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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
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
Stock Markets
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
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
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
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.
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 :
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
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
MARS




y                                           y




    c-(x, )=[ (x )]   c+(x, )=[ (x )]           c-(x, )=[ (x )]   c+(x,egressionx ith)]
                                                                      r )=[ ( w

                                        x                                                 x
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
                                                                          .
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
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 ( )
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 ( )
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
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:




                                Optimization Problem


                                       Representation Problem




             or   stochastic control
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:

                          Parameter Estimation


                                Optimization Problem


                                       Representation Problem




             or   stochastic control
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:




                                Optimization Problem


                                       Representation Problem


                                         Parameter Estimation

             or   stochastic control
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:




                                Optimization Problem


                                       Representation Problem


                                         Parameter Estimation

             or   stochastic control
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).
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)).
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
Appendix

C-MARS


           cluster




           cluster




                     robust optimization
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

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Parameter Estimation in Stochastic Differential Equations by Continuous Optimization

  • 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
  • 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