Gerhard-Wilhelm Weber Institute of Applied Mathematics Middle East Technical University, Ankara, Turkey New Mathematical Tools for the Financial Sector 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 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
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 5. More info at http://summerschool.ssa.org.ua
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Gerhard-Wilhelm Weber
Institute of Applied Mathematics Middle East Technical University, Ankara, Turkey
New Mathematical Tools
for the Financial Sector
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,
Important new class of (Generalized) Partial Linear Models:
, ,
( , ) GPLM ( ) ( )LM MARS= +
TE Y X T G X T
X T X T
e.g.,
2 2* * * *
2 2X L
x
y
+( , )=[ ( )]c x x( , )=[ ( )]-c x x
CMARS
A. Özmen, G.-W. Weber, I.Batmaz
Application
Evaluation of the models based on performance values:
• CMARS performs better than Tikhonov regularization with respect to all the measures for both data sets.
• On the other hand, GLM with CMARS (GPLM) performs better than both Tikhonov regularization and CMARS with respect to all the measures for both data sets.
Identifying Stochastic Differential Equations
F. Yerlikaya Özkurt, G.-W. Weber, P. Taylan
Robust CMARS:
RCMARSsemi-length of confidence interval
.. ..outlier outlier
confidence interval
. ......( )jT
... . .. ... .. .... .. . . . ..
Identifying Stochastic Differential Equations
A. Özmen, G.-W. Weber, I.Batmaz
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Portfolio Optimization
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization
non-default
casesdefault
cases
score value
TP
F, sensitiv
ity
FPF, 1-specificity
ROC curve
cut-off value
c
c = cut-off value
Prediction of Credit Default
K. Yildirak, E. Kürüm, E., G.-W. Weber
Simultaneously obtain the thresholds and parameters a and b
that maximize AUC,
while balancing the size of the classes (regularization),
and guaranteeing a good accuracy
discretization of integral
nonlinear regression problem
Optimization problem:
Prediction of Credit Default
( ) ( )( 1) M ( )s k s kE k E k C
( ) : ( ( 1))
1 if ( )( ( )) :
0 else
B
i i
i
s k F Q E k
E kQ E k
Eco-Finance Networks
θ1,1 θ1,2
θ2,1
θ2,2
( ) ( )( 1) M ( )s k s kE k E k C
( 1) IM ( )kIE k IE k
( ) ( )IE t IM IE t
( ) ( ) ( )( ) ( ) ( )s t s t s tE t M E t C E t D
Eco-Finance Networks
modules
( 1) IM ( )kIE k IE k
( ) ( )IE t IM IE t
Eco-Finance Networks
)))(((:)( tEQFts
1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t
1,)( ii tE
2,1, )( iii tE
)(, tEidi i
where
0 for
1 for( ( )) :
...
for
i
i
Q E t
dparameter estimation:
(i) estimation of thresholds
(ii) calculation of matrices and
vectors describing the system
between thresholds
( ) ( ) ( )( ) ( ) ( )s t s t s tE t M E t C E t D
Eco-Finance Networks
min
( ), ( ), ( )ij i im c d
1
1
1
,min
( 1, ..., )
( 1, ..., )
( , ) ( )
( , ) ( )
( , ) ( )
&
n
ij ij j
i
n
i i
i
n
i i
i
ii i
j n
m
p m y y
q c y y
d y y
m
overall box constraints
( ( , ))y Y C D
( 1,..., )i n
subject to
21
0
l
M E C E D E
Eco-Finance Networks
: structurally stable
global local global
)(
nIR
asymptotic
effect
)(
homeom.
:),(),( 0C
),(
Generalized Semi-Infinite Programming
Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 46
Motivations
• Present a new method for optimal control
of Stochastic Hybrid Systems.
• More flexible than Hamilton-Jacobi,
because handles more problem formulations.
• In implementation, up to dimension 4-5
in the continuous state.
4-55
Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 47
Problem Formulation:
• standard Brownian motion
• continuous state
Solves an SDE whose jumps are governed by the discrete state.
• discrete state
Continuous time Markov chain.
• control
5
Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 48
Applications:
• Systems biology: Parameter identification.
• Finance: Optimal portfolio selection.
• Engineering: Maintain dynamical system in safe domain for maximum time.
5
Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 49
Method: 1st step
1. Derive a PDE satisfied by the objective function in terms of the generator:
• Example 1:
If
then
• Example 2:
If
then
5
Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 50
Method: 2nd and 3rd step
2. Rewrite original problem as deterministic PDE optimization program
3. Solve PDE optimization program using adjoint method
Simple and robust …
5
Aster, A., Borchers, B., Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.
Boyd, S., Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004.
Buja, A., Hastie, T., 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,