Gaussian, Markov and stationary processes

Transcription

1 Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester November 16, 2018 Introduction to Random Processes Gaussian, Markov and stationary processes 1

2 Introduction and roadmap Introduction and roadmap Gaussian processes Brownian motion and its variants White Gaussian noise Introduction to Random Processes Gaussian, Markov and stationary processes 2

3 Random processes Random processes assign a function X (t) to a random event Without restrictions, there is little to say about them Markov property simplifies matters and is not too restrictive Also constrained ourselves to discrete state spaces Further simplification but might be too restrictive Time t and range of X (t) values continuous in general Time and/or state may be discrete as particular cases Restrict attention to (any type or a combination of types) Markov processes (memoryless) Gaussian processes (Gaussian probability distributions) Stationary processes ( limit distribution ) Introduction to Random Processes Gaussian, Markov and stationary processes 3

4 Markov processes X (t) is a Markov process when the future is independent of the past For all t > s and arbitrary values x(t), x(s) and x(u) for all u < s P ( X (t) x(t) X (s) x(s), X (u) x(u), u < s ) = P ( X (t) x(t) X (s) x(s) ) Markov property defined in terms of cdfs, not pmfs Markov property useful for same reasons as in discrete time/state But not that useful as in discrete time /state More details later Introduction to Random Processes Gaussian, Markov and stationary processes 4

5 Gaussian processes X (t) is a Gaussian process when all prob. distributions are Gaussian For arbitrary n > 0, times t 1, t 2,..., t n it holds Values X (t 1 ), X (t 2 ),..., X (t n ) are jointly Gaussian RVs Simplifies study because Gaussian distribution is simplest possible Suffices to know mean, variances and (cross-)covariances Linear transformation of independent Gaussians is Gaussian Linear transformation of jointly Gaussians is Gaussian More details later Introduction to Random Processes Gaussian, Markov and stationary processes 5

6 Markov processes + Gaussian processes Markov (memoryless) and Gaussian properties are different Will study cases when both hold Brownian motion, also known as Wiener process Brownian motion with drift White noise Linear evolution models Geometric brownian motion Arbitrages Risk neutral measures Pricing of stock options (Black-Scholes) Introduction to Random Processes Gaussian, Markov and stationary processes 6

7 Stationary processes Process X (t) is stationary if probabilities are invariant to time shifts For arbitrary n > 0, times t 1, t 2,..., t n and arbitrary time shift s P (X (t 1 + s) x 1, X (t 2 + s) x 2,..., X (t n + s) x n ) = P (X (t 1 ) x 1, X (t 2 ) x 2,..., X (t n ) x n ) System s behavior is independent of time origin Follows from our success studying limit probabilities Study of stationary process Study of limit distribution Will study Spectral analysis of stationary random processes Linear filtering of stationary random processes More details later Introduction to Random Processes Gaussian, Markov and stationary processes 7

8 Gaussian processes Introduction and roadmap Gaussian processes Brownian motion and its variants White Gaussian noise Introduction to Random Processes Gaussian, Markov and stationary processes 8

9 Jointly Gaussian random variables Def: Random variables X 1,..., X n are jointly Gaussian (normal) if any linear combination of them is Gaussian Given n > 0, for any scalars a 1,..., a n the RV (a = [a 1,..., a n ] T ) Y = a 1 X 1 + a 2 X a n X n = a T X is Gaussian distributed May also say vector RV X = [X 1,..., X n ] T is Gaussian Consider 2 dimensions 2 RVs X 1 and X 2 are jointly normal To describe joint distribution have to specify Means: µ 1 = E [X 1 ] and µ 2 = E [X 2 ] Variances: σ 2 11 = var [X 1] = E [ (X 1 µ 1 ) 2] and σ 2 22 = var [X 2] Covariance: σ 2 12 = cov(x 1, X 2 ) = E [(X 1 µ 1 )(X 2 µ 2 )]= σ 2 21 Introduction to Random Processes Gaussian, Markov and stationary processes 9

10 Pdf of jointly Gaussian RVs in 2 dimensions Define mean vector µ = [µ 1, µ 2 ] T and covariance matrix C R 2 2 C = ( σ 2 11 σ 2 12 σ 2 21 σ 2 22 ) C is symmetric, i.e., C T = C because σ 2 21 = σ2 12 Joint pdf of X = [X 1, X 2 ] T is given by ( 1 f X (x) = 2π det 1/2 (C) exp 1 ) 2 (x µ)t C 1 (x µ) Assumed that C is invertible, thus det(c) 0 If the pdf of X is f X (x) above, can verify Y = a T X is Gaussian Introduction to Random Processes Gaussian, Markov and stationary processes 10

11 Pdf of jointly Gaussian RVs in n dimensions For X R n (n dimensions) define µ = E [X] and covariance matrix σ [ ] 11 2 σ σ 2 1n C := E (X µ)(x µ) T σ21 2 σ σ2n 2 = σn1 2 σn σnn 2 C symmetric, (i, j)-th element is σ 2 ij = cov(x i, X j ) Joint pdf of X defined as before (almost, spot the difference) ( 1 f X (x) = (2π) n/2 det 1/2 (C) exp 1 ) 2 (x µ)t C 1 (x µ) C invertible and det(c) 0. All linear combinations normal To fully specify the probability distribution of a Gaussian vector X The mean vector µ and covariance matrix C suffice Introduction to Random Processes Gaussian, Markov and stationary processes 11

12 Notational aside and independence With x R n, µ R n and C R n n, define function N (x; µ, C) as ( 1 N (x; µ, C) := (2π) n/2 det 1/2 (C) exp 1 ) 2 (x µ)t C 1 (x µ) µ and C are parameters, x is the argument of the function Let X R n be a Gaussian vector with mean µ, and covariance C Can write the pdf of X as f X (x) = N (x; µ, C) If X 1,..., X n are mutually independent, then C = diag(σ 2 11,..., σ2 nn) and f X (x) = n 1 exp ( (x i µ i ) 2 ) 2πσ 2 ii i=1 2σ 2 ii Introduction to Random Processes Gaussian, Markov and stationary processes 12

13 Gaussian processes Gaussian processes (GP) generalize Gaussian vectors to infinite dimensions Def: X (t) is a GP if any linear combination of values X (t) is Gaussian For arbitrary n > 0, times t 1,..., t n and constants a 1,..., a n Y = a 1 X (t 1 ) + a 2 X (t 2 ) a n X (t n ) is Gaussian distributed Time index t can be continuous or discrete More general, any linear functional of X (t) is normally distributed A functional is a function of a function Ex: The (random) integral Y = t2 t 1 X (t) dt is Gaussian distributed Integral functional is akin to a sum of X (t i ), for all t i [t 1, t 2 ] Introduction to Random Processes Gaussian, Markov and stationary processes 13

14 Joint pdfs in a Gaussian process Consider times t 1,..., t n. The mean value µ(t i ) at such times is µ(t i ) = E [X (t i )] The covariance between values at times t i and t j is C(t i, t j ) = E [( X (t i ) µ(t i ) )( X (t j ) µ(t j ) )] Covariance matrix for values X (t 1 ),..., X (t n ) is then C(t 1, t 1) C(t 1, t 2)... C(t 1, t n) C(t 2, t 1) C(t 2, t 2)... C(t 2, t n) C(t 1,..., t n) = C(t n, t 1) C(t n, t 2)... C(t n, t n) Joint pdf of X (t 1 ),..., X (t n ) then given as ( ) f X (t1 ),...,X (t n)(x 1,..., x n) = N [x 1,..., x n] T ; [µ(t 1),..., µ(t n)] T, C(t 1,..., t n) Introduction to Random Processes Gaussian, Markov and stationary processes 14

15 Mean value and autocorrelation functions To specify a Gaussian process, suffices to specify: Mean value function µ(t) = E [X (t)]; and Autocorrelation function R(t 1, t 2 ) = E [ X (t 1 )X (t 2 ) ] Autocovariance obtained as C(t 1, t 2 ) = R(t 1, t 2 ) µ(t 1 )µ(t 2 ) For simplicity, will mostly consider processes with µ(t) = 0 Otherwise, can define process Y (t) = X (t) µ X (t) In such case C(t 1, t 2 ) = R(t 1, t 2 ) because µ Y (t) = 0 Autocorrelation is a symmetric function of two variables t 1 and t 2 R(t 1, t 2 ) = R(t 2, t 1 ) Introduction to Random Processes Gaussian, Markov and stationary processes 15

16 Probabilities in a Gaussian process All probs. in a GP can be expressed in terms of µ(t) and R(t 1, t 2 ) For example, pdf of X (t) is ( ( 1 f X (t) (x t ) = 2π ( R(t, t) µ 2 (t) ) exp xt µ(t) ) ) 2 2 ( R(t, t) µ 2 (t) ) X (t) µ(t) R(t,t) µ2 (t) Notice that is a standard Gaussian random variable ( ) a µ(t) P (X (t) > a) = Φ, where Φ(x) = R(t,t) µ 2 (t) x 1 2π exp ( x 2 2 ) dx Introduction to Random Processes Gaussian, Markov and stationary processes 16

17 Joint and conditional probabilities in a GP For a zero-mean GP X (t) consider two times t 1 and t 2 The covariance matrix for X (t 1 ) and X (t 2 ) is C = ( R(t1, t 1 ) R(t 1, t 2 ) R(t 1, t 2 ) R(t 2, t 2 ) Joint pdf of X (t 1 ) and X (t 2 ) then given as (recall µ(t) = 0) ( 1 f X (t1),x (t 2)(x t1, x t2 ) = 2π det 1/2 (C) exp 1 ) 2 [x t 1, x t2 ] T C 1 [x t1, x t2 ] Conditional pdf of X (t 1 ) given X (t 2 ) computed as f X (t1) X (t 2)(x t1 xt2 ) = f X (t 1),X (t 2)(x t1, x t2 ) f X (t2)(x t2 ) ) Introduction to Random Processes Gaussian, Markov and stationary processes 17

18 Brownian motion and its variants Introduction and roadmap Gaussian processes Brownian motion and its variants White Gaussian noise Introduction to Random Processes Gaussian, Markov and stationary processes 18

19 Brownian motion as limit of random walk Gaussian processes are natural models due to Central Limit Theorem Let us reconsider a symmetric random walk in one dimension x(t) Time interval = h t Walker takes increasingly frequent and increasingly smaller steps Introduction to Random Processes Gaussian, Markov and stationary processes 19

20 Brownian motion as limit of random walk Gaussian processes are natural models due to Central Limit Theorem Let us reconsider a symmetric random walk in one dimension x(t) Time interval = h/2 t Walker takes increasingly frequent and increasingly smaller steps Introduction to Random Processes Gaussian, Markov and stationary processes 20

21 Brownian motion as limit of random walk Gaussian processes are natural models due to Central Limit Theorem Let us reconsider a symmetric random walk in one dimension x(t) Time interval = h/4 t Walker takes increasingly frequent and increasingly smaller steps Introduction to Random Processes Gaussian, Markov and stationary processes 21

22 Random walk, time step h and step size σ h Let X (t) be the position at time t with X (0) = 0 Time interval is h and σ h is the size of each step Walker steps right or left w.p. 1/2 for each direction Given X (t) = x, prob. distribution of the position at time t + h is P (X (t + h) = x + σ h ) X (t) = x = 1/2 P (X (t + h) = x σ h ) X (t) = x = 1/2 Consider time T = Nh and index n = 1, 2,..., N Introduce step RVs Y n = ±1, with P (Y n = ±1) = 1/2 Can write X (nh) in terms of X ((n 1)h) and Y n as ( X (nh) = X ((n 1)h) + σ ) h Y n Introduction to Random Processes Gaussian, Markov and stationary processes 22

23 Central Limit Theorem as h 0 Use recursion to write X (T ) = X (Nh) as (recall X (0) = 0) ( X (T ) = X (Nh) = X (0) + σ ) N h Y n = n=1 ( σ h ) N Y 1,..., Y N are i.i.d. with zero-mean and variance var [Y n ] = E [ Yn 2 ] = (1/2) (1/2) ( 1) 2 = 1 As h 0 we have N = T /h, and from Central Limit Theorem N Y n N (0, N) = N (0, T /h) n=1 X (T ) N ( 0, (σ 2 h) (T /h) ) = N ( 0, σ 2 T ) n=1 Y n Introduction to Random Processes Gaussian, Markov and stationary processes 23

24 Conditional distribution of future values More generally, consider times T = Nh and T + S = (N + M)h Let X (T ) = x(t ) be given. Can write X (T + S) as ( X (T + S) = x(t ) + σ ) N+M h From Central Limit Theorem it then follows N+M n=n+1 n=n+1 Y n Y n N ( 0, (N + M N) ) = N (0, S/h) [X (T + S) ] X (T ) = x(t ) N (x(t ), σ 2 S) Introduction to Random Processes Gaussian, Markov and stationary processes 24

25 Definition of Brownian motion The former analysis was for motivational purposes Def: A Brownian motion process (a.k.a Wiener process) satisfies (i) X (t) is normally distributed with zero mean and variance σ 2 t X (t) N ( 0, σ 2 t ) (ii) Independent increments For disjoint intervals (t 1, t 2 ) and (s 1, s 2 ) increments X (t 2 ) X (t 1 ) and X (s 2 ) X (s 1 ) are independent RVs (iii) Stationary increments Probability distribution of increment X (t + s) X (s) is the same as probability distribution of X (t) Property (ii) Brownian motion is a Markov process Properties (i)-(iii) Brownian motion is a Gaussian process Introduction to Random Processes Gaussian, Markov and stationary processes 25

26 Mean and autocorrelation of Brownian motion Mean function µ(t) = E [X (t)] is null for all times (by definition) µ(t) = E [X (t)] = 0 For autocorrelation R X (t 1, t 2 ) start with times t 1 < t 2 Use conditional expectations to write [ [ R X (t 1, t 2 ) = E [X (t 1 )X (t 2 )] = E X (t1) E X (t2) X (t1 )X (t 2 ) X (t1 ) ]] In the innermost expectation X (t 1 ) is a given constant, then [ [ R X (t 1, t 2 ) = E X (t1) X (t 1 )E X (t2) X (t2 ) X (t 1 ) ]] Proceed by computing innermost expectation Introduction to Random Processes Gaussian, Markov and stationary processes 26

27 Autocorrelation of Brownian motion (continued) The conditional distribution of X (t 2 ) given X (t 1 ) for t 1 < t 2 is [ X (t 2 ) ] ( ) X (t 1 ) N X (t 1 ), σ 2 (t 2 t 1 ) Innermost expectation is E X (t2)[ X (t2 ) X (t 1 ) ] = X (t 1 ) From where autocorrelation follows as R X (t 1, t 2 ) = E X (t1)[ X (t1 )X (t 1 ) ] = E X (t1)[ X 2 (t 1 ) ] = σ 2 t 1 Repeating steps, if t 2 < t 1 R X (t 1, t 2 ) = σ 2 t 2 Autocorrelation of Brownian motion R X (t 1, t 2 ) = σ 2 min(t 1, t 2 ) Introduction to Random Processes Gaussian, Markov and stationary processes 27

28 Brownian motion with drift Similar to Brownian motion, but start from biased random walk Time interval h, step size σ h, right or left with different probs. ( P X (t + h) = x + σ h ) X (t) = x = 1 ( 1 + µ ) h 2 σ P (X (t + h) = x σ h ) X (t) = x = 1 ( 1 µ ) h 2 σ If µ > 0 biased to the right, if µ < 0 biased to the left Definition requires h small enough to make (µ/σ) h 1 Notice that bias vanishes as h, same as step size Introduction to Random Processes Gaussian, Markov and stationary processes 28

29 Mean and variance of biased steps Define step RV Y n = ±1, with probabilities P (Y n = 1) = 1 ( 1 + µ ) h, P (Y n = 1) = 1 ( 1 µ ) h 2 σ 2 σ Expected value of Y n is E [Y n] = 1 P (Y n = 1) + ( 1) P (Y n = 1) = 1 ( 1 + µ ) h 1 ( 1 µ ) h = µ h 2 σ 2 σ σ Second moment of Y n is [ ] E Yn 2 = (1) 2 P (Y n = 1) + ( 1) 2 P (Y n = 1) = 1 [ ] Variance of Y n is var [Y n] = E Yn 2 E 2 [Y n] = 1 µ2 σ h 2 Introduction to Random Processes Gaussian, Markov and stationary processes 29

30 Central Limit Theorem as h 0 Consider time T = Nh, index n = 1, 2,..., N. Write X (nh) as ( X (nh) = X ((n 1)h) + σ ) h Use recursively to write X (T ) = X (Nh) as ( X (T ) = X (Nh) = X (0) + σ ) N h Y n = n=1 Y n ( σ h ) N As h 0 we have N and N n=1 Y n normally distributed As h 0, X (T ) tends to be normally distributed by CLT Need to determine mean and variance (and only mean and variance) n=1 Y n Introduction to Random Processes Gaussian, Markov and stationary processes 30

31 Mean and variance of X (T ) Expected value of X (T ) = scaled sum of E [Y n ] (recall T = Nh) ( E [X (T )] = σ ) ( h N E [Y n ] = σ ) ( µ ) h N h = µt σ Variance of X (T ) = scaled sum of variances of independent Y n ( var [X (T )] = σ 2 h) N var [Yn ] = ( σ 2 h ) ) N (1 µ2 σ 2 h σ 2 T Used T = Nh and 1 (µ 2 /σ 2 )h 1 Brownian motion with drift (BMD) X (t) N ( µt, σ 2 t ) Normal with mean µt and variance σ 2 t Independent and stationary increments Introduction to Random Processes Gaussian, Markov and stationary processes 31

32 Geometric random walk Suppose next state follows by multiplying current by a random factor Compare with adding or subtracting a random quantity Define RV Y n = ±1 with probabilities as in biased random walk P (Y n = 1) = 1 ( 1 + µ ) h, P (Y n = 1) = 1 ( 1 µ ) h 2 σ 2 σ Def: The geometric random walk follows the recursion Z(nh) = Z((n 1)h)e (σ h)y n When Y n = 1 increase Z(nh) by relative amount e (σ h) When Y n = 1 decrease Z(nh) by relative amount e (σ h) Notice e ±(σ h) 1 ± (σ ) h Useful to model investment return Introduction to Random Processes Gaussian, Markov and stationary processes 32

33 Geometric Brownian motion Take logarithms on both sides of recursive definition ( ) ( ) ( log Z(nh) = log Z((n 1)h) + σ ) h Y n ( ) Define X (nh) = log Z(nh), thus recursion for X (nh) is ( X (nh) = X ((n 1)h) + σ ) h Y n As h 0, X (t) becomes BMD with parameters µ and σ 2 Def: Given a BMD X (t) with parameters µ and σ 2, the process Z(t) Z(t) = e X (t) is a geometric Brownian motion (GBM) with parameters µ and σ 2 Introduction to Random Processes Gaussian, Markov and stationary processes 33

34 White Gaussian noise Introduction and roadmap Gaussian processes Brownian motion and its variants White Gaussian noise Introduction to Random Processes Gaussian, Markov and stationary processes 34

35 Dirac delta function Consider a function δ h (t) defined as δ h (t) { 1/h if h/2 t h/2 δ h (t) = 0 else Define delta function as limit of δ h (t) as h 0 { if t = 0 δ(t) = lim δ h (t) = h 0 0 else Q: Is this a function? A: Of course not t Consider the integral of δ h (t) in an interval that includes [ h/2, h/2] b a δ h (t) dt = 1, Integral is 1 independently of h for any a, b such that a h/2, h/2 b Introduction to Random Processes Gaussian, Markov and stationary processes 35

36 Dirac delta function (continued) Another integral involving δ h (t) (for h small) b f (t)δ h (t) dt h/2 a h/2 f (0) 1 dt f (0), h a h/2, h/2 b Def: The generalized function δ(t) is the entity having the property b a { f (0) if a < 0 < b f (t)δ(t) dt = 0 else A delta function is not defined, its action on other functions is Interpretation: A delta function cannot be observed directly But can be observed through its effect on other functions Delta function helps to define derivatives of discontinuous functions Introduction to Random Processes Gaussian, Markov and stationary processes 36

37 Heaviside s step function and delta function Integral of delta function between and t t { } 0 if t < 0 δ(u) du = := H(t) 1 if t > 0 H(t) is called Heaviside s step function Define the derivative of Heaviside s step function as H(t) t = δ(t) Maintains consistency of fundamental theorem of calculus H(t) δ(t) t Introduction to Random Processes Gaussian, Markov and stationary processes 37

38 White Gaussian noise Def: A white Gaussian noise (WGN) process W (t) is a GP with Zero mean: µ(t) = E [W (t)] = 0 for all t Delta function autocorrelation: R W (t 1, t 2 ) = σ 2 δ(t 1 t 2 ) To interpret W (t) consider time step h and process W h (nh) with (i) Normal distribution W h (nh) N (0, σ 2 /h) (ii) W h (n 1h) and W h (n 2h) are independent for n 1 n 2 White noise W (t) is the limit of the process W h (nh) as h 0 W (t) = lim n W h(nh), with n = t/h Process W h (nh) is the discrete-time representation of WGN Introduction to Random Processes Gaussian, Markov and stationary processes 38

39 Properties of white Gaussian noise For different times t 1 and t 2, W (t 1 ) and W (t 2 ) are uncorrelated E [W (t 1 )W (t 2 )] = R W (t 1, t 2 ) = 0, t 1 t 2 But since W (t) is Gaussian uncorrelatedness implies independence Values of W (t) at different times are independent WGN has infinite power E [ W 2 (t) ] = R W (t, t) = σ 2 δ(0) = WGN does not represent any physical phenomena However WGN is a convenient abstraction Approximates processes with large power and independent samples Some processes can be modeled as post-processing of WGN Cannot observe WGN directly But can model its effect on systems, e.g., filters Introduction to Random Processes Gaussian, Markov and stationary processes 39

40 Integral of white Gaussian noise Consider integral of a WGN process W (t) X (t) = t 0 W (u) du Since integration is linear functional and W (t) is GP, X (t) is also GP To characterize X (t) just determine mean and autocorrelation The mean function µ(t) = E [X (t)] is null [ t ] t µ(t) = E W (u) du = E [W (u)] du = The autocorrelation R X (t 1, t 2 ) is given by (assume t 1 < t 2 ) [( t1 ) ( t2 )] R X (t 1, t 2 ) = E W (u 1 ) du 1 W (u 2 ) du Introduction to Random Processes Gaussian, Markov and stationary processes 40

41 Integral of white Gaussian noise (continued) Product of integral is double integral of product R X (t 1, t 2) = E [ t1 t2 Interchange expectation and integration R X (t 1, t 2) = 0 t1 t W (u 1)W (u 2) du 1du 2 ] E [W (u 1)W (u 2)] du 1du 2 Definition and value of autocorrelation R W (u 1, u 2) = σ 2 δ(u 1 u 2) R X (t 1, t 2) = = = t1 t2 0 0 t1 t1 0 t1 0 0 σ 2 δ(u 1 u 2) du 1du 2 t1 t2 σ 2 δ(u 1 u 2) du 1du 2 + σ 2 δ(u 1 u 2) du 1du 2 0 t 1 σ 2 du 1 = σ 2 t 1 Same mean and autocorrelation functions as Brownian motion Introduction to Random Processes Gaussian, Markov and stationary processes 41

42 White Gaussian noise and Brownian motion GPs are uniquely determined by mean and autocorrelation functions The integral of WGN is a Brownian motion process Conversely the derivative of Brownian motion is WGN With W (t) a WGN process and X (t) Brownian motion t 0 W (u) du = X (t) X (t) t = W (t) Brownian motion can be also interpreted as a sum of Gaussians Not Bernoullis as before with the random walk Any i.i.d. distribution with same mean and variance works This is all nice, but derivatives and integrals involve limits What are these derivatives and integrals? Introduction to Random Processes Gaussian, Markov and stationary processes 42

43 Mean-square derivative of a random process Consider a realization x(t) of the random process X (t) Def: The derivative of (lowercase) x(t) is x(t) t = lim h 0 x(t + h) x(t) h When this limit exists Limit may not exist for all realizations Can define sure limit, a.s. limit, in probability,... Notion of convergence used here is in mean-squared sense Def: Process X (t)/ t is the mean-square sense derivative of X (t) if [ (X ) ] 2 (t + h) X (t) lim E X (t) = 0 h 0 h t Introduction to Random Processes Gaussian, Markov and stationary processes 43

44 Mean-square integral of a random process Likewise consider the integral of a realization x(t) of X (t) b a (b a)/h x(t)dt = lim hx(a + nh) h 0 n=1 Limit need not exist for all realizations Can define in sure sense, almost sure sense, in probability sense,... Again, adopt definition in mean-square sense Def: Process b X (t)dt is the mean square sense integral of X (t) if a ( (b a)/h b ) 2 lim hx (a + nh) X (t)dt = 0 h 0 E n=1 Mean-square sense convergence is convenient to work with GPs a Introduction to Random Processes Gaussian, Markov and stationary processes 44

45 Linear state model example Def: A random process X (t) follows a linear state model if X (t) t = ax (t) + W (t) with W (t) WGN, autocorrelation R W (t 1, t 2) = σ 2 δ(t 1 t 2) Discrete-time representation of X (t) X (nh) with step size h Solving differential equation between nh and (n + 1)h (h small) X ((n + 1)h) X (nh)e ah + (n+1)h nh W (t) dt Defining X (n) := X (nh) and W (n) := (n+1)h W (t) dt may write nh X (n + 1) (1 + ah)x (n) + W (n) Where E [ W 2 (n) ] = σ 2 h and W (n 1) independent of W (n 2) Introduction to Random Processes Gaussian, Markov and stationary processes 45

46 Vector linear state model example Def: A vector random process X(t) follows a linear state model if X(t) t = AX(t) + W(t) with W(t) vector WGN, autocorrelation R W (t 1, t 2) = σ 2 δ(t 1 t 2)I Discrete-time representation of X(t) X(nh) with step size h Solving differential equation between nh and (n + 1)h (h small) X((n + 1)h) X(nh)e Ah + (n+1)h nh W(t) dt Defining X(n) := X(nh) and W(n) := (n+1)h W(t) dt may write nh X(n + 1) (I + Ah)X(n) + W(n) Where E [ W 2 (n) ] = σ 2 hi and W(n 1) independent of W(n 2) Introduction to Random Processes Gaussian, Markov and stationary processes 46

47 Glossary Markov process Gaussian process Stationary process Gaussian random vectors Mean vector Covariance matrix Multivariate Gaussian pdf Linear functional Autocorrelation function Brownian motion (Wiener process) Brownian motion with drift Geometric random walk Geometric Brownian motion Investment returns Dirac delta function Heaviside s step function White Gaussian noise Mean-square derivatives Mean-square integrals Linear (vector) state model Introduction to Random Processes Gaussian, Markov and stationary processes 47