The document discusses advancements in parameter estimation within stochastic differential equations and various statistical models. It covers topics such as nonlinear regression, conic quadratic programming, and portfolio optimization. The work highlights the balance between accuracy and stability in estimation methods, drawing from various mathematical principles and techniques.

1. 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
Parameter Estimation in
Stochastic Differential Equations
by Continuous Optimization
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
* Faculty of Economics, Management and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal
Universiti Teknologi Malaysia, Skudai, Malaysia

2. Outline
• 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

3. Stock Markets

4. Stochastic Differential Equations
dX t a( X t , t )dt b( X t , t )dWt
drift and diffusion term
Wt N (0, t ) (t [0, T ])
Wiener process

5. Stochastic Differential Equations
dX t a( X t , t )dt b( X t , t )dWt
drift and diffusion term
Ex.: price, wealth, interest rate, volatility
processes
Wt N (0, t ) (t [0, T ])
Wiener process

6. Regression
T
Input vector X X 1 , X 2 ,..., X m and output variable Y ;
linear regression :
m
Y E (Y X 1 ,..., X m ) 0 Xj j ,
j 1
T
0, 1 ,..., m which minimizes
N 2
T
RSS : yi x
i
i 1
ˆ X X T 1
XT y,
ˆ
Cov( β ) XTX
1
2

7. Generalized Additive Models
m
E Yi xi1 , xi 2 ,..., xi m 0 f j xi j
j 1
f j are estimated by a smoothing on a single coordinate.
Standard convention : E f j xij 0.
• Backfitting algorithm (Gauss-Seidel)
ri j yi ˆ
f k xik ,
0
k j
it “cycles” and iterates.

8. Generalized Additive Models
• Given data ( yi , xi ) (i = 1,2,...,N ),
• penalized residual sum of squares
2 b
N m m 2
PRSS ( 0 , f1 ,..., f m ) : yi 0 f j ( xij ) μj ''
f (t j ) dt j
j
i 1 j 1 j 1 a
j 0.
• New estimation methods for additive model with CQP :

9. Generalized Additive Models
min t
t , β0 , f
2
N m
subject to yi β0 f j ( xij ) t2, t 0,
i=1 j 1
2
''
f (t j ) dt j
j Mj (j 1, 2,..., m),
dj
j
splines: f j ( x) l hl j ( x).
l 1
By discretizing, we get
min t
t , β0 , f
2
subject to W( 0, ) 2
t2, t 0,
2
Vj ( 0, ) Mj (j 1,..., m).
2

10. Generalized Additive Models
min t
t , β0 , f
2
N m
subject to yi β0 f j ( xij ) t2, t 0,
i=1 j 1
2
''
f (t j ) dt j
j Mj (j 1, 2,..., m),
dj
j
splines: f j ( x) l hl j ( x).
l 1
By discretizing, we get
min t
t , β0 , f
2
subject to W( 0, ) 2
t2, t 0,
2
Vj ( 0, ) Mj (j 1,..., m).
2

11. MARS
y y
c-(x, )=[ (x )] c+(x, )=[ (x )] c-(x, )=[ (x )] c+(x,egressionx ith)]
r )=[ ( w
x x

12. C-MARS
N M max 2
2 2
PRSS : yi f ( xi ) μm 2
m Dr ,s m (t ) d t m
m
i 1 m 1 1 r s
( 1, 2 ) r ,s V ( m)
Tradeoff between both accuracy and complexity.
V ( m) : m
j | j 1, 2,..., K m Dr ,s m (t m ) : m
1
trm 2
tsm (t m )
t m := (tm1 , tm2 ,..., tm K )T
m
( 1, 2 )
: 1 2 , where 1 , 2 0,1

13. C-MARS
Tikhonov regularization:
2 2
PRSS y (d ) L 2
2
L 2
Conic quadratic programming:
y (d )
2
min t,
t,
subject to (d ) y t,
2
L 2
M

14. Stochastic Differential Equations Revisited
dX t a( X t , t )dt b( X t , t )dWt
drift and diffusion term
Ex.: price, wealth, interest rate, volatility,
processes
Wt N (0, t ) (t [0, T ])
Wiener process

15. Stochastic Differential Equations Revisited
dX t a( X t , t )dt b( X t , t )dWt
drift and diffusion term
Ex.: bioinformatics, biotechnology
(fermentation, population dynamics)
Universiti Teknologi Malaysia
Wt N (0, t ) (t [0, T ])
Wiener process

16. Stochastic Differential Equations Revisited
dX t a( X t , t )dt b( X t , t )dWt
drift and diffusion term
Ex.: price, wealth, interest rate, volatility,
processes
Wt N (0, t ) (t [0, T ])
Wiener process

17. Stochastic Differential Equations
Milstein Scheme :
ˆ ˆ ˆ ˆ 1 ˆ
Xj 1 Xj a ( X j , t j )(t j 1 t j ) b( X j , t j )(W j 1 Wj ) (b b)( X j , t j ) (W j 1 W j ) 2 (t j 1 tj)
2
and, based on our finitely many data:
Wj ( W j )2
Xj a ( X j , t j ) b( X j , t j ) 1 2(b b)( X j , t j ) 1 .
hj hj

18. Stochastic Differential Equations
• step length hj tj 1 tj : tj
Xj 1 Xj
, if j 1, 2,..., N 1
hj
Xj :
XN XN 1
, if j N
hN
• Wt N (0, t ), W j (independent), Var( W j ) tj
• Wj Zj tj , Zj N (0,1)
Zj 1
Xj a ( X j , t j ) b( X j , t j ) (b b)( X j , t j ) Z j2 1
hj 2

19. Stochastic Differential Equations
• More simple form:
Xj Gj H j cj ( H j H j )d j ,
where
G j : a( X j , t j ), H j : b( X j , t j ),
cj : Z j hj , d j : 1 2 Z j2 1 .
• Our problem:
N 2
min X j (G j H jc j ( H j H j )d j )
y 2
j 1
y is a vector which comprises a subset of all the parameters.

20. Stochastic Differential Equations
g
2 2 dp
l l
Gj a( X j , t j ) 0 f p (U j , p ) 0 p B p (U j , p )
p 1 p 1 l 1
2 2 d rh
m
H jc j b ( X j , t j )c j 0 g r (U j ,r ) 0 r Crm (U j ,r )
r 1 r 1 m 1
2 2 d sf
n
Fj d j b b( X j , t j )d j 0 hs (U j ,s ) 0 s Dsn (U j ,s )
s 1 s 1 n 1
where
Uj U j ,1 ,U j ,2 : X j , tj ;
• k th order base spline B ,k : a polynomial of degree k − 1, with knots, say x ,
1, x x x 1
B ,1 ( x)
0, otherwise
x x x k x
B ,k ( x) B ,k 1 ( x) B 1, k 1 ( x)
x k 1 x x k x 1

21. Stochastic Differential Equations
• penalized sum of squares PRRS
N 2 2
2
PRSS ( f , g , h) : Xj Gj H jc j Fj d j p f p (U p ) dU p
j 1 p 1
2 2
2 2
r g r (U r ) dU r s hs (U s ) dU s
r 1 s 1
b
• p , r , s 0 (smoothing parameters), ( p, r , s )
a
• large values of p , r , s yield smoother curves,
smaller ones allow more fluctuation
N 2
Xj Gj H jc j Fj d j
j 1
2
N 2 dh
p 2 d rg 2 d sf
l
Xj 0 p B lp (U j , p ) 0 r
m
Crm (U j ,r ) 0
n
s Dsn (U j ,s )
j 1 p 1 l 1 r 1 m 1 s 1 n 1

22. Stochastic Differential Equations
T
T T T
, , ,
T g T
T T 1 2 dp
0 , 1 , 2 , p p , p ,..., p ( p 1, 2),
T T
T T 1 2 d rh
0 , 1 , 2 , r r , r ,..., r ( r 1, 2),
T T
T T 1 2 d sf
0, 1 , 2 , s s , s ,..., s ( s 1, 2).
N 2 2 T
• A A1T , A2 ,..., AN
T T
Then, Xj Aj X A .
j 1 2 T
X X 1 , X 2 ,..., X N
• Furthermore,
b 2 N 1 2
f p (U p ) dU p f p (U jp ) (U j 1, p U jp )
a j 1
g 2
N 1 dp
l
p B lp (U jP )u j .
j 1 l 1

23. Stochastic Differential Equations
2 2 2 2 2 2 2
B C
PRSS ( f , g , h) X A p A
p p 2 r A
r r 2 s AsD s 2
2 p 1 r 1 s 1
2
• If p r s : :
2
2 2
PRSS ( f , g , h) X A L 2
,
2
where
0 A1B 0 0 0 0 0 0 0
0 0 A2B 0 0 0 0 0 0
0 0 0 0 A1C 0 0 0 0 T T T
T
L: , , , .
0 0 0 0 0 A2C 0 0 0
0 0 0 0 0 0 0 A1D 0
0 0 0 0 0 0 0 0 A2D

24. Stochastic Differential Equations
2
2
min X A μ L 2
Tikhonov regularization
2
min t, Conic quadratic programming
t,
subject to A X t,
2
L 2
M
Interior Point Methods

25. Stochastic Differential Equations
min t
t,
0N A t X
subject to : ,
1 0T
m 0 primal problem
06( N 1) L t 06( N 1)
: ,
0 0T
m M
LN 1 , L6( N 1) 1
LN 1
: x ( x1 , x2 ,..., xN )T R N 1 | xN+1 x12 2 2
x2 ... xN
max ( X T , 0) 1 0T N 1) ,
6( M 2
0T
N 1 0T N
6( 1) 0 1
subject to 1 2 , dual problem
AT 0m LT 0m 0m
1 LN 1 , 2 L6 ( N 1) 1

26. Stochastic Differential Equations
(t , , , , 1 , 2 ) is a primal dual optimal solution if and only if
0N A t X
: ,
1 0T
m 0
06( N 1) L t 06( N 1)
:
0 0T
m M
0T
N 1 0T N
6( 1) 0 1
1 2
AT 0m LT 0m 0m
T T
1 0, 2 0
1 LN 1 , 2 L6( N 1) 1
LN 1 , L6( N 1) 1
.

27. Stochastic Differential Equations
Ex.:
dVt t
T
( μ rt ) + rt Vt dt ct dt t
T
σVt dWt ,
drt α R rt dt t rt τ dWt ,
dX t t , X t , Z t dt t , X t , Z t dWt .
nonlinear regression

28. Nonlinear Regression
N 2
min f dj g xj ,
j 1
N
: f j2
j 1
T
F( ) : f1 ( ),..., f N ( )
min f ( ) F T ( )F ( )

29. Nonlinear Regression
k 1 : k qk
• Gauss-Newton method :
T
F( ) F ( )q F ( )F ( )
• Levenberg-Marquardt method :
0
T
F( ) F( ) Ip q F ( )F ( )

30. Nonlinear Regression
alternative solution
min t,
t,q
T
subject to F( ) F( ) Ip q F ( )F ( ) t, t 0,
2
|| Lq || 2 M
conic quadratic programming
interior point methods

31. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
or stochastic control

32. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Parameter Estimation
Optimization Problem
Representation Problem
or stochastic control

33. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
Parameter Estimation
or stochastic control

34. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
Parameter Estimation
or stochastic control

35. References
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).

36. 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)).

37. Appendix
Generalized Additive Models
a b
I1
(3a)
a b
I1 I2 I3 I4 I5 I6 I7 I8
(3b)
a b
I1 I2 I3 I4 I5 I6
(3c)
.
. . . . . ... . .. . .
. .. . . . . . . .. .
Ind j : = d j ( D j ) v j (V j )
a b

38. Appendix
C-MARS
cluster
cluster
robust optimization

39. Appendix
Nonlinear Regression
alternative solution
min t,
t,q
T
subject to F( ) F( ) Ip q F ( )F ( ) t, t 0,
2
|| Lq || 2 M
1
min Q(q) := f ( ) + qT F ( ) F ( ) + qT F ( ) T
F ( )q
q 2
subject to q2
trust region