processi stocastici (3): processi gaussiani -...

Corso di Modelli di Computazione Affettiva

Prof. Giuseppe Boccignone

Dipartimento di InformaticaUniversità di Milano

[email protected]@unimi.it

http://boccignone.di.unimi.it/CompAff2016.html

Processi stocastici (3): processi gaussiani

Processi stocastici // Classi di processi

Table of content 25

Fig. 14 A conceptual mapof stochastic processes thatare likely to play a role ineye movement modelling andanalyses

.

P(2) =! propagator"!!!#$!!!%

P(2 | 1) P(1)dx1 (27)

where P(2 | 1) serves as the evolution kernel or propagator P(1→ 2).A stochastic process whose joint pdf does not change when shifted in time is

called a (strict sense) stationary process:

P(x1, t1; x2, t2; · · · ; xn , tn) = P(x1, t1 + τ; x2, t2 + τ; · · · ; xn , tn + τ) (28)

τ > 0 being a time shift. Analysis of a stationary process is frequently much simplerthan for a similar process that is time-dependent: varying t, all the random variablesXt have the same law; all the moments, if they exist, are constant in time; the distri-bution of X(t1) and X(t2) depends only on the difference τ = t2 − t1 (time lag), i.e,P(x1, t1; x2, t2) = P(x1,x2; τ).

A conceptual map of main kinds of stochastic processes that we will discuss inthe remainder of this Chapter is presented in Figure 14.

5 How to leave the past behind: Markov Processes

The most simple kind of stochastic process is the Purely Random Process in whichthere are no correlations. From Eq. (27):

P(1,2) = P(1)P(2) (29)P(1,2,3) = P(1)P(2)P(3)

P(1,2,3,4) = P(1)P(2)P(3)P1(3)

· · ·

One such process can be obtained for example by repeated coin tossing. Thecomplete independence property can be written explicitly as:

• Processi Gaussiani: prendendo un qualsiasi numero finito di variabili aleatorie, dall’ensemble che forma il processo aleatorio stesso, esse hanno una distribuzione di probabilità congiunta gaussiana (multivariata)

• Più precisamente: è Gaussiano se per qualunque scelta di si ha che


Chapter 3

Gaussian Processes andStochastic DifferentialEquations

3.1 Gaussian Processes

Gaussian processes are a reasonably large class of processes that havemany applications, including in finance, time-series analysis and machinelearning. Certain classes of Gaussian processes can also be thought of asspatial processes. We will use these as a vehicle to start considering moregeneral spatial objects.

Definition 3.1.1 (Gaussian Process). A stochastic process {Xt}t≥0 is Gaus-sian if, for any choice of times t1, . . . , tn, the random vector (Xt1 , . . . , Xtn)has a multivariate normal distribution.

3.1.1 The Multivariate Normal Distribution

Because Gaussian processes are defined in terms of the multivariate nor-mal distribution, we will need to have a pretty good understanding of thisdistribution and its properties.

Definition 3.1.2 (Multivariate Normal Distribution). A vectorX = (X1, . . . , Xn)is said to be multivariate normal (multivariate Gaussian) if all linear combi-nations of X, i.e. all random variables of the form

n!

k=1

αkXk

have univariate normal distributions.

89

Chapter 3








n!

k=1

αkXk


89

3.1. GAUSSIAN PROCESSES 93

Simulating Multivariate Normals

One possible way to simulate a multivariate normal random vector isusing the Cholesky decomposition.

Lemma 3.1.11. If Z ∼ N(0, I), Σ = AA! is a covariance matrix, andX = µ+ AZ, then X ∼ N(µ,Σ).

Proof. We know from lemma 3.1.4 that AZ is multivariate normal and sois µ + AZ. Because a multivariate normal random vector is completelydescribed by its mean vector and covariance matrix, we simple need to findEX and Var(X). Now,

EX = Eµ+ EAZ = µ+ A0 = µ

and

Var(X) = Var(µ+ AZ) = Var(AZ) = AVar(Z)A! = AIA! = AA! = Σ.

Thus, we can simulate a random vector X ∼ N(µ,Σ) by

(i) Finding A such that Σ = AA!.

(ii) Generating Z ∼ N(0, I).

(iii) Returning X = µ+ AZ.

Matlab makes things a bit confusing because its function ‘chol’ producesa decomposition of the form B!B. That is, A = B!. It is important tobe careful and think about whether you want to generate a row or columnvector when generating multivariate normals.

The following code produces column vectors.

Listing 3.1: Matlab code

1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);

The following code produces row vectors.


1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;

92 CHAPTER 3. GAUSSIAN PROCESSES ETC.

If A is also symmetric, then A is called symmetric positive semi-definite(SPSD). Note that if A is an SPSD then there is a real-valued decomposition

A = LL!,

though it is not necessarily unique and L may have zeroes on the diagonals.

Lemma 3.1.9. Covariance matrices are SPSD.

Proof. Given an n × 1 real-valued vector x and a random vector Y withcovariance matrix Σ, we have

Var (x!Y) = x!Var (Y)x.

Now this must be non-negative (as it is a variance). That is, it must be thecase that

x!Var (Y)x ≥ 0,

so Σ is positive semi-definite. Symmetry comes from the fact that Cov(X, Y ) =Cov(Y,X).

Lemma 3.1.10. SPSD matrices are covariance matrices.

Proof. Let A be an SPSD matrix and Z be a vector of random variables withVar(Z) = I. Now, as A is SPSD, A = LL!. So,

Var(LZ) = LVar(Z)L! = LIL! = A,

so A is the covariance matrix of LZ.

COVARIANCE POS DEF ...

Densities of Multivariate Normals

If X = (X1, . . . , Xn) is N(µ,Σ) and Σ is positive definite, then X has thedensity

f(x) = f(x1, . . . , xn) =1!

(2π)n|Σ|exp

"−1

2(x− µ)!Σ−1(x− µ)

#.

Chapter 3








n!

k=1

αkXk


89



A = LL!,






x!Var (Y)x ≥ 0,









f(x) = f(x1, . . . , xn) =1!

(2π)n|Σ|exp

"−1

2(x− µ)!Σ−1(x− µ)

#.



A = LL!,






x!Var (Y)x ≥ 0,









f(x) = f(x1, . . . , xn) =1!

(2π)n|Σ|exp

"−1

2(x− µ)!Σ−1(x− µ)

#.



A = LL!,






x!Var (Y)x ≥ 0,









f(x) = f(x1, . . . , xn) =1!

(2π)n|Σ|exp

"−1

2(x− µ)!Σ−1(x− µ)

#.


• Processi Gaussiani: prendendo un qualsiasi numero finito di variabili aleatorie, dalla collezione che forma il processo aleatorio stesso, esse hanno una distribuzione di probabilità congiunta gaussiana (multivariata)

• Sono processi continui nel tempo e nello spazio degli stati

• Esempio: il rumore bianco è un processo Gaussiano i.i.d


This is quite a strong definition. Importantly, it implies that, even if all ofits components are normally distributed, a random vector is not necessarilymultivariate normal.

Example 3.1.3 (A random vector with normal marginals that is not multi-variate normal). Let X1 ∼ N(0, 1) and

X2 =

!X1 if |X1| ≤ 1

−X1 if |X1| > 1.

Note that X2 ∼ N(0, 1). However, X1 + X2 is not normally distributed,because |X1+X2| ≤ 2, which implies X1+X2 is bounded and, hence, cannotbe normally distributed.

Linear transformations of multivariate normal random vectors are, again,multivariate normal.

Lemma 3.1.4. Suppose X = (X1, . . . , Xn) is multivariate normal and A isan m× n real-valued matrix. Then, Y = AX is also multivariate normal.

Proof. Any linear combination of Y1, . . . , Ym is a linear combination of linearcombinations of X1, . . . , Xn and, thus, univariate normal.

Theorem 3.1.5. A multivariate normal random vector X = (X1, . . . , Xn)is completely described by a mean vector µ = EX and a covariance matrixΣ = Var(X).

Proof. The distribution of X is described by its characteristic function whichis

Eexp {iθ!X} = Eexp

!

in"

i=1

θiXi

#

.

Now, we know$n

i=1 θiXi is a univariate normal random variable (becauseX is multivariate normal). Let m = E

$ni=1 θiXi and σ2 = Var (

$ni=1 θiXi).

Then,

E

!

in"

i=1

θiXi

#

= exp

%im− 1

2σ2

&.

Now

m = E

n"

i=1

θiXi =n"

i=1

θiµi

and

σ2 = Var

'n"

i=1

θiXi

(

=n"

i=1

n"

j=1

θiθjCov(Xi, Xj) =n"

j=1

n"

j=1

θiθjΣi,j .

So, everything is specified by µ and Σ.


• Processi Gaussiani: prendendo un qualsiasi numero finito di variabili aleatorie, dalla collezione che forma il processo aleatorio stesso, esse hanno una distribuzione di probabilità congiunta gaussiana (multivariata)

• Un processo Gaussiano è completamente specificato mediante media e matrice di covarianza







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;


• Processi Gaussiani: moto Browniano

Può essere considerato il limite continuo del random walk semplice


Table of content 25


.

P(2) =! propagator"!!!#$!!!%

P(2 | 1) P(1)dx1 (27)








P(1,2) = P(1)P(2) (29)P(1,2,3) = P(1)P(2)P(3)

P(1,2,3,4) = P(1)P(2)P(3)P1(3)

· · ·





2.2. IL MOTO BROWNIANO 27

la formalizzazione precisa di questo risultato alla Proposizione 2.25, dopo che avremodiscusso la nozione di �-algebra associata a un processo.

2.2. Il moto browniano

Ricordiamo l’Osservazione 1.2: fissato uno spazio di probabilità (⌦,F ,P), scriveremo “q.c.[. . . ]” come abbreviazione di “esiste A 2 F , con P(A) = 1, tale che per ogni ! 2 A [. . . ]”.

Definiamo ora il moto browniano, detto anche processo di Wiener, che costituiscel’oggetto centrale di questo corso. Si tratta dell’esempio più importante di processostocastico a tempo continuo. Esso può essere visto come l’analogo a tempo continuo diuna passeggiata aleatoria reale con incrementi gaussiani. In effetti, come discuteremo piùavanti, il moto browniano può essere ottenuto come un opportuno limite di qualunquepasseggiata aleatoria con incrementi di varianza finita (cf. il sottoparagrafo 2.7.1).

Definizione 2.3 (Moto browniano). Si dice moto browniano qualunque processostocastico reale B = {B

t

}t2[0,1)

che soddisfa le seguenti proprietà:

(a) B0

= 0 q.c.;(b) B ha incrementi indipendenti, cioè per ogni scelta di k � 2 e 0 t

0

< t1

< . . . <tk

< 1 le variabili aleatorie {Bt

i

�Bt

i�1}1ik

sono indipendenti;(c) B ha incrementi stazionari gaussiani centrati : più precisamente, per ogni scelta

di t > s � 0 si ha (Bt

�Bs

) ⇠ N (0, t� s);(d) q.c. B ha traiettorie continue, cioè q.c. la funzione t 7! B

t

è continua.

Nella definizione è sottinteso lo spazio di probabilità (⌦,F ,P) su cui è definito ilprocesso B, per cui si ha B

t

= Bt

(!) con ! 2 ⌦. La dipendenza da ! verrà quasi sempreomessa, ma è importante essere in grado di esplicitarla quando è necessario. Per esempio,la proprietà (d) si può riformulare nel modo seguente: esiste A 2 F con P(A) = 1 taleche per ogni ! 2 A la funzione t 7! B

t

(!) è continua. Oltre a essere una richiesta moltonaturale dal punto di vista fisico, la continuità delle traiettorie è una proprietà di basilareimportanza anche da un punto di vista matematico (si veda il sottoparagrafo § 2.2.2).

Talvolta parleremo di moto browniano con insieme dei tempi ristretto a un intervalloT = [0, t

0

], dove t0

2 (0,1) è fissato, intendendo naturalmente con ciò un processo{B

t

}t2T che soddisfa le condizioni della Definizione 2.3 per t ristretto a T.

Nella Figura 2.1 sono mostrate tre traiettorie illustrative del moto browniano.Veniamo ora al primo risultato fondamentale sul moto browniano, dimostrato per la

prima volta da Norbert Wiener nel 1923. A dispetto delle apparenze, si tratta di unrisultato non banale.

Teorema 2.4 (Wiener). Il moto browniano esiste.

Sono possibili diverse dimostrazioni di questo teorema. Un metodo standard, basatosu un teorema molto generale dovuto a Kolmogorov, consiste nel costruire sullo spazio







t

}t2[0,1)


(a) B0


0

< t1

< . . . <tk


i

�Bt

i�1}1ik



�Bs


t

è continua.


t

= Bt


t



0

], dove t0


t






26 2. MOTO BROWNIANO

prodotto (EI , EI

), che è detto talvolta spazio delle traiettorie del processo X. Comeogni variabile aleatoria, X induce sullo spazio d’arrivo (EI , EI

) la sua legge µX

: questaprobabilità è detta legge del processo.

Se C = {x 2 EI

: xt1 2 A

1

, . . . , xt

k

2 Ak

} è un sottoinsieme cilindrico di EI , siha µ

X

(C) = P(X 2 C) = P((Xt1 , . . . , Xt

k

) 2 A1

⇥ · · · ⇥ Ak

), dunque la probabilitàµX

(C) può essere calcolata conoscendo le leggi finito-dimensionali di X. Ricordando che isottoinsiemi cilindrici sono una base della �-algebra EI , segue che la legge µ

X

del processoX sullo spazio delle traiettorie (EI , EI

) è determinata dalle leggi finito-dimensionali di X.(Per questa ragione, con il termine legge del processo X si indica talvolta la famiglia delleleggi finito dimensionali.) In particolare, due processi X = {X

t

}t2I , X 0

= {X 0t

}t2I con lo

stesso insieme degli indici I e a valori nello stesso spazio (E, E) hanno la stessa legge se esolo se hanno le stesse leggi finito-dimensionali.

2.1.3. Processi gaussiani. Un processo vettoriale X = {Xt

}t2I a valori in Rd, con

Xt

= (X(1)

t

, . . . , X(d)

t

), può essere sempre visto come un processo stocastico reale a pattodi ampliare l’insieme degli indici: infatti basta scrivere X = {X(i)

t

}(i,t)2{1,...,d}⇥I

. Perquesta ragione, quando risulta conveniente, è possibile limitare la trattazione ai processireali, senza perdita di generalità. Questo è quello che faremo sempre nel caso dei processigaussiani, che ora definiamo.

Definizione 2.2. Un processo stocastico reale X = {Xt

}t2I è detto gaussiano se,

per ogni scelta di t1

, . . . , tn

2 I, il vettore aleatorio (Xt1 , . . . , Xt

n

) è normale, cioè sequalunque combinazione lineare finita delle X

t

è una variabile aleatoria normale.

I processi gaussiani costituiscono una generalizzazione dei vettori aleatori normali.Si noti infatti che, quando I = {t

1

, . . . , tk

} è un insieme finito, un processo gaussianoX = {X

t

}t2I = (X

t1 , . . . , Xt

k

) non è altro che un vettore aleatorio normale a valori in Rk.Come per i vettori normali, dato un processo gaussiano X = {X

t

}t2I introduciamo le

funzioni media µ(t) := E(Xt

) e covarianza K(s, t) := Cov(Xs

, Xt

), ben definite in quantoX

t

2 L2 per ogni t 2 I (perché?). Si noti che la funzione K(·, ·) è simmetrica e semi-definita positiva, nel senso seguente: per ogni scelta di n 2 N, t

1

, . . . , tn

2 I e di u 2 Rn siha

P

n

i,j=1

K(ti

, tj

)ui

uj

� 0; infatti {Kij

:= K(ti

, tj

)}1i,jn

è la matrice di covarianzadel vettore (X

t1 , . . . , Xt

n

). Si può mostrare (non lo faremo) che, assegnate arbitrariamentedue funzioni µ : I ! R e K : I ⇥ I ! R, con K simmetrica e semi-definita positiva, esisteun processo gaussiano {X

t

}t2I che ha µ e K come funzioni media e covarianza.

Una proprietà fondamentale è che le leggi finito-dimensionali di un processo gaussianosono univocamente determinate dalle sue funzioni media µ(·) e covarianza K(·, ·). Questosegue immediatamente dal fatto che ogni vettore della forma (X

t1 , . . . , Xt

k

) è per defini-zione normale a valori in Rk e dunque la sua funzione caratteristica, espressa dalla formula(1.9), è una funzione del vettore (µ(t

1

), . . . , µ(tk

)) e della matrice {Kij

:= K(ti

, tj

)}1i,jk

.Anche la proprietà basilare per cui variabili congiuntamente normali sono indipendenti

se e solo se sono scorrelate, cf. il Lemma 1.14, si estende ai processi gaussiani. Rimandiamo







t

}t2[0,1)


(a) B0


0

< t1

< . . . <tk


i

�Bt

i�1}1ik



�Bs


t

è continua.


t

= Bt


t



0

], dove t0


t






26 2. MOTO BROWNIANO

prodotto (EI , EI

), che è detto talvolta spazio delle traiettorie del processo X. Comeogni variabile aleatoria, X induce sullo spazio d’arrivo (EI , EI

) la sua legge µX

: questaprobabilità è detta legge del processo.

Se C = {x 2 EI

: xt1 2 A

1

, . . . , xt

k

2 Ak

} è un sottoinsieme cilindrico di EI , siha µ

X

(C) = P(X 2 C) = P((Xt1 , . . . , Xt

k

) 2 A1

⇥ · · · ⇥ Ak

), dunque la probabilitàµX

(C) può essere calcolata conoscendo le leggi finito-dimensionali di X. Ricordando che isottoinsiemi cilindrici sono una base della �-algebra EI , segue che la legge µ

X

del processoX sullo spazio delle traiettorie (EI , EI

) è determinata dalle leggi finito-dimensionali di X.(Per questa ragione, con il termine legge del processo X si indica talvolta la famiglia delleleggi finito dimensionali.) In particolare, due processi X = {X

t

}t2I , X 0

= {X 0t

}t2I con lo

stesso insieme degli indici I e a valori nello stesso spazio (E, E) hanno la stessa legge se esolo se hanno le stesse leggi finito-dimensionali.

2.1.3. Processi gaussiani. Un processo vettoriale X = {Xt

}t2I a valori in Rd, con

Xt

= (X(1)

t

, . . . , X(d)

t

), può essere sempre visto come un processo stocastico reale a pattodi ampliare l’insieme degli indici: infatti basta scrivere X = {X(i)

t

}(i,t)2{1,...,d}⇥I

. Perquesta ragione, quando risulta conveniente, è possibile limitare la trattazione ai processireali, senza perdita di generalità. Questo è quello che faremo sempre nel caso dei processigaussiani, che ora definiamo.

Definizione 2.2. Un processo stocastico reale X = {Xt

}t2I è detto gaussiano se,

per ogni scelta di t1

, . . . , tn

2 I, il vettore aleatorio (Xt1 , . . . , Xt

n

) è normale, cioè sequalunque combinazione lineare finita delle X

t

è una variabile aleatoria normale.

I processi gaussiani costituiscono una generalizzazione dei vettori aleatori normali.Si noti infatti che, quando I = {t

1

, . . . , tk

} è un insieme finito, un processo gaussianoX = {X

t

}t2I = (X

t1 , . . . , Xt

k

) non è altro che un vettore aleatorio normale a valori in Rk.Come per i vettori normali, dato un processo gaussiano X = {X

t

}t2I introduciamo le

funzioni media µ(t) := E(Xt

) e covarianza K(s, t) := Cov(Xs

, Xt

), ben definite in quantoX

t

2 L2 per ogni t 2 I (perché?). Si noti che la funzione K(·, ·) è simmetrica e semi-definita positiva, nel senso seguente: per ogni scelta di n 2 N, t

1

, . . . , tn

2 I e di u 2 Rn siha

P

n

i,j=1

K(ti

, tj

)ui

uj

� 0; infatti {Kij

:= K(ti

, tj

)}1i,jn

è la matrice di covarianzadel vettore (X

t1 , . . . , Xt

n

). Si può mostrare (non lo faremo) che, assegnate arbitrariamentedue funzioni µ : I ! R e K : I ⇥ I ! R, con K simmetrica e semi-definita positiva, esisteun processo gaussiano {X

t

}t2I che ha µ e K come funzioni media e covarianza.

Una proprietà fondamentale è che le leggi finito-dimensionali di un processo gaussianosono univocamente determinate dalle sue funzioni media µ(·) e covarianza K(·, ·). Questosegue immediatamente dal fatto che ogni vettore della forma (X

t1 , . . . , Xt

k

) è per defini-zione normale a valori in Rk e dunque la sua funzione caratteristica, espressa dalla formula(1.9), è una funzione del vettore (µ(t

1

), . . . , µ(tk

)) e della matrice {Kij

:= K(ti

, tj

)}1i,jk

.Anche la proprietà basilare per cui variabili congiuntamente normali sono indipendenti

se e solo se sono scorrelate, cf. il Lemma 1.14, si estende ai processi gaussiani. Rimandiamo

• Moto Browniano: gli incrementi sono genericamente Gaussiani

• Processo di Wiener: se gli incrementi sono Gaussiani centrati (media 0, varianza 1)


Example 3.1.28 (Ornstein-Uhlenbeck Process). The Ornstein-Uhlenbeckprocess has

µi = EXti = 0

withσi,i+1 = exp{α|ti − ti+1|/2}

and σi,i = 1. So, our updating formula is

Xti+1 = exp{−α|ti − ti+1|/2}Xti +!"

1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...

8 + sqrt(1 - exp(-alpha * h))*randn;

9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion

One of the most fundamental stochastic processes. It is a Gaussian pro-cess, a Markov process, a Levy process, a Martingale and a process that isclosely linked to the study of harmonic functions (do not worry if you do notknow all these terms). It can be used as a building block when consideringmany more complicated processes. For this reason, we will consider it inmuch more depth than any other Gaussian process.

Definition 3.2.1 (Brownian Motion). A stochastic process {Wt}t≥0 is calledBrownian motion (a Wiener process) if:

(i) It has independent increments.

(ii) It has stationary increments.

(iii) Wt ∼ N(0, t) for all t ≥ 0.

• Poichè gli incrementi sono Gaussiani




116 Stochastic Simulation and Applications in Finance

7.3.3 Wiener Process

Let us examine the limit case of a random walk

X (t) =g(t)!

k=1

Bk, where g(t) = Int"

t!t

#. (7.44)

We have seen that mX(t) = 0 and σ 2X (t) = g(t)δ2. Suppose that the jump δ is proportional to the

duration of the interval !t, δ =√

!t , then σ 2X (t) = g(t)!t. When !t goes to 0, g(t)!t tends

to t. Thus, taking the limit,

σ 2X (t) = t. (7.45)

On the other hand, when !t goes to 0, X(t) is the sum of a large number of independentbinary random variables. By virtue of the central limit theorem, it converges to a Gaussianrandom variable.

Taken to the limit, the random walk tends toward a Gaussian random process with zeromean, variance σ 2

X (t) = t and independent increments. Such a process is by definition a standardWiener process.

Definition 3.1 W(t) is a standard Wiener process if

(i) W(0) = 0,(ii) W(t) has independent increments, which means that

∀ t1 < t2 < t3 < t4, W (t4) − W (t3) and W (t2) − W (t1)

are independent,(iii) Z(t) = W(t) − W(t0) is Gaussian with zero mean and variance

σ 2Z = t − t0, t0 ≤ t.

From this definition we can deduce the following basic properties for a standard Wienerprocess.

Property 3.2 Var(W(t)) = t and mW(t) = E[W(t)] = 0. Since its variance increases with timet, the trajectory of this process moves away from the horizontal axis when t increases.

Property 3.3 From the law of large numbers, we can prove that:

W (t)t

−→t→∞

0. (7.46)



µi = EXti = 0




1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...


9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion









X (t) =g(t)!

k=1


t!t

#. (7.44)





σ 2X (t) = t. (7.45)








σ 2Z = t − t0, t0 ≤ t.




W (t)t

−→t→∞

0. (7.46)




X (t) =g(t)!

k=1


t!t

#. (7.44)





σ 2X (t) = t. (7.45)








σ 2Z = t − t0, t0 ≤ t.




W (t)t

−→t→∞

0. (7.46)

La varianza cresce linearmente nel tempo e il processo si sposta lontano dall’asse t

• La funzione di auto-correlazione è






µi = EXti = 0




1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...


9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion






Introduction to Random Processes 117

Property 3.4 W’s trajectory, even if it is continuous, varies in a very irregular way thatmakes it non differentiable. Heuristically, this result comes from !W(t) = W(t + !t) − W(t)being proportional to

√!t . This property is often called Levy’s oscillatory property.

Property 3.5 The autocorrelation function of W(t) is given by:

RW W (t1, t2) = min(t1, t2). (7.47)

Proof: Indeed,

RW W (t1, t2) = E [W (t1)W (t2)] . (7.48)

Without loss of generality, we suppose t1 < t2 then

RW W (t1, t2) = E!(W (t2) − W (t1)) W (t1) + W (t1)2" . (7.49)

Since W(t) has independent increments,

E [(W (t2) − W (t1)) W (t1)] = E [W (t1)] E [W (t2) − W (t1)] = 0. (7.50)

Which implies that

RW W (t1, t2) = E!W (t1)2" = Var(W (t1)) = t1 = min(t1, t2). (7.51)

Property 3.6 The standard Wiener process exhibits a very important property called themartingale property, that is:

E [W (tn)|W (tn−1), W (tn−2), . . . , W (t1)] = W (tn−1), t1 < t2 < . . . < tn. (7.52)

Proof: Since W(t) has independent increments, we have:

E [W (tn)|W (tn−1), . . . , W (t1)] (7.53)

or

E [W (tn) − W (tn−1) + W (tn−1)|W (tn−1), . . . , W (t1)] . (7.54)

Moreover, the expression

E [W (tn) − W (tn−1)|W (tn−1), . . . , W (t1)] (7.55)

can be rewritten as

E [W (tn) − W (tn−1)|W (tn−1)] = E [W (tn)|W (tn−1)]

−E [W (tn−1)|W (tn−1)] . (7.56)


• Moto Browniano: è un processo Gaussiano Markoviano


andµi = EXti .

By theorem 3.1.24, we know that (Xti , Xti+1)! has a multivariate normal

distribution. In particular,(

Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).

Using theorem 3.1.25, we have

Xti+1 |Xti = xi ∼ N

(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).

Algorithm 3.1.2 (Generating a Markovian Gaussian Process).

(i) Draw Z ∼ N(0, 1). Set Xt1 = µ1 +√σi,iZ.

(ii) For i = 1, . . . ,m− 1 draw Z ∼ N(0, 1) and set

Xti+1 = µi +σi,i+1

σi, i(Xti − µi) +

⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.

Example 3.1.27 (Brownian Motion). Consider Brownian motion, which isa Markov process. Now,

µi = EXti = 0

andσi,i+1 = min(ti, ti+1) = ti.

So, our updating formula is

Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;

3 X(1) = sqrt(h)*randn;

4

5 for i = 1:m-1

6 X(i+1) = X(i) + sqrt(h) * randn;

7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);

x1 x2 x3 x4

t1 t2 t3 t4

P(xt | xt-1 ) = Gaussiana

Processi stocastici // Simulazione di processi Gaussiani • Matrice definita positiva: una matrice A (n x n) è PD se e solo se

• Se è anche simmetrica, allora è simmetrica definita positiva (SPD)

• E’ semidefinita positiva (SPSD) se e solo se

• Nota bene: le matrici di covarianza sono SPSD (e viceversa)


INDEPENDENCEThere is nothing too difficult about dealing with the mean vector µ. How-

ever, the covariance matrix makes things pretty difficult, especially when wewish to simulate a high dimensional random vector. In order to simulateGaussian processes effectively, we need to exploit as many properties of co-variance matrices as possible.

Symmetric Positive Definite and Semi-Positive Definite Matrices

Covariance matrices are members of a family of matrices called symmetricpositive definite matrices.

Definition 3.1.6 (Positive Definite Matrices (Real-Valued)). An n×n real-valued matrix, A, is positive definite if and only if

x!Ax > 0 for all x = 0.

If A is also symmetric, then A is called symmetric positive definite (SPD).Some important properties of SPD matrices are

(i) rank(A) = n.

(ii) |A| > 0.

(iii) Ai,i > 0.

(iv) A−1 is SPD.

Lemma 3.1.7 (Necessary and sufficient conditions for an SPD). The fol-lowing are necessary and sufficient conditions for an n × n matrix A to beSPD

(i) All the eigenvalues λ1, . . . ,λn of A are strictly positive.

(ii) There exists a unique matrix C such that A = CC!, where C is a real-valued lower-triangular matrix with positive diagonal entries. This iscalled the Cholesky decomposition of A.

Definition 3.1.8 (Positive Semi-definite Matrices (Real-Valued)). An n×nreal-valued matrix, A, is positive semi-definite if and only if

x!Ax ≥ 0 for all x = 0.









(i) rank(A) = n.

(ii) |A| > 0.

(iii) Ai,i > 0.

(iv) A−1 is SPD.














(i) rank(A) = n.

(ii) |A| > 0.

(iii) Ai,i > 0.

(iv) A−1 is SPD.






Decomposizione di Cholesky

• Simulazione : utilizziamo la decomposizione di Cholesky

Processi stocastici // Simulazione di processi Gaussiani

Decomposizione di Cholesky







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;

1. Trovare A tale che

2.

3.








and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;

Simulazione:







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;

Genera vettori colonna:

Genera vettori riga:







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;

• Simulazione : utilizziamo la decomposizione di Cholesky







and









1 mu = [1 2 3]’;

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + A’ * randn(3,1);



1 mu = [1 2 3];

2 Sigma = [3 1 2; 1 2 1; 2 1 5];

3 A = chol(Sigma);

4 X = mu + randn(1,3) * A;

Processi stocastici // Simulazione di processi Gaussiani • Simulazione di un moto Browniano:


The complexity of the Cholesky decomposition of an arbitrary real-valuedmatrix is O(n3) floating point operations.

3.1.2 Simulating a Gaussian Processes Version 1

Because Gaussian processes are defined to be processes where, for anychoice of times t1, . . . , tn, the random vector (Xt1 , . . . , Xtn) has a multivariatenormal density and a multivariate normal density is completely describe bya mean vector µ and a covariance matrix Σ, the probability distribution ofGaussian process is completely described if we have a way to construct themean vector and covariance matrix for an arbitrary choice of t1, . . . , tn.

We can do this using an expectation function

µ(t) = EXt

and a covariance function

r(s, t) = Cov(Xs, Xt).

Using these, we can simulate the values of a Gaussian process at fixed timest1, . . . , tn by calculating µ where µi = µ(t1) and Σ, where Σi,j = r(ti, tj) andthen simulating a multivariate normal vector.

In the following examples, we simulate the Gaussian processes at evenlyspaced times t1 = 1/h, t2 = 2/h, . . . , tn = 1.

Example 3.1.12 (Brownian Motion). Brownian motion is a very importantstochastic process which is, in some sense, the continuous time continuousstate space analogue of a simple random walk. It has an expectation functionµ(t) = 0 and a covariance function r(s, t) = min(s, t).


1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4

5 %Make the mean vector

6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10

11 %Make the covariance matrix

12 Sigma = zeros(n,n);






µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10




13 for i = 1:n

14 for j = 1:n

15 Sigma(i,j) = min(t(i),t(j));

16 end

17 end

18

19 %Generate the multivariate normal vector

20 A = chol(Sigma);

21 X = mu + randn(1,n) * A;

22

23 %Plot

24 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

Xt

Figure 3.1.1: Plot of Brownain motion on [0, 1].

Example 3.1.13 (Ornstein-Uhlenbeck Process). Another very importantGaussian process is the Ornstein-Uhlenbeck process. This has expectationfunction µ(t) = 0 and covariance function r(s, t) = e−α|s−t|/2.


1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4 %paramter of OU process

5 alpha = 10;

6






µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10



Processi stocastici // Simulazione di processi Gaussiani • Simulazione di un moto Browniano:






µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10




13 for i = 1:n

14 for j = 1:n


16 end

17 end

18


20 A = chol(Sigma);

21 X = mu + randn(1,n) * A;

22

23 %Plot

24 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

Xt




1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);


5 alpha = 10;

6


13 for i = 1:n

14 for j = 1:n


16 end

17 end

18


20 A = chol(Sigma);

21 X = mu + randn(1,n) * A;

22

23 %Plot

24 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

Xt




1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);


5 alpha = 10;

6






µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10




Table of content 25


.

P(2) =! propagator"!!!#$!!!%

P(2 | 1) P(1)dx1 (27)








P(1,2) = P(1)P(2) (29)P(1,2,3) = P(1)P(2)P(3)

P(1,2,3,4) = P(1)P(2)P(3)P1(3)

· · ·








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10




13 for i = 1:n

14 for j = 1:n


16 end

17 end

18


20 A = chol(Sigma);

21 X = mu + randn(1,n) * A;

22

23 %Plot

24 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

Xt




1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);


5 alpha = 10;

6


13 for i = 1:n

14 for j = 1:n


16 end

17 end

18


20 A = chol(Sigma);

21 X = mu + randn(1,n) * A;

22

23 %Plot

24 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

Xt




1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);


5 alpha = 10;

6



8 mu = zeros(1,n);

9 for i = 1:n

10 mu(i) = 0;

11 end

12



15 for i = 1:n

16 for j = 1:n

17 Sigma(i,j) = exp(-alpha * abs(t(i) - t(j)) / 2);

18 end

19 end

20


22 A = chol(Sigma);

23 X = mu + randn(1,n) * A;

24

25 %Plot

26 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

Xt

Figure 3.1.2: Plot of an Ornstein-Uhlenbeck process on [0, 1] with α = 10.

Example 3.1.14 (Fractional Brownian Motion). Fractional Brownian mo-tion (fBm) is a generalisation of Brownian Motion. It has expectation func-tion µ(t) = 0 and covariance function Cov(s, t) = 1/2(t2H + s2H − |t− s|2H),

• Simulazione di un processo di Ornstein-Uhlenbeck (O-U):

Processi stocastici // Simulazione di processi Gaussiani • Simulazione di un processo di Ornstein-Uhlenbeck (O-U):


13 for i = 1:n

14 for j = 1:n


16 end

17 end

18


20 A = chol(Sigma);

21 X = mu + randn(1,n) * A;

22

23 %Plot

24 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t

Xt




1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);


5 alpha = 10;

6



8 mu = zeros(1,n);

9 for i = 1:n

10 mu(i) = 0;

11 end

12



15 for i = 1:n

16 for j = 1:n


18 end

19 end

20


22 A = chol(Sigma);

23 X = mu + randn(1,n) * A;

24

25 %Plot

26 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

Xt





8 mu = zeros(1,n);

9 for i = 1:n

10 mu(i) = 0;

11 end

12



15 for i = 1:n

16 for j = 1:n


18 end

19 end

20


22 A = chol(Sigma);

23 X = mu + randn(1,n) * A;

24

25 %Plot

26 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

Xt




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

4

t

Xt


where H ∈ (0, 1) is called the Hurst parameter. When H = 1/2, fBm re-duces to standard Brownian motion. Brownian motion has independent in-crements. In contrast, for H > 1/2 fBm has positively correlated incrementsand for H < 1/2 fBm has negatively correlated increments.


1 %Hurst parameter

2 H = .9;

3

4 %use mesh size h

5 m = 1000; h = 1/m; t = h : h : 1;

6 n = length(t);

7


9 mu = zeros(1,n);

10 for i = 1:n

11 mu(i) = 0;

12 end

13



16 for i = 1:n

17 for j = 1:n

18 Sigma(i,j) = 1/2 * (t(i)^(2*H) + t(j)^(2*H)...

19 - (abs(t(i) - t(j)))^(2 * H));


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

4

t

Xt




1 %Hurst parameter

2 H = .9;

3

4 %use mesh size h

5 m = 1000; h = 1/m; t = h : h : 1;

6 n = length(t);

7


9 mu = zeros(1,n);

10 for i = 1:n

11 mu(i) = 0;

12 end

13



16 for i = 1:n

17 for j = 1:n

18 Sigma(i,j) = 1/2 * (t(i)^(2*H) + t(j)^(2*H)...

19 - (abs(t(i) - t(j)))^(2 * H));



8 mu = zeros(1,n);

9 for i = 1:n

10 mu(i) = 0;

11 end

12



15 for i = 1:n

16 for j = 1:n


18 end

19 end

20


22 A = chol(Sigma);

23 X = mu + randn(1,n) * A;

24

25 %Plot

26 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

Xt




Table of content 25


.

P(2) =! propagator"!!!#$!!!%

P(2 | 1) P(1)dx1 (27)








P(1,2) = P(1)P(2) (29)P(1,2,3) = P(1)P(2)P(3)

P(1,2,3,4) = P(1)P(2)P(3)P1(3)

· · ·


Processi stocastici // Simulazione di processi Gaussiani • Simulazione di moto Browniano frazionario (fBm):






µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10








µ(t) = EXt







1 %use mesh size h

2 m = 1000; h = 1/m; t = h : h : 1;

3 n = length(t);

4


6 mu = zeros(1,n);

7 for i = 1:n

8 mu(i) = 0;

9 end

10





8 mu = zeros(1,n);

9 for i = 1:n

10 mu(i) = 0;

11 end

12



15 for i = 1:n

16 for j = 1:n


18 end

19 end

20


22 A = chol(Sigma);

23 X = mu + randn(1,n) * A;

24

25 %Plot

26 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

Xt



Coefficiente di Hurst


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

4

t

Xt




1 %Hurst parameter

2 H = .9;

3

4 %use mesh size h

5 m = 1000; h = 1/m; t = h : h : 1;

6 n = length(t);

7


9 mu = zeros(1,n);

10 for i = 1:n

11 mu(i) = 0;

12 end

13



16 for i = 1:n

17 for j = 1:n

18 Sigma(i,j) = 1/2 * (t(i)^(2*H) + t(j)^(2*H)...

19 - (abs(t(i) - t(j)))^(2 * H));


20 end

21 end

22


24 A = chol(Sigma);

25 X = mu + randn(1,n) * A;

26

27 %Plot

28 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Xt

Figure 3.1.4: Plot of fractional Brownian motion on [0, 1] with H = 0.1.

3.1.3 Stationary and Weak Stationary Gaussian Pro-cesses

Gaussian processes (and stochastic processes in general) are easier to workwith if they are stationary stochastic processes.

Definition 3.1.15 (Stationary Stochastic Process). A stochastic process{Xt}t≥0 is said to be stationary if the random vectors (Xt1 , Xt2 , . . . , Xtn)and (Xt1+s, Xt2+s, . . . , Xtn+s) have the same distribution for all choices ofs, n and t1, t2, . . . , tn.

An example of such a process would be an irreducible positive recurrentcontinuous time Markov chain started from its stationary distribution.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

4

t

Xt




1 %Hurst parameter

2 H = .9;

3

4 %use mesh size h

5 m = 1000; h = 1/m; t = h : h : 1;

6 n = length(t);

7


9 mu = zeros(1,n);

10 for i = 1:n

11 mu(i) = 0;

12 end

13



16 for i = 1:n

17 for j = 1:n

18 Sigma(i,j) = 1/2 * (t(i)^(2*H) + t(j)^(2*H)...

19 - (abs(t(i) - t(j)))^(2 * H));


20 end

21 end

22


24 A = chol(Sigma);

25 X = mu + randn(1,n) * A;

26

27 %Plot

28 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Xt







20 end

21 end

22


24 A = chol(Sigma);

25 X = mu + randn(1,n) * A;

26

27 %Plot

28 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Xt







0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

t

Xt


The conditions for a stochastic process to be stationary are quite strict.Often, a slightly weaker version of stationarity, called weak stationarity isrequired instead.

Definition 3.1.16 (Weak Stationary Stochastic Process). A stochastic pro-cess {Xt}t≥0 is said to be weak stationary (sometimes wide sense stationaryor second order stationary) if EXt = c for all t ≥ 0 and Cov(Xt, Xt+s) doesnot depends on t.

An example of a weak stationary process is the Ornstein-Uhlenbeck pro-cess, as EXt = µ(t) = 0 and Cov(Xt, Xt+s) = r(t, t + s) = e−α|t−(t+s)|/2 =e−αs/2 .

Lemma 3.1.17. Gaussian processes that are weak stationary are stationary.

Proof. We know from theorem 3.1.5 that Gaussian distributions are entirelydetermined by their mean vector and covariance matrix. Since the mean andcovariance of a weakly stationary process do not change when the times areall shifted by s, a weakly stationary Gaussian process is stationary.

An Ornstein-Uhlenbeck process is a weak stationary Gaussian process, soit is a stationary stochastic process.

The covariance matrix of a stationary Gaussian process evaluated atequidistant times (e.g. t0 = 0, t1 = 1/h, t2 = 2/h, . . . , tn = 1) has a par-


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

t

Xt










20 end

21 end

22


24 A = chol(Sigma);

25 X = mu + randn(1,n) * A;

26

27 %Plot

28 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Xt








20 end

21 end

22


24 A = chol(Sigma);

25 X = mu + randn(1,n) * A;

26

27 %Plot

28 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Xt







0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

t

Xt










0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

t

Xt










20 end

21 end

22


24 A = chol(Sigma);

25 X = mu + randn(1,n) * A;

26

27 %Plot

28 plot(t,X);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Xt






H = 1/2 moto Browniano

H > 1/2 incrementi positivamente correlati

H < 1/2 incrementi negativamente correlati


Table of content 25


.

P(2) =! propagator"!!!#$!!!%

P(2 | 1) P(1)dx1 (27)








P(1,2) = P(1)P(2) (29)P(1,2,3) = P(1)P(2)P(3)

P(1,2,3,4) = P(1)P(2)P(3)P1(3)

· · ·


Processi stocastici // Simulazione di processi Gaussiani Markoviani • Simulazione : possiamo sfruttare la Markovianità per rendere le simulazioni

più efficienti• Ricordiamo alcune proprietà delle Gaussiane MV


Discretization error needs to be considered when we decide on a sam-pling budget. Consider, for example, an evenly spaced mesh of points t1 =1/m, t2 = 2/m, . . . tn = 1. As m gets bigger, the mesh gets finer and thediscretization error gets smaller. We still have statistical error, however, sowe need to make sure we generate a large enough sample of realizations ofthe stochastic process (determined by the sample size N). The total workdone by our simulation is then given by

work = number of samples× work to make one sample = Nf(m).

Usually, f grows at least linearly in m. In the case of Cholesky decompo-sition, for example, it is O(m3). For a fixed level of work, we need to decideon how much effort to allocate to reducing discretization error (how large mshould be) and how much effort to allocate to reducing statistical error (howlarge N should be). Finding the optimal tradeoff can be difficult. We willconsider such tradeoffs in a number of situations.

3.1.5 Marginal and Conditional Multivariate NormalDistributions

The multivariate normal distribution has many attractive properties. Inparticular, its marginal distributions are also multivariate normal. In addi-tion, if we condition on part of the a multivariate normal vector, the remain-ing values are also multivariate normal.

To see this, we need to write normal random vectors in the appropriateform. LetX ∼ N(µ,Σ). We can decomposeX into two parts,X = (X1,X2)!.We can then write µ as (µ1,µ2)! and Σ as

Σ =

!Σ11 Σ12

Σ21 Σ22

".

Theorem 3.1.24 (Multivariate Normal Marginal Distributions). GivenX =(X1,X2) ∼ N(µ,Σ),

X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).

Theorem 3.1.25 (Multivariate Normal Conditional Distributions). GivenX = (X1,X2) ∼ N(µ,Σ), X2 conditional on X1 is multivariate normal with

E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)

andVar(X2 |X1) = Σ22 − Σ21Σ

−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.








Σ =

!Σ11 Σ12

Σ21 Σ22

".


X1 ∼ N(µ1,Σ11),

andX2 ∼ N(µ2,Σ22).


E[X2 |X1] = µ2 + Σ21Σ−111 (X1 − µ1)


−111 Σ12.

Processi stocastici // Simulazione di processi Gaussiani Marsovini • Simulazione : possiamo sfruttare la Markovianità per rendere le simulazioni

più efficienti• Processo Markoviano:


19 t_new = 1/2^level : 2/2^level : (2^level-1)/(2^level);

20 n_new = length(t_new);

21

22 %Record the time points for the whole process

23 t_temp = [t t_new];

24 n_temp = length(t_temp);

25

26 %Make a covariance matrix for the whole thing

27 Sigma_temp = zeros(n_temp,n_temp);

28 for i = 1:n_temp

29 for j = 1:n_temp

30 Sigma_temp(i,j) = min(t_temp(i),t_temp(j));

31 end

32 end

33

34 %Make the separate Sigma components

35 Sigma_11 = Sigma;

36 Sigma_21 = Sigma_temp(n+1:n_temp, 1:n);

37 Sigma_12 = Sigma_temp(1:n, n+1:n_temp);

38 Sigma_22 = Sigma_temp(n+1:n_temp, n+1:n_temp);

39

40 temp_mean = Sigma_21 * inv(Sigma_11) * X;

41 Sigma_new = Sigma_22 - Sigma_21 * inv(Sigma_11) * Sigma_12;

42 X_new = temp_mean + chol(Sigma_new)’ * randn(n_new,1);

43 X = [X; X_new];

44 t = t_temp;

45 n = n_temp;

46 Sigma = Sigma_temp;

47 [dummy index] = sort(t);

48 another_dummy = waitforbuttonpress;

49

50 plot([0; t(index)’],[0; X(index)]);

51 axis([0 1 -2.5 2.5]);

52 end

3.1.7 Markovian Gaussian Processes

If a Gaussian process, {Xt}t≥0, is Markovian, we can exploit this structureto simulate the process much more efficiently. Because {Xt}t≥0 is Markovian,we can use we only need to know the value of Xti in order to generate Xti+1 .Define

σi,i+1 = Cov(Xti , Xti+1)


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);

Gaussianità


• Moto Browniano: è un processo Gaussiano Markoviano


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


• Processo O-U: è un processo Gaussiano Markoviano


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);


andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);



µi = EXti = 0




1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...


9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion








µi = EXti = 0




1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...


9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion








µi = EXti = 0




1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...


9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion








µi = EXti = 0




1− exp{α|ti − ti+1|}#Z.


1 alpha = 50; m = 10^4;

2

3 X = zeros(m,1); h = 1/m;

4 X(1) = randn;

5

6 for i = 1:m-1

7 X(i+1) = exp(-alpha * h / 2)*X(i)...


9 end

10

11 plot(h:h:1,X);

3.2 Brownian Motion







andµi = EXti .



Xti

Xti+1

)∼ N

((µi

µi+1

),

(σi,i σi,i+1

σi,i+1 σi+1,i+1

)).



(µi +

σi,i+1

σi, i(xi − µi) , σi+1,i+1 −

σ2i,i+1

σi,i

).






⎛

⎝√

σi+1,i+1 −σ2i,i+1

σi,i

⎞

⎠Z.


µi = EXti = 0



Xti+1 = Xti +(√

ti+1 − ti)Z.


1 m = 10^4; X = zeros(m,1);

2 h = 1/m;


4

5 for i = 1:m-1


7 end

8

9 plot(h:h:1,X);

mean-reverting

processi stocastici (3): processi gaussiani -...

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