An introduction to using SDEs - UCL Computer Science · An introduction to using SDEs L.M.Harrison....

An introduction to using SDEs

L.M.Harrison

Outline

• Ito-Taylor expansion• Basic numerical schemes & convergence• Parameter estimation• Filtering• Local linearization

• Next meeting – Valdes Sosa (1999) paper

Resources

• Kloeden & Platen• “An algorithmic introduction to numerical

simulation of SDEs” by Higham• “MAPLE and MATLAB for SDEs in finance” by

Higham & Kloeden• Jimenez. 1999 J. of Stat. Phys. Vol 94• Ozaki. 1992 Statistica Sinica 2 pp113-135• Ghahramani & Hinton. 1996 Technical report

CRG-TR-96-2

Weiner process

• Properties

• Update equation0)))(((

~)(,0)(

1234

2

=!!

=

tttt

tt

WWWWE

tWEWE

T = 1; N = 500; dt = T/N; dW = sqrt(dt)*randn(1,N); % increments W = cumsum(dW); % cumulative sum plot([0:dt:T],[0,W],'r-') % plot W against t

0),1,0(~,0111=+=

+++ tttttWNtdWdWWW

nnnn

!

Normally distributed

Independent incrementsTtttt !<<<!

43210

Weiner process

• Properties

• Update equation

0),1,0(~,0111=+=

+++ tttttWNtdWdWWW

nnnn

!

T = 1; N = 500; dt = T/N; dW = sqrt(dt)*randn(1,N); % increments W = cumsum(dW); % cumulative sum plot([0:dt:T],[0,W],'r-') % plot W against t

0)))(((

~)(,0)(

1234

2

=!!

=

tttt

tt

WWWWE

tWEWE Normally distributed

Independent incrementsTtttt !<<<!

43210

Sample path of a Weiner process

Scalar SDEs

• Integral form

• Differential form (symbolic)

!! ++=

t

t

ss

t

t

sttdWXbdsXaXX

00

0)()(

Stochastic integral

ttttdWXbdtXadX )()( +=

Stochastic integrals

• Ito

• Stratonovich

)()()(1

1

00

nn tt

N

n

n

T

sWWthdWth !"

+#$!

=

)()()(1

1

02

1

0

nn tt

N

n

nt

nt

T

sWWhdWth !"

+#$!

=

++

Example of difference

!"#

$%& '() TWdWW

T

T

ss

2

2

1

0

2

2

1

0

T

T

ssWdWW !"

• Ito

• Stratonovich

Additional term

Same as classicalcalculus

Ito’s formula

• Function of SDE solution

• Ito’s formula

• Partial differential operators

xbL

xb

xa

tL

!

!=

!

!+

!

!+

!

!=

1

2

2

2

2

10

ttttttdWXbdtXadXXVV )()(),( +==

Additional termc.f. classical calculus

ttttdWXtVLdtXtVLdV ),(),( 10

+=

Ito’s formula

• Ito stochastic integral equation

!! ++=

t

t

ss

t

t

sttdWXsVLdsXsVLVV

00

0),(),( 10

Ito’s formula


!! ++=

t

t

ss

t

t

sttdWXsVLdsXsVLVV

00

0),(),( 10

Stochastic integral

Ito’s formula


!! ++=

t

t

ss

t

t

sttdWXsVLdsXsVLVV

00

0),(),( 10

Stochastic integralCoefficient functions

Ito’s formula - example

• SDE

• Change of variabletttt

dWXdtXdX !" +#= )(

XXV =)(

Ito’s formula - example

• SDE

• Change of variable

• Ito’s formula

( )t

VVt

tttt

dWdtdV

Xx

VX

x

V

t

V

dWXVLdtXVLdV

!!"

2

1

2

1

8

24

2

3

4

1

2

2

2

1

2

1

10

,,0

)()(

+=

#=$

$=

$

$=

$

$

+=

##

##

ttttdWXdtXdX !" +#= )(

XXV =)(

Higham’s code - chain.m

alpha = 2; beta = 1; T = 1; N = 200; dt = T/N; % Problem parameters Xzero = 1; Xzero2 = sqrt(Xzero); % Dt = dt; % EM steps of size Dt = dt Xem1 = zeros(1,N); Xem2 = zeros(1,N); % preallocate for efficiency Xtemp1 = Xzero; Xtemp2 = Xzero2; for j = 1:N Winc = sqrt(dt)*randn; f1 = (alpha-Xtemp1); g1 = beta*sqrt(abs(Xtemp1)); Xtemp1 = Xtemp1 + Dt*f1 + Winc*g1; Xem1(j) = Xtemp1; f2 = (4*alpha-beta^2)/(8*Xtemp2) - Xtemp2/2; g2 = beta/2; Xtemp2 = Xtemp2 + Dt*f2 + Winc*g2; Xem2(j) = Xtemp2; end

plot([0:Dt:T],[sqrt([Xzero,abs(Xem1)])],'b-',[0:Dt:T],[Xzero,Xem2],'ro')

ttttdWXdtXdX !" +#= )(




( )t

VVt

dWdtdV !!"

2

1

2

1

8

24 += #

#




ttXXV =)(

tV

Compare V(t) using dX & dV

Ito-Taylor expansion

• Iterated application of Ito’s formula

• Generalization of Taylor expansion

• Convenient notation– Coefficient functions & stochastic integrals

(Kloeden & Platen ch. 5)


• SDE

• Ito-Taylor expansion

!! ++=

t

t

ss

t

t

sttdWXbdsXaXX

00

0)()(

!"#

=k

tttk XtfItX$

$$ ,0 00)],([)(

Series (hierarchy) of stochastic integrals


• SDE

• Ito-Taylor expansion

!"#

=k

tttk XtfItX$

$$ ,0 00)],([)(


Coefficient functions

Hierarchical set

!! ++=

t

t

ss

t

t

sttdWXbdsXaXX

00

0)()(

Truncated expansion

• SDE

• e.g.

)1,1()1,1()0,1()0,1()1,0()1,0()0,0()0,0(

)1()1()0()0( ...0

IfIfIfIf

IfIfXX tt

+++

+++=

!! ++=

t

t

ss

t

t

sttdWXbdsXaXX

00

0)()(


• Differential operators

• e.g.

xbL

xb

xa

tL

!

!=

!

!+

!

!+

!

!=

1

2

2

2

2

10,

bbidLLfabidLLf

bbbaidLLfabaaidLLf

xx

xx

′==′==

′′+′==′′+′==

11)1,1(

01)0,1(

22110

)1,0(2

2100

)0,0(

,

,

bidLfaidLf

xidxtf

xx

x

====

==

1

)1(

0

)0( ,

),(


Euler-Maruyama

• Differential operators

• e.g.

xbL

xb

xa

tL

!

!=

!

!+

!

!+

!

!=

1

2

2

2

2

10,

MilsteinbbidLLfabidLLf

bbbaidLLfabaaidLLf

xx

xx

′==′==

′′+′==′′+′==

11)1,1(

01)0,1(

22110

)1,0(2

2100

)0,0(

,

,

bidLfaidLf

xidxtf

xx

x

====

==

1

)1(

0

)0( ,

),(


• Examples

( ){ }nn

t

t

tn

t

t

t

t

t

s

ntt

t

t

s

nnn

t

t

WdWWdWdWI

WWWdWI

ttdsI

n

n

n

n n

nn

n

n

n

n

!"!=!=##

$

%

&&

'

(=

!="==

!="==

)) )

)

)

++

+

+

+

+

2

)1,1(

)1(

1)0(

2

111

1

1

1


• Examples

( ){ }nn

t

t

tn

t

t

t

t

t

s

ntt

t

t

s

nnn

t

t

WdWWdWdWI

WWWdWI

ttdsI

n

n

n

n n

nn

n

n

n

n

!"!=!=##

$

%

&&

'

(=

!="==

!="==

)) )

)

)

++

+

+

+

+

2

)1,1(

)1(

1)0(

2

111

1

1

1 Euler-Maruyama

Milstein

Basic numerical schemes &convergence

• Schemes based on truncated Ito-Taylorexpansion– Euler-Maruyama– Milstein

SDE schemes – truncated Ito-Taylor expansion

)1,1()1,1()1()1()0()0(0IfIfIfXX tt +++=

Euler-Maruyama

Milstein

Order 0.5, 1 strong, weak convergence resp.

Order 1, 1 strong, weak convergence resp.

Euler-Maruyama

• SDE

• Truncated Ito-Taylor expansion

• Update equation

))()()(()( 11 nntttttWtWXbtXaXX

nnnn

!+"+= ++

{ })1(),0(,,)1()1()0()0(0!" =++= IfIfXX tt

TtdWXbdsXaXX

t

t

ss

t

t

stt!!++= "" 0,)()(

00

0

Euler-Maruyama

• SDE


• Update equation

))()()(()( 11 nntttttWtWXbtXaXX

nnnn

!+"+= ++

{ })1(),0(,,)1()1()0()0(0!" =++= IfIfXX tt

TtdWXbdsXaXX

t

t

ss

t

t

stt!!++= "" 0,)()(

00

0

Hierarchical set

Higham’s code – em.m

ntntttWXXXX

nnnn

!+!+=+

µ"1

ttttdWXdtXdX µ! +=

lambda = 2; mu = 1; Xzero = 1; % problem parameters T = 1; N = 2^8; dt = T/N; dW = sqrt(dt)*randn(1,N); % Brownian increments W = cumsum(dW); % discretized Brownian path Xtrue = Xzero*exp((lambda-0.5*mu^2)*([dt:dt:T])+mu*W); R = 4; Dt = R*dt; L = N/R; % L EM steps of size Dt = R*dt Xem = zeros(1,L); % preallocate for efficiency Xtemp = Xzero; for j = 1:L Winc = sum(dW(R*(j-1)+1:R*j)); Xtemp = Xtemp + Dt*lambda*Xtemp + mu*Xtemp*Winc; Xem(j) = Xtemp; end

Approximate

!+1n

n

t

t

sdW

Higham’s code – em.m

ntntttWXXXX

nnnn

!+!+=+

µ"1

ttttdWXdtXdX µ! +=

lambda = 2; mu = 1; Xzero = 1; % problem parameters T = 1; N = 2^8; dt = T/N; dW = sqrt(dt)*randn(1,N); % Brownian increments W = cumsum(dW); % discretized Brownian path Xtrue = Xzero*exp((lambda-0.5*mu^2)*([dt:dt:T])+mu*W); R = 4; Dt = R*dt; L = N/R; % L EM steps of size Dt = R*dt Xem = zeros(1,L); % preallocate for efficiency Xtemp = Xzero; for j = 1:L Winc = sum(dW(R*(j-1)+1:R*j)); Xtemp = Xtemp + Dt*lambda*Xtemp + mu*Xtemp*Winc; Xem(j) = Xtemp; end

Exact solution ))exp(( 2

2

1

0 tttWtXX µµ! +"=

Exact solution• SDE

• Transformation

• Ito’s formula

• Solution

ttttdWXdtXdX µ! +=

XV ln=

( )tt

tt

WtXX

dWdtdV

Xx

VX

x

V

t

V

µµ!

µµ!

+"=

+"=

"=#

#=

#

#=

#

# ""

)(exp

)(

,,0

2

2

1

0

2

2

1

2

2

21

ttttdWXVLdtXVLdV )()( 10

+=

Compare exact with numericalntnttt

WXXXXnnnn

!+!+=+

µ"1

Convergence

Strong convergence of truncatedIto-Taylor expansion

• Pathwise criteria

( ) ],0[,)( TtntCXXEnnn

!"="#$ %% &

( )

tqCe

tCe

TtLTXXEe

strong

t

strong

t

L

strong

t

!+=

!"

=!#=

!

!

!

logloglog

,)(:

$

Error @end point

!"qestimate

Measures rate of decay of “mean of the error” as

Higham’s code – emstrong.m

lambda = 2; mu = 1; Xzero = 1; % problem parameters T = 1; N = 2^9; dt = T/N; % M = 1000; % number of paths sampled Xerr = zeros(M,5); % preallocate array for s = 1:M, % sample over discrete Brownian paths dW = sqrt(dt)*randn(1,N); % Brownian increments W = cumsum(dW); % discrete Brownian path Xtrue = Xzero*exp((lambda-0.5*mu^2)+mu*W(end)); for p = 1:5 R = 2^(p-1); Dt = R*dt; L = N/R; % L Euler steps of size Dt = R*dt Xtemp = Xzero; for j = 1:L Winc = sum(dW(R*(j-1)+1:R*j)); Xtemp = Xtemp + Dt*lambda*Xtemp + mu*Xtemp*Winc; end Xerr(s,p) = abs(Xtemp - Xtrue); % store the error at t = 1 end end

Variablestep size

Approximate

!+1n

n

t

t

sdW

ttttdWXdtXdX µ! +=

Weak convergence of truncatedIto-Taylor expansion

• statistical criteria],0[,)))((())(( TtntCXpEXpE nnn !"="#$ %% &

tqCe

tCe

TtLTXEXEe

weak

t

weak

t

L

weak

t

!+=

!"

=!#=

!

!

!

logloglog

,))(()(:

$

Error @end point

!"qestimate

Measures rate of decay of “error of the means” as

Higham’s code – emweak.m

lambda = 2; mu = 0.1; Xzero = 1; T = 1; % problem parameters M = 50000; % number of paths sampled Xem = zeros(5,1); % preallocate arrays for p = 1:5 % take various Euler timesteps Dt = 2^(p-10); L = T/Dt; % L Euler steps of size Dt Xtemp = Xzero*ones(M,1); for j = 1:L Winc = sqrt(Dt)*randn(M,1); Xtemp = Xtemp + Dt*lambda*Xtemp + mu*Xtemp.*Winc; end Xem(p) = mean(Xtemp); end Xerr = abs(Xem - exp(lambda));

Note: can use two point random variable to generate random numbers for weakconvergence (see Higham p.13)

ttttdWXdtXdX µ! +=

Milstein

( ) ))()()(()(

...))()()(()(

2

12

1

11

ttWtWXbXb

tWtWXbtXaXX

nntxt

nntttt

nn

nnnn

!""

+"+!+=

+#

#

++

{ })1,1(),1(),0(,,)1,1()1,1()1()1()0()0(0!" =+++= IfIfIfXX tt

TtdWXbdsXaXX

t

t

ss

t

t

stt!!++= "" 0,)()(

00

0

• Integral form


• Update equation

Milstein

( ) ))()()(()(

...))()()(()(

2

12

1

11

ttWtWXbXb

tWtWXbtXaXX

nntxt

nntttt

nn

nnnn

!""

+"+!+=

+#

#

++

{ })1,1(),1(),0(,,)1,1()1,1()1()1()0()0(0!" =+++= IfIfIfXX tt

TtdWXbdsXaXX

t

t

ss

t

t

stt!!++= "" 0,)()(

00

0

• Integral form


• Update equation

double stochastic integral

Hierarchical set

Higham’s code – milstrong.m

tttttdWXdtXKrXdX !+"= )(

r = 2; K = 1; beta = 0.25; Xzero = 0.5; % problem parameters T = 1; N = 2^(11); dt = T/N; % M = 500; % number of paths sampled R = [1; 16; 32; 64; 128]; % Milstein stepsizes are R*dt dW = sqrt(dt)*randn(M,N); % Brownian increments Xmil = zeros(M,5); % preallocate array for p = 1:5 Dt = R(p)*dt; L = N/R(p); % L timesteps of size Dt = R dt Xtemp = Xzero*ones(M,1); for j = 1:L Winc = sum(dW(:,R(p)*(j-1)+1:R(p)*j),2); Xtemp = Xtemp + Dt*r*Xtemp.*(K-Xtemp) + beta*Xtemp.*Winc ... + 0.5*beta^2*Xtemp.*(Winc.^2 - Dt); end Xmil(:,p) = Xtemp; % store Milstein solution at t =1 end

( ){ }nntntntttt

WXWXXKrXXXnnnnnn

!"!+!+!"+=+

22

2

1)(

1##

Linear SDEs

• Moment equations• Parameter estimation• Filtering

Mean, variance & auto-covariance

• Scalar SDE

• Exact solution (using integrating factor)

( )

!

!

""+"=

="

=

t

st

t

st

tt

dWstabXatX

dWasbXXat

dWatbXatdt

d

0

0

0

0

)(exp()exp(

)exp()exp(

)exp()exp(

tttbdWdtaXdX +!=

( )

a

bXE

ata

bmXE

t

tt

2))((

)2exp(12

))((

22

22

=

!!=!

Mean, variance & auto-covariance

• Mean

• Auto-covariance

• Variance

• Stationary variance

ttmXatXE =!= 0)exp()(

( ))2(exp()exp(2

))22(exp()))(((

2

0

2

staasa

b

tdtstabmXmXE

t

ststtt

+!!!=

""!+!=!! #++

Alternatively …

• SDE

• 1st and 2nd moments

• Using result from next slide

( ))2exp(12

2

)exp(

22

0

ata

bPbaP

dt

dP

matmamdt

dm

t

t

!!="+!=

!="!=

tttbdWdtaXdX +!=

… 2nd moment

• Transformation

• Expectation

22 baP

dt

dP

UP

+!=

=

bXdWdtbaXdU

bXULbaXUL

x

Ux

x

U

t

U

UdWLUdtLdU

XxtU

t

t

2)2(

2,2

2,2,0

),(

22

1220

2

2

10

2

++!=

=+!=

="

"=

"

"=

"

"

+=

=

Expectation iszero

Solve usingintegrating

factor

Parameter estimation

• With or without observation noise i.e. nofiltering

• Use transition probability density tocompute likelihood of data points

• Maximize likelihood

Transition probability density

• Density

• Mean

• Variance

!!"

#$$%

& ''=+

'

s

ss

V

mXVtXstXp

2

)(exp)2(),|,(

2

22/1

12 (

( ))2exp(12

)exp(

2

1

asa

bV

Xasm

s

s

!!=

!=

Log-likelihood

)(log)2log())exp((

2

1

)(log)|(log})({log

01

1

1

01

1

2

1

t

N

i

t

t

tt

t

N

i

tt

XpVV

XtaX

XpXXpXp

i

i

ii

ii

+!!"

#

$$%

&+

'(((=

+=

)

)

=

=

(

(

(

(

*

Initialdistribution

Transition probabilitydensity

!=

"=

N

i

tt iiXXpXpXp

1

0 )|()(})({1

Filtering

• Observation noise• 2 level model• Update equations for conditional moments• Kalman gain

Filtering

• Discretized state and observation equations

• Can include inputs (see Ghahramani &Hinton 1996)

),0(~,

),0(~,1

RNvvCxy

QNwwAxx

tttt

tttt

+=

+=+Hidden states

Observed data

ttttwBuAxx ++=+1

p

t

k

t

y

x

!"

!"

Filtering: joint likelihood

!"

#$%

&'''=

!"

#$%

&'''=

''

'

''

''

'

''+

)()(2

1exp)2()|(

)()(2

1exp)2()|(

1

1

1

2/12/

1

1

1

2/12/

1

tt

T

tt

p

tt

tt

T

tt

k

tt

CxyRCxyRxyp

AxxQAxxQxxp

(

(

! != =

+=N

t

N

t

tttt xypxxpxpyxp0 0

10 )|()|()(}){},({

Conditional moments

• Update equations

• where

( )11

11

111

1

1

1

1

1

1

)(

!!

!!

!!!

!

!

!

!

!

!

!=

!+=

+=

+=

=

t

tt

t

t

t

t

t

ttt

t

t

t

t

Tt

t

Tt

tt

Tt

t

t

t

t

t

t

t

CVKVV

CxyKxx

RCCVCVK

QAAVV

Axx

)}{|(),}{|( 11

!!!!yxVarVyxEx tttt ==

Nonlinear SDEs

• Local linearization (Ozaki 1992)– Ito-Taylor expansion– Analytic solution– Ito-integral– Discretized equation

• Uses– Simulation– Parameter estimation– Filtering

Local linearization for deterministicdynamics – basic intuition

))(()1)(exp()()(

))(()1)(exp()()()(

))(()exp()()exp()(

)(

))((

1

1

0

)(

txftJJtxttx

txftJJtxttxdtx

txfJtxJtx

xJx

x

xfJ

txfx

tt

tt

t

tt

t

txx

t

!"+="+

!"=!"+=+

==+

=

#

#=

=

!

!

"

=

$ %%

%%%

&

&&

&&&

&

Value of x @ t+dt asfunction of x @ t

Discretization

• Nonlinear SDE

• Discretized update equation

);,();,(

))(,()(

1hXthXtXX

gdwdttxtftdx

nnnn tntntt !+"+=

+=

+

nnnnttttwBuAxx ++=

+1

Increment term

Noise term

c.f.

Ito-Taylor expansion – f(s,x(s))and x(s)

• wherexxxxt

s

t

s

t

s

t

s

t

gffLfgffffL

udwgduftxsx

udwtxtfLdutxtfLtxtfsxsf

gdwdttxtftdx

=++=

++!

++!

+=

""

""

12

2

10

10

,

)()()(

)())(,())(,())(,())(,(

))(,()(

Re-arrange

( )

( )

( )

( ) xftsfgffsta

ftA

sgdwdsstasXtAsdX

txsxftsfgftxtf

udwgduffdufgftxtfsxsf

xxxt

x

xxxt

s

t

s

t

x

s

t

xxt

!!++=

=

++"

!+!++"

##$

%&&'

(++++" )))

)(),(

)(

)(),()()()(

))()(()())(,(

)())(,())(,(

2

2

1

2

2

1

2

2

1

)(, tXt

Evaluated @

Exact solution

• Linear SDE

• Exact solution

!!"

#$$%

&''+''+

(=+

) )+ +ht

t

ht

t

udWugtutAdutuatutAtX

htAhtX

)()))((exp(),()))((exp()(

))(exp()(

( ) ttt dWtgdttaXtAdX )()()( ++=

Re-write

• Using (see Jimenez 1999)

( )

( ) ( )( )

( ) !=

"#

$%&

'+(

+=)

+)+=+

a

n

n

xxt

tntntt

duuMuaMr

fgfhtArhtAhr

tXtfhtArhtXt

hXthXtXXnnnn

0

2

10

0

)exp(,

2

1),(),(

...))(,(),());(,(

);,();,(1

*

Approximate noise

• Substitute Gaussian noise term

• Substitute into discretized update equation

( )))(2exp(1)(2

),0(~

));(,(

2

htAtA

g

N

htXt

t

t

!="

"

#

$

$$

%$

nnnntttt

XX !+"+=+1

Summary

• Ito-Taylor expansion• Basic numerical schemes & convergence• Parameter estimation• Filtering• Local linearization

• Next meeting – Valdes Sosa (1999) paper

If you notice any typos (or mistakes!) please email [email protected]

An introduction to using SDEs - UCL Computer Science · An introduction to using SDEs L.M.Harrison....

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