the mathematical foundations of stochastic analysis. In 1942, Ito began to reconstruct from scratch the concept of stochastic integrals, and its associated theory of analysis. He created the theory of stochastic differential equations, which describe motion due to random events. Albert Einstein's (in his 1905 paper) and Marian Smoluchowski's (1906) independent research of the problem that brought the solution to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules.

• Stochastic ProcessesSOLORandom Walk Assume the process of walking on a straight line at discrete intervals T. At each timewe walk a distance s , randomly, to the left or to the right, with the same probability p=1/2. In this way we created a Stochastic Process called Random Walk. (This experiment is equivalent to tossing a coin to get, randomly, Head or Tail). Assume that at t = n T we have taken k steps to the right and n-k steps to the left, then the distance traveled isx (nT) is a Random Walk, taking the values r s, wherer equals n, n-2,, -(n-2),-nTherefore

x (t) Random Walk

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• Stochastic ProcessesSOLORandom Walk (continue 1)We have at step i the event xi: P {xi = +s} = p = 1/2 and P {xi = - s} = 1-p = 1/2 For large r and

x (t) Random Walk

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• Stochastic ProcessesSOLORandom Walk (continue 2) For n1 > n2 > n3 > n4 the number of steps to the right from n2T to n1T interval is independent of the number of steps to the right between n4T to n3T interval. Hence x (n1T) x (n2T) is independent of x (n4T) x (n3T).

x (t) Random Walk

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• SOLOStochastic ProcessesSmoluchowski EquationIn physics, the Diffusion Equation with drift term is often called Smoluchowski equation (after Marian von Smoluchowski). Let w(r, t) be a density, D a diffusion constant, a friction coefficient, and U(r, t) a potential. Then the Smoluchowski equation states that the density evolves according to

• SOLOStochastic ProcessesEinstein-Smoluchowski Equation In physics (namely, in kinetic theory) the Einstein relation (also known as EinsteinSmoluchowski relation) is a previously unexpected connection revealed independently by Albert Einstein in 1905 and by Marian Smoluchowski (1906) in their papers on Brownian motion. Two important special cases of the relation are:(diffusion of charged particles) ("EinsteinStokes equation", for diffusion of spherical particles through liquid with low Reynolds number)Where

(x,t) density of the Brownian particlesD is the diffusion constant,q is the electrical charge of a particle,q, the electrical mobility of the charged particle, i.e. the ratio of the particle's terminal drift velocity to an applied electric field,kB is Boltzmann's constant,T is the absolute temperature, is viscosityr is the radius of the spherical particle.The more general form of the equation is:where the "mobility" is the ratio of the particle's terminal drift velocity to an applied force, = vd / F.Einsteins EquationFor Brownian Motion

• Langevin EquationSOLOStochastic Processes Langevin equation (Paul Langevin, 1908) is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic natu