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

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
New Mathematical Tools
for the Financial Sector
Gerhard-Wilhelm Weber
Institute of Applied Mathematics
Middle East Technical University, Ankara, 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 Identification
• Uncertainty , Ellipsoidal Calculus
• Bubbles
• Programming Aspects
• Portfolio Optimization
• Hybrid Control
• 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
Ex.: price, wealth, interest rate, volatility
processes
Wt N (0, t ) (t [0, T ])
Wiener process

5. 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

6. Example: Technology Emissions-Means Model
E E E
( k 1) (k ) (k )
M (k )
M M M
E E E
( k 1) (k ) (k )
(k ) 0
M
M M M u(k )
IE( k +1) IM( k ) IE( k )

7. Gene-Environmental and Financial Dynamics
d Ei(t ) ai (E(t ) , t ) dt bi (E(t ) , t ) dWi (t )
. (k )
Wi ( k ) 1 ( Wi ( k ) )2
Ei ai (E ( k ) , t ( k ) ) bi (E ( k ) , t ( k ) ) ( k ) (b'bi )(E ( k ) , t ( k ) )
i (k )
1
h 2 h
IE( k +1) IM( k ) IE( k )

8. Gene-Environment Networks Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ2
θ1

9. Gene-Environment Networks Errors and Uncertainty
Identify groups (clusters) of jointly acting
genetic and environmental variables
disjoint
overlapping
stable clustering

10. Gene-Environment Networks Errors and Uncertainty
2) Interaction of Genetic Clusters

11. Gene-Environment Networks Errors and Uncertainty
3) Interaction of Environmental Clusters

12. Gene-Environment Networks Errors and Uncertainty
3) Interaction of Genetic & Environmental Clusters
Determine the degree of connectivity

13. Gene-Environment Networks Errors and Uncertainty
Clusters and Ellipsoids:
Genetic clusters: C1,C2,…,CR
Environmental clusters: D1,D2,…,DS
Genetic ellipsoids: X1,X2,…,XR Xi = E (μi,Σi)
Environmental ellipsoids: E1,E2,…,ES, Ej = E (ρj,Πj)

14. Gene-Environment Networks Ellipsoidal Calculus
r 1

15. Gene-Environment Networks Ellipsoidal Calculus
r=1

16. Gene-Environment Networks Ellipsoidal Calculus
The Regression Problem:
measurement
Maximize (overlap of ellipsoids)
T R R
ˆ
X r( )
X r( ) ˆ
Er( )
Er( )
1 r 1 r 1
prediction

17. Gene-Environment Networks Ellipsoidal Calculus
Measures for the size of intersection:
• Volume → ellipsoid matrix determinant
• Sum of squares of semiaxes → trace of configuration matrix
• Length of largest semiaxes → eigenvalues of configuration matrix
E r , r
r
semidefinite programming
interior point methods

18. What is a Bubble?
A situation in which prices for securities, especially stocks, rise
far above their actual value.
When does it burst?
When investors realize how far prices have risen from actual values,
the bubble bursts.

19. Shape of a Bubble
Dimensions of the ellipsoid Intersection of the two bubbles

20. Our Goals
Modelling Bubbles
Lin, L., and Sornette, D., Diagnostics of rational expectation financial bubbles
with stochastic mean-reverting termination times, Cornell University Library,
2009.
Abreu, D., and Brunnermeier, M.K., Bubbles and crashes, Econometrica 71(1),
174-203, 2003.
Brunnermeier, M.K., Asset Pricing Under Asymmetric Information, Oxford
University Press, 2001.
Binswanger, M., Stock Market Speculative Bubbles and Economic Growth,
Edward Elgar Publishing Limited, 1999.
Garber, P.M., and Flood, R.P., Speculative Bubbles Speculative Attacks, and
Policy Switching, The MIT Press, 1997.
Developing a method to contract a bubble to one point or shrink them,
e.g., as soon as possible.

21. Homotopy
transition between bubbles
?
concept of homotopy
In topology, two continuous functions from one topological space to another are called
homotopic if the first can be "continuously deformed" into the second one.
Such a deformation is called a homotopy between the two functions.
Formally, a homotopy between two continuous functions f and g from a topological space X
to a topological space Y is defined to be a continuous function
H : X × [0,1] → Y such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).
If we think of the second parameter of H as time, then H describes a
continuous deformation of f into g:
at time 0 we have the function f and
at time 1 we have the function g.

22. In topology, two continuous functions from one topological space to another are called
homotopic if the first can be "continuously deformed" into the second one.
Such a deformation is called a homotopy between the two functions.
Formally, a homotopy between two continuous functions f and g from a topological space X
to a topological space Y is defined to be a continuous function
H : X × [0,1] → Y such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).
If we think of the second parameter of H as time, then H describes a
continuous deformation of f into g:
at time 0 we have the function f and
at time 1 we have the function g.

23. Homotopy
transition between bubbles
?
concept of homotopy
In topology, two continuous functions from one topological space to another are called
homotopic if the first can be "continuously deformed" into the second one.
Such a deformation is called a homotopy between the two functions.
Formally, a homotopy between two continuous functions f and g from a topological space X
to a topological space Y is defined to be a continuous function
H : X × [0,1] → Y such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).
If we think of the second parameter of H as time, then H describes a
continuous deformation of f into g:
at time 0 we have the function f and
at time 1 we have the function g.

24. First Bubble Model
m m
dp p (1 ( p, t )) dt p dW
( p, t ) 0 dp p m dt
p K (tc t )
( m 1)
1 (m 1) K ( ) tc p0 (m 1)
( p, t ) ~ (t ) 1 m
tc 2
[ p(t ) ] m 1
2
d~
tc ~ dt (
tc ) dW

25. Second Bubble Model
y(t ) ln p(t ) :
dy x (1 ( x , t )) dt ( ) x dW
dx x m (1 ( x , t )) dt xmdW
( x , t ) 0: dy x dt ( ) x dW showing that x (t ) dt E [dy]
m
( x, t ) (x , t) 0: d2y dy
y (t ) A B ( Tc t ) 1
dt 2 dt
1/
dp 1
1 (m 1) Tc ( ) B ( ) A p (Tc )
dt t t0 1
2
~ (t ) m 1
( x, t ) ~ (t ) 1 m
tc 2
[ x (t ) ] m 1
( x, t ) tc [ x (t ) ]
2 2

26. Jump Process
t
Φ ( s) d X ( s) The integrator X may have jumps.
0
X (t ) X (0) I (t ) R(t ) J (t ) X ( 0 ) is a nonrandom initial condition.
t
I (t ) Γ ( s) dW ( s) Ito integral of an adapted process (s) .
0
t
R(t ) ( s) ds Riemann integral for some adapted process (s) .
0
J (t ) lim J ( s ) Adapted right-continuous pure Jump process, J (0) 0 .
s t

27. Clarke’s Subdifferential as “Bubble”
path nowhere differentiable,
we discretize
-2 -1 0 1
This constitutes a “homotopy of bubbles”.
fC (t ) co ξ | lim f (tk ), tk t (k ), tk D f (k IN)
k

28. Identifying 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

29. Identifying 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 1 )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

30. Identifying Stochastic Differential Equations
A. Özmen, G.-W. Weber, I.Batmaz
Important new class of (Generalized) Partial Linear Models:
E Y X ,T G XT T , e.g.,
GPLM (X ,T ) = LM (X ) + MARS (T )
y
c-(x, )=[ (x )] c+(x, )=[ (x )]
x
X * * 2
L* * 2 CMARS
2 2

31. Identifying Stochastic Differential Equations
Application F. Yerlikaya Özkurt, G.-W. Weber, P. Taylan
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.

32. Identifying Stochastic Differential Equations
A. Özmen, G.-W. Weber, I.Batmaz
Robust CMARS:
confidence interval
(T j )
... ...................... .
outlier
... outlier
semi-length of confidence interval
RCMARS

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

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

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

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

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

38. Prediction of Credit Default
non-default default
cases cases
cut-off value
ROC curve
c score value
TPF, sensitivity
c = cut-off value
K. Yildirak, E. Kürüm, E., G.-W. Weber
FPF, 1-specificity

39. Prediction of Credit Default
Optimization problem:
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

40. Eco-Finance Networks
E (k 1) M s ( k ) E (k ) Cs ( k )
s(k ) : FBQ( E (k 1))
1 if Ei (k ) i
Qi ( E (k )) :
0 else
θ2,2
θ2,1
θ1,1 θ1,2

41. Eco-Finance Networks
E (k 1) M s ( k ) E (k ) Cs ( k )
IE (k 1) IMk IE (k )
IE (t ) IM IE (t )
E (t ) M s (t ) E (t ) Cs (t ) E (t ) Ds (t )

42. Eco-Finance Networks
IE (k 1) IMk IE (k )
IE (t ) IM IE (t )
modules

43. Eco-Finance Networks
E (t ) M s (t ) E (t ) Cs (t ) E (t ) Ds (t )
where
s(t ) : F (Q( E(t )))
Q( E(t )) (Q1 ( E(t )),..., Qn ( E(t )))
0 for Ei (t ) i ,1
1 for i ,1 Ei (t ) i,2
Qi ( E (t )) :
...
parameter estimation:
di for i ,di Ei (t )
(i) estimation of thresholds
(ii) calculation of matrices and
vectors describing the system
between thresholds

44. Eco-Finance Networks
2
l 1
min M E C E D E
0
(mij ), (ci ), (di )
subject to
n
pij ( mij , y ) j ( y) (j 1, ..., n )
i 1
n
qi (ci , y ) ( y) ( 1, ..., m ) ( y Y (C , D ))
i 1
n
i (di , y ) ( y)
i 1
mii i ,min (i 1,..., n)
& overall box constraints

45. Generalized Semi-Infinite Programming
( ), ( , ) C 0 :
( )
(, )
homeom.
asymptotic
()
: structurally stable effect
IR n
global local global

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.
Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 4-5
546

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
Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5
47

48. Applications:
• Engineering: Maintain dynamical system in safe domain for maximum time.
• Systems biology: Parameter identification.
• Finance: Optimal portfolio selection.
Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5
48

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
Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5
49

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 …
Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5
50

51. References
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,
Sage Publications, 2002.
Friedman, J.H., Multivariate adaptive regression splines, Annals of Statistics 19, 1 (1991) 1-141.
Friedman, J.H., Stuetzle, W., Projection pursuit regression, J. Amer. Statist Assoc. 76 (1981) 817-823.
Hastie, T., Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310.
Hastie, T., Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc.
82, 398 (1987) 371-386.
Hastie, T., Tibshirani, R., Friedman, J.H., The Element of Statistical Learning, Springer, 2001.
Hastie, T.J., Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990.
Kloeden, P.E, Platen, E., Schurz, H., Numerical Solution of SDE Through Computer Experiments,
Springer Verlag, New York, 1994.
Korn, R., Korn, E., Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics,
Oxford University Press, 2001.
Nash, G., Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996.
Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).

52. References
Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).
Nesterov, Y.E , 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., 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., Beck, A., 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., 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., 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)).

53. Appendix TEM Model

54. Appendix The mixed-integer problem
: nxn constant matrix with entries representing the effect
which the expression level of gene has on the change of expression of gene
Genetic regulation network
mixed-integer nonlinear optimization problem (MINLP):
subject to
: constant vector representing the lower bounds for
the decrease of the transcript concentration.
in order to bound the indegree of each node, introduce
binary variables :
is a given parameter.

55. Appendix Numerical Example
Ö. Defterli, A. Fügenschuh, G.-W. Weber
Data
Gebert et al. (2004a)
Apply 3rd-order Heun method
Take
using the modeling language Zimpl 3.0, we solve
by SCIP 1.2 as a branch-and-cutframework,
together with SOPLEX 1.4.1 as our LP-solver

56. Appendix Numerical Example
Apply 3rd-order Heun’s time discretization :