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AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 8.More info at http://summerschool.ssa.org.ua
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Gerhard-Wilhelm Weber *
Nüket Erbil, Ceren Can, Vefa Gafarova, Azer Kerimov
Institute of Applied Mathematics Middle East Technical University, Ankara, Turkey
Pakize Taylan Dept. Mathematics, Dicle University, Diyarbakır, Turkey
Parameter Estimation in
Stochastic Differential Equations
by Continuous Optimization
* Faculty of Economics, Management and Law, University of Siegen, GermanyCenter for Research on Optimization and Control, University of Aveiro, Portugal
Universiti Teknologi Malaysia, Skudai, Malaysia
5th International Summer School
Achievements and Applications of Contemporary Informatics,
Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 3-15, 2010
• Stochastic Differential Equations
• Parameter Estimation
• Various Statistical Models
• C-MARS
• Accuracy vs. Stability
• Tikhonov Regularization
• Conic Quadratic Programming
• Nonlinear Regression
• Portfolio Optimization
• Outlook and Conclusion
Outline
Stock Markets
drift and diffusion term
Wiener process
( , ) ( , )t t t tdX a X t dt b X t dW
(0, ) ( [0, ])tW N t t T
Stochastic Differential Equations
drift and diffusion term
Wiener process
( , ) ( , )t t t tdX a X t dt b X t dW
(0, ) ( [0, ])tW N t t T
Stochastic Differential Equations
Ex.: price, wealth, interest rate, volatility
processes
Input vector and output variable Y ;
linear regression :
which minimizes
1 2, ,...,T
mX X X X
1 0
1
( ,..., ) ,m
m j j
j
Y E Y X X X
0 1, ,...,T
m
2
1
:N
T
i i
i
RSS y x
1ˆ ,T TX X X y
12ˆCov( ) Tβ X X
Regression
are estimated by a smoothing on a single coordinate.
Standard convention .
• Backfitting algorithm (Gauss-Seidel)
it “cycles” and iterates.
0ˆ ,i i kj
j
ik
k
r y f x
jf
: 0j ijE f x
Generalized Additive Models
1 2 0
1
, ,...,i i i i m ij j
m
j
E x fx x xY
• Given data
• penalized residual sum of squares
• New estimation methods for additive model with CQP :
2
2''
0 0
1 1 1
( , ,..., ) : ( ) ( )
bN m m
1 m i j ij j j j
i j j a
PRSS f f y f x f t dtjμ
0.j
( , ) ( = 1,2,..., ),i iy x i N
Generalized Additive Models
splines:
By discretizing, we get
0, ,
2
2
0
1 1
2''
min
subject to ( ) , 0,
( ) ( 1,2,..., ),
t β f
N m
i j ij
i= j
j j j j
t
y β f x t t
f t dt M j m
1
( ) ( ).jd
j j
j l l
l
f x h x
0, ,
2 2
0 2
2
0 2
min
subject to ( , ) , 0,
( , ) ( 1,..., ).
t β f
j j
t
W t t
V M j m
Generalized Additive Models
splines:
By discretizing, we get
0, ,
2
2
0
1 1
2''
min
subject to ( ) , 0,
( ) ( 1,2,..., ),
t β f
N m
i j ij
i= j
j j j j
t
y β f x t t
f t dt M j m
1
( ) ( ).jd
j j
j l l
l
f x h x
0, ,
2 2
0 2
2
0 2
min
subject to ( , ) , 0,
( , ) ( 1,..., ).
t β f
j j
t
W t t
V M j m
Generalized Additive Models
MARS
x
y
+( , )=[ ( )]c x x( , )=[ ( )]-c x x
x
y
+( , )=[ ( )]c x x( , )=[ ( )]-c x x r egression w ith
Tradeoff between both accuracy and complexity.
1 2
1 2
1 2 1 2
( ) : | 1, 2,...,
: ( , ,..., )
( , )
: , , 0,1
Km
m
j m
m T
m m m
V m j K
t t tt =
where
1 2
, ( ) : ( )m m m m
r s m m r sD t tt t
max
1 2
222 2
,
1 1 1, ( )( , )
: ( ) ( )MN
m m
i i m r s m
i m r sr s V m
PRSS y f D dmx t tμ
C-MARS
Tikhonov regularization:
2 2
22( )PRSS y d L
Conic quadratic programming:
,
2
2
subject to
min ,
( ) ,
tt
td y
ML
2L
2( )y d
C-MARS
drift and diffusion term
Wiener process
( , ) ( , )t t t tdX a X t dt b X t dW
(0, ) ( [0, ])tW N t t T
Stochastic Differential Equations Revisited
Ex.: price, wealth, interest rate, volatility,
processes
drift and diffusion term
Wiener process
( , ) ( , )t t t tdX a X t dt b X t dW
(0, ) ( [0, ])tW N t t T
bioinformatics, biotechnology
(fermentation, population dynamics)
Universiti Teknologi Malaysia
Ex.:
Stochastic Differential Equations Revisited
drift and diffusion term
Wiener process
( , ) ( , )t t t tdX a X t dt b X t dW
(0, ) ( [0, ])tW N t t T
Stochastic Differential Equations Revisited
Ex.: price, wealth, interest rate, volatility,
processes
Milstein Scheme :
and, based on our finitely many data:
2
1 1 1 1 1
1ˆ ˆ ˆ ˆ ˆ( , )( ) ( , )( ) ( )( , ) ( ) ( )2
j j j j j j j j j j j j j j j jX X a X t t t b X t W W b b X t W W t t
2( )( , ) ( , ) 1 2( )( , ) 1 .
j j
j j j j j j j
j j
W WX a X t b X t b b X t
h h
Stochastic Differential Equations
• step length
• (independent),
•
1 :j j j jh t t t
1
1
, if 1,2,..., 1
:
, if
j
j j
j
N N
N
X Xj N
hX
X Xj N
h
jW Var( )j jW t
21( , ) ( , ) ( )( , ) 1
2
j
j j j j j j j j
j
ZX a X t b X t b b X t Z
h
(0, ),tW N t
, (0,1)j j j jW Z t Z N
Stochastic Differential Equations
• More simple form:
where
• Our problem:
is a vector which comprises a subset of all the parameters.
2
: ( , ) , : ( , ),
: , : 1 2 1 .
j j j j j j
j j j j j
G a X t H b X t
c Z h d Z
y
( ) ,j j j j j j jX G H c H H d
2
21
min ( ( ) )N
j j j j j j jy
j
X G H c H H d
Stochastic Differential Equations
where
• k th order base spline : a polynomial of degree k − 1, with knots, say
2 2
0 , 0 ,
1 1 1
2 2
0 , 0 ,
1 1 1
2 2
0 , 0 ,
1 1 1
( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
gp
hr
fs
d
l l
j j j p j p p p j p
p p l
dm m
j j j j j r j r r r j r
r r m
dn n
j j j j j s j s s s j s
s s n
G a X t f U B U
H c b X t c g U C U
F d b b X t d h U D U
,1 ,2, : , ;j j j j jU U U X t
,kB ,x
1
,1
, , 1 1, 1
1 1
1,( )
0, otherwise
( ) ( ) ( )k
k k k
k k
x x xB x
x x x xB x B x B x
x x x x
Stochastic Differential Equations
• penalized sum of squares PRRS
• (smoothing parameters),
• large values of yield smoother curves,
smaller ones allow more fluctuation
2
1
22 2 2
0 , 0 , 0 ,
1 1 1 1 1 1 1
( ) ( ) ( )
h fgp sr
N
j j j j j j
j
d ddNl l m m n n
j p p j p r r j r s s j s
j p l r m s n
X G H c F d
X B U C U D U
22 2
1 1
2 22 2
1 1
( , , ) : ( )
( ) ( )
N
j j j j j j p p p p
j p
r r r r s s s s
r s
PRSS f g h X G H c F d f U dU
g U dU h U dU
, , 0p r s
, ,p r s
( , , )
b
a
p r s
Stochastic Differential Equations
• Then,
• Furthermore,
1 2
0 1 2
1 2
0 1 2
1 2
0 1 2
,
, , , , ,..., ( 1,2),
, , , , ,..., ( 1
, ,
, 2),
, , , , ,..., ( 1,2).
gp
hr
fs
TT dT T
p p p p
TTdT T
r r r r
TTdT T
s
T T
s
T
s s
T
p
r
s
22
21
.N
j j
j
X A X A 1 2
1 2
, ,...,
, ,...,
TT T T
N
T
N
A A A A
X X X X
12 2
1,
1
21
1 1
( ) ( ) ( )
( ) .
gp
b N
p p p p jp j p jp
ja
dNl l
p p jP j
j l
f U dU f U U U
B U u
Stochastic Differential Equations
• If
where
2 2 22 2 2 2
2 2 22 1 1 1
( , , ) B C D
p p p r r r s s s
p r s
PRSS f g h X A A A A
2: :p r s
222
22
( , , ) ,PRSS f g h X A L
, , .T
T T T
1
2
1
2
1
2
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0: ,
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
B
B
C
C
D
D
A
A
AL
A
A
A
Stochastic Differential Equations
22
22
min X A Lμ Tikhonov regularization
,
2
2
subject to
min ,
,
tt
A X t
L M
Stochastic Differential Equations
Conic quadratic programming
Interior Point Methods
Stochastic Differential Equations
,
6( 1)6( 1)
1 6( 1) 1
min
0subject to : ,
1 0 0
00: ,
0 0
,
t
N
T
m
NN
T
m
N N
t
tA X
t
M
L L
L
1 1 2 2 2
1 2 1 1 2: ( , ,..., ) | ...N T N
N N+ NL x x x x x x x xR
6( 1) 1
1 6( 1) 2
6( 1)
1 2
1
1 2
max ( ,0) 0 ,
10 1 0 0subject to ,
00 0
, N
T T
N
T T
N N
T Tmm m
N
X M
A
L L
Ldual problem
primal problem
is a primal dual optimal solution if and only if 1 2( , , , , , )t
6( 1)6( 1)
6( 1)
1 2
1 2
1 6( 1) 1
1 2
1 6( 1) 1
0: ,
1 0 0
00:
0 0
10 1 0 0
00 0
0, 0
,
, .
N
T
m
NN
T
m
T T
N N
T Tmm m
T T
N N
N N
tA X
t
M
A
L L
L L
L
L
Stochastic Differential Equations
Stochastic Differential Equations
Ex.:
nonlinear regression
, , , ,t t t t t tdX t X Z dt t X Z dW .
( ) + ,T T
t t t t t t t t tdV V dt cr dr t V dWμ σ
,t t t t td R r dt rr dWτα
min ( ) ( ) ( )
Tf F F
1( ) : ( ),..., ( )T
NF f f
2
,
1
2
1
min
:
N
j j
j
N
j
j
f d g x
f
Nonlinear Regression
• Gauss-Newton method :
• Levenberg-Marquardt method :
( ) ( ) ( ) ( )T qF F F F
( ) ( ) I ( ) ( )T
p qF F F F
0
1 :k k kq
Nonlinear Regression
,
2
2
min ,
subject to ( ) ( ) I ( ) ( ) , 0,
|| ||
t
T
p
qt
F F F F
qL
q t t
M
Nonlinear Regression
interior point methods
alternative solution
conic quadratic programming
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.
Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004.
Buja, A., Hastie, T., and Tibshirani, R., Linear smoothers and additive models, The Ann. Stat. 17, 2 (1989)
453-510.
Fox, J., Nonparametric regression, Appendix to an R and S-Plus Companion to Applied Regression,
Sage Publications, 2002.
Friedman, J.H., Multivariate adaptive regression splines, Annals of Statistics 19, 1 (1991) 1-141.
Friedman, J.H., and Stuetzle, W., Projection pursuit regression, J. Amer. Statist Assoc. 76 (1981) 817-823.
Hastie, T., and Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310.
Hastie, T., and Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc.
82, 398 (1987) 371-386.
Hastie, T., Tibshirani, R., and Friedman, J.H., The Element of Statistical Learning, Springer, 2001.
Hastie, T.J., and Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990.
Kloeden, P.E, Platen, E., and Schurz, H., Numerical Solution of SDE Through Computer Experiments,
Springer Verlag, New York, 1994.
Korn, R., and Korn, E., Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics,
Oxford University Press, 2001.
Nash, G., and Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996.
Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).
References
Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).
Nesterov, Y.E , and Nemirovskii, A.S., Interior Point Methods in Convex Programming, SIAM, 1993.
Önalan, Ö., Martingale measures for NIG Lévy processes with applications to mathematical finance,
presentation in: Advanced Mathematical Methods for Finance, Side, Antalya, Turkey, April 26-29, 2006.
Taylan, P., Weber G.-W., and Kropat, E., Approximation of stochastic differential equations by additive
models using splines and conic programming, International Journal of Computing Anticipatory Systems 21
(2008) 341-352.
Taylan, P., Weber, G.-W., and A. Beck, New approaches to regression by generalized additive models
and continuous optimization for modern applications in finance, science and techology, in the special issue
in honour of Prof. Dr. Alexander Rubinov, of Optimization 56, 5-6 (2007) 1-24.
Taylan, P., Weber, G.-W., and Yerlikaya, F., A new approach to multivariate adaptive regression spline
by using Tikhonov regularization and continuous optimization, to appear in TOP, Selected Papers at the
Occasion of 20th EURO Mini Conference (Neringa, Lithuania, May 20-23, 2008) 317- 322.
Seydel, R., Tools for Computational Finance, Springer, Universitext, 2004.
Stone, C.J., Additive regression and other nonparametric models, Annals of Statistics 13, 2 (1985) 689-705.
Weber, G.-W., Taylan, P., Akteke-Öztürk, B., and Uğur, Ö., Mathematical and data mining contributions
dynamics and optimization of gene-environment networks, in the special issue Organization in Matter
from Quarks to Proteins of Electronic Journal of Theoretical Physics.
Weber, G.-W., Taylan, P., Yıldırak, K., and Görgülü, Z.K., Financial regression and organization, to appear
in the Special Issue on Optimization in Finance, of DCDIS-B (Dynamics of Continuous, Discrete and
Impulsive Systems (Series B)).
References
I1
a b
(3a)
I1 I2 I3 I4
a bI5 I6 I7 I8
(3b)
I2 I4I3 I5 I6I1
a b
(3c)
a b
.
.
.. ...... ......
. .......
..
: ( ) ( )j j j j jInd = d D v V
Generalized Additive Models
Appendix
cluster
cluster
robust optimization
C-MARS
Appendix
,
2
2
min ,
subject to ( ) ( ) I ( ) ( ) , 0,
|| ||
t
T
p
qt
F F F F
qL
q t t
M
Nonlinear Regression
alternative solution
2
1min ( ) := ( ) + ( ) ( ) + ( ) ( )
2
subject to
T T T
q
Q q f q F F q F F q
q
trust region
Appendix
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