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# Stochastic Processes - part 2

Lecture notes of Prof. H.Amindavar.
Professor of Electrical engineering at Amirkabir university of technology.

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### Stochastic Processes - part 2

1. 1. Wide sense cyclostationarity: µX(t + mT) = µ(t) RX(t + mT + τ, t + mT) = RX(t + τ, t) The ACF of WS cyclostationary process is periodic on the diagonal plane: RX(t1 + mT, t2 + mT) = RX(t1, t2) AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.1/46
2. 2. Example: R(t + τ, t) = E{X(t1)X(t2)} = 1, (n − 1)T ≤ t, t + τ ≤ nT 0, otherwise R̄(τ) = 1 T Z T 0 R(t + τ, t)dt ⇒ 0 ≤ t ≤ T If τ 0 then RX(t + τ, t) = 1 iff τ T and t T − τ ⇒ AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.2/46
3. 3. R̄(τ) = 1 T Z T−τ 0 1 dt = 1 − τ T , 0 ≤ τ ≤ T If τ 0 then RX(t + τ, t) = 1 iff τ −T and t T + τ ⇒ R̄(τ) = 1 T Z T+τ 0 1 dt = 1 + τ T , −T ≤ τ ≤ 0 =⇒ R̄(τ) = 1 − |τ| T , |τ| ≤ T AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.3/46
4. 4. Example: Y (t) = X2 (t), X(t) ∼ zero-mean Gaussian of ACF RX(τ) What is RY (τ)? E{X2 1 X2 2 } = E{X2 1 }E{X2 2 } + 2E2 {X1X2}, this is true if X1, X2 are jointly Gaussian. ⇒ AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.4/46
5. 5. f(X1, X2) = exp −0.5(X1 X2)C−1 (X1 X2)T 2π √ ∆ C = RX(0) RX(τ) RX(τ) RX(0) , ∆ = det C Using eigenvalue-eigenvector decomposition, we have: RY (τ) = R2 X(0) + 2R2 X(τ) AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.5/46
6. 6. Example: The Gaussian process X(t) is an input to a hard limiter, find the ACF of the output Y (t). RY (τ) = E{Y (t)Y (t + τ)} ⇒ Y (t)Y (t + τ) = 1 −1 = 1 . P{X(t)X(t + τ) 0} − 1 . P{X(t)X(t + τ) 0} X1 = X(t), X2 = X(t + τ) AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.6/46
7. 7. If Z = X/Y then XY 0 ⇒ Z 0 and XY 0 ⇒ Z 0 f(X1, X2) = exp −0.5(X1 X2)C−1 (X1 X2)T 2π √ ∆ C = RX(0) RX(τ) RX(τ) RX(0) , ∆ = det C AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.7/46
8. 8. P{Z 0} = P{X1X2 0} = 1 − Z ∞ 0 Z Y Z=0 −∞ f(X1, X2)dX1dX2 + Z 0 −∞ Z −∞ Y Z=0 f(X1, X2)dX1dX2 P{Z 0} = 0.5 + α π , P{Z 0} = 0.5 − α π α = arcsin RX(τ) RX(0) AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.8/46
9. 9. RY (τ) = 2 π arcsin RX(τ) RX(0) AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.9/46
10. 10. In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that has stationary independent incre- ments. The most well-known examples are the Wiener pro- cess and the Poisson process. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.10/46
11. 11. A continuous-time stochastic process assigns a random variable Xt to each point t = 0 in time. In effect it is a random function of t. The increments of such a process are the differences Xs −Xt between its values at different times s t. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.11/46
12. 12. To call the increments of a process independent means that increments Xs − Xt and Xu − Xv are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments as- signed to pairwise non-overlapping time intervals are mutu- ally (not just pairwise) independent. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.12/46
13. 13. To call the increments stationary means that the probability distribution of any increment Xs − Xt depends only on the length s−t of the time interval; increments with equally long time intervals are identically distributed. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.13/46
14. 14. In the Wiener process, the probability distribution of Xs −Xt is normal with expected value 0 and variance s − t. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.14/46
15. 15. In the Poisson process, the probability distribution of Xs−Xt is a Poisson distribution with expected value (s − t)λ, where λ 0 is the intensity or rate of the process. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.15/46
16. 16. The probability distributions of the increments of any Lévy process are infinitely divisible. There is a Lévy process for each infinitely divisible probability distribution. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.16/46
17. 17. The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter, space, time, money, or abstract mathematical objects. This theory is exposed in Plato’s dia- logue Timaeus and was also supported by Aristotle. Atom- ism denies that matter is infinitely divisible. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.17/46
18. 18. To say that a probability distribution F on the real line is infinitely divisible means that if X is any random vari- able whose distribution is F, then for every positive integer n there exist n independent identically distributed random variables X1, · · · , Xn whose sum is X (those n other ran- dom variables do not usually have the same probability dis- tribution that X has (but do sometimes, as in the case of the Cauchy distribution)). AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.18/46
19. 19. The Poisson distributions, the normal distributions, and the gamma distributions are infinitely divisible probability distributions. Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic pro- cess Xt : t = 0 with stationary independent increments (stationary means that for s t, the probability distribution of Xs − Xt depends only on s − t; AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.19/46
20. 20. independent increments means that difference is independent of the corresponding difference on any interval not overlapping with [t, s], and similarly for any finite number of intervals). This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.20/46
21. 21. In any Lévy process with finite moments, the nth moment µn(t) = E(Xn t ) is a polynomial function of t; these functions satisfy a binomial identity: µn(t) = n X k=0 n k µk(t)µn−k(t) AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.21/46
22. 22. Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with inde- pendent increments used in modelling Brownian motion and some random phenomena observed in finance. It is one of the best-known Lévy processes. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.22/46
23. 23. For each positive number t, denote the value of the process at time t by Wt. Then the process is characterized by the following two conditions: If 0 t s Ws − Wt ∼ N(0, s − t) 0 ≤ t s ≤ u v ⇒ Ws − Wt and Wv − Wu are independent RVs AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.23/46
24. 24. The erratic motion, visible through a microscope, of small grains suspended in a fluid. The motion results from collisions between the grains and atoms or molecules in the fluid. Brownian motion was first explained by Albert Einstein, who considered it direct proof of the exis- tence of atoms. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.24/46
25. 25. A geometric Brownian motion (GBM) (occasionally, expo- nential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, or, perhaps more pre- cisely, a Wiener process. It is appropriate to mathematical modelling of some phenomena in financial markets. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.25/46
26. 26. A stochastic process St is said to follow a GBM if it satisfies the following stochastic differential equation: dSt = uStdt + vStdWt where {Wt} is a Wiener process or Brownian motion and u (’the percentage drift’) and v (’the percentage volatility’) are constants. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.26/46
27. 27. The equation has an analytic solution: St = S0 exp (u − v2 /2)t + vWt for an arbitrary initial value S0. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.27/46
28. 28. The correctness of the solution can be verified using Itô’s lemma. The random variable log(St/S0) is normally dis- tributed with mean (u − v2 /2)t and variance (v.v).t, which reflects the fact that increments of a GBM are normal rela- tive to the current price, which is why the process has the name ’geometric’. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.28/46
29. 29. In mathematics, Itô’s lemma is a theorem of stochastic cal- culus that shows that second order differential terms of Wiener processes become deterministic under stochastic integration. It is somewhat analogous in stochastic calculus to the chain rule in ordinary calculus. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.29/46
30. 30. Let x(t) be an Itô (or generalized Wiener) process. That is let dx(t) = a(x, t)dt + b(x, t)dWt where Wt is a Wiener process, and let f(x, t) be a function with continuous second derivatives. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.30/46
31. 31. Then f(x(t), t) is also an Itô process, and df(x(t), t) = a(x, t) ∂f ∂x + ∂f ∂t + 1 2 b(x, t)2 ∂2 f ∂x2 + b(x, t) ∂f ∂x dWt AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.31/46
32. 32. A formal proof of the lemma requires us to take the limit of a sequence of random variables. Expanding f(x, t) from above in a Taylor series in x and t we have df = ∂f ∂x dx + ∂f ∂t dt + 1 2 ∂2 f ∂x2 dx2 + · · · AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.32/46
33. 33. and substituting (adt + bdW) for dx gives df = ∂f ∂x (adt + bdW) + ∂f ∂t dt + 1 2 ∂2 f ∂x2 a2 dt2 +2abdtdW + b2 dW2 Reordering this to combine like differential terms, we have df = a ∂f ∂x + ∂f ∂t dt + 1 2 b2 ∂2 f ∂x2 dW2 + b ∂f ∂x dW + e AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.33/46
34. 34. where e (an error term) is ab ∂2 f ∂x2 dtdW + 1 2 a2 ∂2 f ∂x2 dt2 + · · · . In the limit as dt tends to 0, the dt2 and dt dW terms disappear but the dW2 term tends to dt. The latter can be shown if we prove that dW2 → E(dW2 ), since E(dW2 ) = dt. The proof of this statistical property is however beyond the scope of this course. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.34/46
35. 35. Substituting this dt in and reordering the terms so that the dt and dW terms are collected, we obtain df = a ∂f ∂x + ∂f ∂t + 1 2 b2 ∂2 f ∂x2 dt + b ∂f ∂x dW AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.35/46
36. 36. A random walk is a formalization of the intuitive idea of tak- ing successive steps, each in a random direction. A random walk is a simple stochastic process. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.36/46
37. 37. The simplest random walk is a path constructed according to the following rules: There is a starting point. The distance from one point in the path to the next is a constant. The direction from one point in the path to the next is chosen at random, and no direction is more probable than another. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.37/46
38. 38. Imagine now a bug walking around in the city. The city is infinite and completely ordered, and at every corner he chooses one of the four possible routes (including the one he came from) with equal probability. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.38/46
39. 39. Formally, this is a random walk on the set of all points in the plane with integer coordinates. Will the bug ever get back to his home from the bar? It turns out that he will. This is the high dimensional equivalent of the level crossing problem discussed before. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.39/46
40. 40. However, the similarity stops here. In dimensions 3 and above, this no longer holds. In other words, a drunk bird might forever wander around, never finding its nest. The formal terms to describe this phenomenon is that random walk in dimensions 1 and 2 is recurrent while in dimension 3 and above it is transient. This was proved by Pólya in 1921. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.40/46
41. 41. The trajectory of a random walk is the collection of sites it visited, considered as a set with disregard to when the walk arrived at the point. In 1 dimension, the trajectory is simply all points between the minimum height the walk achieved and the maximum (both are, on average, on the order of √ n). AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.41/46
42. 42. In higher dimensions the set has interesting geometric prop- erties. In fact, one gets a discrete fractal, that is a set which exhibits stochastic self-similarity on large scales, but on small scales one can observe “jugginess” resulting from the grid on which the walk is performed. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.42/46
43. 43. A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be ’broken up’ in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.43/46
44. 44. Fractals are said to possess infinite detail, and some of them have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.44/46
45. 45. A self-similar object is exactly or approximately similar to a part of itself. A curve is said to be self-similar if, for every piece of the curve, there is a smaller piece that is similar to it. For instance, coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.45/46
46. 46. Self-similarity is a typical property of fractals. Coastlines hypothetically can be divided into two halves, each of which is similar to the whole. AKU-EE/Stochastic/HA, 1st Semester, 85-86 – p.46/46