The stochastic modeling

Transcription

1 February 211 Gdansk

2 A schedule of the lecture Generation of randomly arranged numbers Monte Carlo Strong Law of Large Numbers SLLN We use SAS 9 For convenience programs will be given in an appendix Programs will be also available on my personal website

3 Random number Create a sequence of random numbers using Program ,71448, 37498, 1574, 12514, 793, 13727, 61135, 47887, 63348, 643, 49142, 83783, 57892, 879, 64154, 13433, 58922, 8279, 6935, 85657, 9582, 6178, 4394, Numbers are chosen completely irregularly from the interval [, 1] In 1927 random number tables was created by Tippett Here we use a random number generator Random numbers are drawn from a uniform distribution It means that percentage of random numbers drawn from any interval [a, b] [, 1] approximately is equal to (b a)

4 Uniform Distribution Arrange a haphazard collection of n = 1 random numbers u 1, u 2,, u n use Program 1a To make sure that random numbers are indeed drawn from a uniform distribution take n = 1 and n = 1 in Program 1a and see the histogram

5 Application of the random numbers, Monte Carlo Such sequences of randomly numbers are helpful to calculate value of integrals Let us consider 1 sin(x) dx We can use the random numbers to do the following approximation 1 sin(x) dx 1 n n sin(u j ) Use Program 2 to get an estimated value of the integral Change n to obtain better accuracy Here are some results If n = 1 1 sin(x) dx If n = 1 1 sin(x) dx

6 Error of estimation, Central limit theorem If we use several times (m-times) a program 2 for n = 1 we see a distribution of results If we take the simulation together, ie we create a histogram we can see that the distribution is similar to a normal distribution Tests conrms guess, hypothesis, see Program 21

7 Random numbers drawn from Beta distribution Note that 1 sin(x)4(1 x) 3 dx 1 n n sin(u j )4(1 u j ) 3 Use Program 2a to obtain an estimated value of the above integral But 1 4(1 x) 3 dx = 1 This function is an example of a density in a class of Beta distributions, (SAS9 Beta(1,4)) Create the sequence of randomly arranged numbers ξ 1, ξ 2,, ξ n drawn from Beta distribution Use Program 3 Observe the histogram Use Program 4 to calculate the value of the integral by formula 1 sin(x)4(1 x) 3 dx 1 n n sin(ξ j )

8 The probability density function of Beta distribution The probability density function (density) of beta distribution with parameter a, b > is given by β a,b (x) = 1 B(a, b) x a 1 (1 x) b 1, x 1, where B(a, b) is a normalizing constant Note that the uniform distribution is an example of beta distribution for a = b = 1 For almost all parameters a, b > the probability density function of beta distribution is a continuous function To make a better estimation of the density than given by histogram apply Program 5

9 Estimation of density of Beta distribution Notice the role of parameter bmw (so called bandwidth parameter ) in smoothing of histogram Change bmw in scale from 1 to 2 If bmw=1 then If bmw=2 then

10 The expected value and variance of Beta distribution The expected value (expectation, mean) and variance of a Beta distribution radome variable X is given by: VarX = 1 EX = 1 xβ a,b (x)dx = x 2 β a,b (x)dx (EX ) 2 = a a + b ab (a + b) 2 (a + b + 1) Choose any a, b > in Program 6 to check above formula Notice the dierence between true and simulated mean and variance

11 SLLN Kolmogorov, Marcinkiewicz, Zygmund In the above method (called Monte Carlo) we use the following Theorem Theorem Let η, η 1, η 2, η 3 be a sequence of ( iid) independent and identically distributed random variables, x any p (, 2)Then 1 n 1/p n η j converges as(almost surely) if and only if E η p < and either p 1 or Eη = In that case the limit equals Eη for p = 1 and is otherwise

12 Applications Monte Carlo Theorem If we take p = 1 and ξ, ξ 1, ξ 2, ξ 3 a sequence of (iid rvs) such that E f (ξ) < then puting η j = f (ξ j ) we obtain for n that 1 n n f (ξ j ) Ef (ξ) as

13 Application 2 Theorem If X j, j = 1, 2 and 1 n n X j does not converge then EX =!!! Example Let W t be a Wiener process For W t we denote by T a the rst hitting time of a level (a barrier) a > by W t, ie T a = inf{t > : W t = a} Program 39 We know that for almost all scenarios ω Ω a trajectory W t (ω) reaches the level a We observe that in given time a trajectory is reluctant to reach level a We suspect that ET a = How to verify it? Program 4 shows a distribution of T a Change time in which we observe the Wiener process and calculate 1 n n X j Program 4aTake p = 1

14 Application 3 If we suspect that there is < p < 1 such that for all p < p E X p < and E X p = for p > p then using SLLN we can estimate p Program 4a Take p = 4; p = 5; p = 6

15 SLLN - equivalent formulation Theorem Let η, η 1, η 2, η 3 be a sequence of (iid rvs) independent and identically distributed random variables, x any p (, 2)Then 1 n η n 1/p j an as(almost surely) (1) for some real a if and only if In that case we have a = E η p < { p < 1, Eη p [1, 2)

16 Problem Let Note that 1 n 1/p E η p < n η j an = n 1 1/p 1 n n η j a What happens if p 2 ie 1 1/p 1/2? Let η, η 1, η 2, η 3 be a sequence of (iid rvs) independent and identically distributed random variables, let p = 2 n 1 n η j a?? (2) n

17 SLLN for order statistics Theorem of this type was proved by van Zwet, Sen For a random variable X with E X < and the distribution function F X, expected shortfall at condance level α [, 1) is dened as ES α = 1 1 α 1 where Q u (F X ) is the quantile function of F X Theorem α q u (F X )du, Let X, X 1, X 2, be random variables iid with above assumptions We have [n(1 α)] 1 lim L j,n = ES α ae, n [n(1 α)] where L 1,n L 2,n L n,n are the order statistics of L 1, L 2,, L n and where [n(1 α)] denotes the largest integer not exceeding n(1 α)