AACIMP 2011 Summer School. Operational Research stream. Lecture by Gerhard-Wilhelm Weber.

1. 6th International Summer School National University of Technology of the Ukraine Kiev, Ukraine, August 8-20, 2011 Let Us Predict Dynamicsunder Various Assumptions on Time and Uncertainty Gerhard-Wilhelm Weber 1*, Özlem Defterli 2, Linet Özdamar3, Zeynep Alparslan-Gök 4, Chandra Sekhar Pedamallu5, Büşra Temoçin1, 6,Azer Kerimov 1, Ceren Eda Can7, Efsun Kürüm1, 1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Mathematics and Computer Science, Cankaya Uniiversity, Ankara, Turkey 3 Department of Systems Engineering, Yeditepe University, Istanbul, Turkey 4 Department of Mathematics, Süleyman Demirel University, Isparta, Turkey 5 Department of Medical Oncology, Dana-Farber Cancer Institute, Boston, MA, USA, The Broad Institute of MIT and Harvard, Cambridge, MA, USA 6 Department of Statistics, Ankara University, Ankara, Turkey 7 Department of Statistics, Hacettepe 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 Advisor to EURO Conferences

2. Outline Introduction and Motivation System Dynamics Primary Education (PE) Simulation in PE in Turkey and Its Results Simulation in PE in India and Its Results Transmission of HIV in Developing Countries Time-Continuous and Time-Discrete Models Gene-Environment Networks Numerical Example and Results Regulatory Networks under Uncertainty Ellipsoidal Model Stochastic Model Stochastic Hybrid Systems Portfolio Optimization subject to SHSs Lévy Processes, Simulation and Asset Price Dynamics Conclusion

3. Introduction and Motivation

4. System Dynamics Primary Education Education Index (according to 2007/2008, Human Development Report) 101 million children of primary school age are out of school Number of primary-school-agechildren not in school, by region (2007) http://www.childinfo.org/education.html

8. logistic regression models (Admassu(2008)),

9. Poisson regression models(Admassu(2008)),

10. system dynamics models(Altamirano and van Daalen (2004),Karadeli et al. (2001), Pedamallu (2001), Terlouet al. (1999)),

11. behavioral models (Benson (1995), Hanusheket al. (2008)),in the contexts of different countries.

12. Several factors have been identified which influence the school enrollment and drop outs.

16. System variables are bounded:

17. A variable increases or decreases according to whether the net impact of the other variables is positive or negative. Julius (2002)

20. Some changes in attribute levels may be desirablewhile others may not be so.

21. Each attribute influences several others, thus creating a web of complex interactions that eventually determine system behaviour. In other terms, attributes are variables that vary from time to time.

22. They can vary in an unsupervisedway in the system.

23. However, variables can be controlleddirectly or indirectly, and partially by introducing new intervention policies.

25. System Dynamics 4 Steps of Implementation d. The impact of infrastructural facilities on primary school enrollments and progression become visible by running the simulation model. e. An exemplary partial cross-impact matrix with the attributes and their hypothetical values above is illustrated as follows:

27. Lack of parental interest

28. Lack of adequate shelter, food

30. System Dynamics Problems with Migrant Primary School Students in Turkey Attributes under each entity Entity 1: Student: F1.1Not believing in education F1.2Dislike of school books F1.3Lack of good Turkish language skills F1.4Dislike of school F1.5Weak academic self confidence F1.6Frequent disciplinary problem Entity 2: Teacher: F2.1Relating to guidance teacher Entity 3: Family: F3.1Economic difficulty F3.2Child labor F3.3Large families F3.4Malnutrition F3.5Homes with poor infrastructure (e.g., lack of heating, lack of room) F3.6Different home language F3.7Parent interest Entity 4: School environment: F4.1 Violence at school F4.2Alien school environment F4.3Lecture not meeting student needs

31. System Dynamics Problems with Migrant Primary School Students in Turkey The Proposed Policies to remediate the system P1. Daily distribution of milk P2. Distribution of coal, food, etc. to Turkish green card holders P3. The three-children per family aspiration by government P4. School infrastructure - renovation P5. Providing adult education for poor parents in migrant communities P6. Providing one year of Turkish language class before migrant student attends urban school P7.Combined policy: P2 + P4 + P5 + P6 While the mere basics of survival in the city are maintained by P2, policy P5would enable the parents to find better employment and improve the migrant family’s economic conditions. Policies P2and P5support parentsand, hence, their impacts on their offspring are indirect. On the other hand, policies P4and P6target the academic performance of students directly. The combined policy P7would naturally produce the best overall impact on the system if the budget of the Ministry of Education can afford it.

32. System Dynamics Problems with Migrant Primary School Students in Turkey Implementation Results

33. System Dynamics Problems with Primary Education in India The following simulationis based on data from Gujarat (India).

36. DNA experiments Ex.:yeastdata http://genome-www5.stanford.edu/

37. Analysis of DNA experiments

38. Modeling & Prediction least squares – max likelihood statistical learning prediction, anticipation Expression expression data matrix-valued function – metabolic reaction

39. Modeling & Prediction Ex.: Ex.: Euler, Runge-Kutta M We analyze the influence of em-parameters on the dynamics (expression-metabolic).

40. Genetic Network Ex. :

41. Genetic Network 0.4x1 gene2 gene1 0.2 x2 1 x1 gene3 gene4

42. Gene-Environment Networks if gene j regulates gene i otherwise

43. The Model Class is the firstly introduced time-autonomous form, where d-vector of concentration levels of proteins and of certain levels of environmental factors continuous change in the gene-expression data in time nonlinearities initial values of the gene-exprssion levels : experimental data vectors obtained from microarray experiments and environmental measurements : the gene-expression level (concentration rate) of the i th gene at time t denotes anyone of the first n coordinates in the d-vector of genetic and environmental states. Weber et al. (2008c), Chen et al. (1999), Gebert et al. (2004a), Gebert et al. (2006), Gebert et al. (2007), Tastan (2005), Yilmaz (2004), Yilmaz et al. (2005), Sakamoto and Iba (2001), Tastan et al. (2005) is the set of genes.

44. The Model Class (i)is an constant (nxn)-matrix is an (nx1)-vector of gene-expression levels represents and t the dynamical system of the n genes and their interaction alone. : (nxn)-matrix with entries as functions of polynomials, exponential, trigonometric, splines or wavelets containing some parameters to be optimized. (iii) Weber et al. (2008c), Tastan (2005), Tastan et al. (2006), Ugur et al. (2009), Tastan et al. (2005), Yilmaz (2004), Yilmaz et al. (2005), Weber et al. (2008b), Weber et al. (2009b) environmental effects (*) n genes , m environmental effects are (n+m)-vector and (n+m)x(n+m)-matrix, respectively.

45. The Model Class In general, in the d-dimensional extended space, with : (dxd)-matrix : (dx1)-vectors Ugur and Weber (2007), Weber et al. (2008c), Weber et al. (2008b), Weber et al. (2009b)

46. The Time-Discretized Model - Euler’s method, - Runge-Kutta methods, e.g., 2nd-order Heun's method 3rd-order Heun's method is introduced byDefterli et al. (2009) we rewrite it as where Ergenc and Weber (2004), Tastan (2005), Tastan et al. (2006), Tastan et al. 2005)

47. The Time-Discretized Model (**) in the extended spacedenotes the DNA microarray experimental data and the dataof environmental items obtained at the time-level the approximations obtainedby the iterative formula above initial values kthapproximation or prediction is calculated as:

48. Matrix Algebra are (nxn)- and (nxm)-matrices, respectively (n+m)x(n+m) -matrix are (n+m)-vectors Applying the 3rd-order Heun’s method to the eqn. (*) gives the iterative formula (**), where

49. Matrix Algebra Final canonical block form of : = .

50. Optimization Problem mixed-integer least-squares optimization problem: Boolean variables subject to Ugur and Weber (2007), Weber et al.(2008c), Weber et al. (2008b), Weber et al. (2009b), Gebert et al. (2004a), Gebert et al. (2006), Gebert et al. (2007). , : th : the numbers of genes regulated by gene (its outdegree), by environmental item , or by the cumulative environment, resp..

51. The Mixed-Integer Problem : constant (nxn)-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 vectorrepresenting the lower bounds for the decrease of the transcript concentration. Binary variables :

52. Numerical Example Consider our MINLP for the following 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-cut framework, together with SOPLEX 1.4.1 as our LP-solver

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

54. ____ gene A ........ gene B _ . _ . gene C - - - - gene D Results of Euler Method for all genes:

55. ____ gene A ........ gene B _ . _ . gene C - - - - gene D Results of 3rd-order Heun Method for all genes:

57. Regulatory NetworksunderUncertainty θ2 θ1

58. Regulatory NetworksunderUncertainty θ2 θ1

59. Regulatory NetworksunderUncertainty θ2 θ1

60. Regulatory NetworksunderUncertainty θ2 θ1

61. Regulatory NetworksunderUncertainty Interaction of Target & Environmental Clusters Determine the degree of connectivity.

62. Regulatory NetworksunderUncertainty Clusters and Ellipsoids: Target clusters: C1,C2,…,CREnvironmental clusters: D1,D2,…,DS Target ellipsoids: X1,X2,…,XRXi = E(μi , Σi) Environmental ellipsoids: E1,E2,…,ESEj = E(ρj ,Πj) Center Covariancematrix

63. Regulatory NetworksunderUncertainty Time-Discrete Model Target  Target Environment  Target ( R ) ( S ) Targetcluster TT (k) (k+1) ET (k) X ξ X A + + = E A j r r j0 j s j s r =1 s =1 ( R ) ( S ) Environmental cluster TE (k) (k+1) EE (k) X ζ E A + + = E A i r r i0 is i s r =1 s =1 Target  Environment Environment  Environment Determine system matrices and intercepts.

64. Regulatory NetworksunderUncertainty

65. Regulatory NetworksunderUncertainty

66. Regulatory NetworksunderUncertainty Time-Discrete Model controlfactor ( ( ) ) TT (k) (k+1) ET (k) Emissionclusters X ξ X A + + = E A + 0 j r r j0 j s j s r =1 s =1 ( R ) ( S ) Environ-mental cluster (k) TE (k) (k+1) EE (k) u + X ζ E A + + = E A i r r i0 is i s r =1 s =1  Predictionof CO2-emissions

67. Regulatory NetworksunderUncertainty The Mixed-Integer Regression Problem: R S T Σ Σ Σ − − ^ (k) (k) (k)  (k) +  ^ E E X Maximize X ∩ ∩ s r r s r = 1 s = 1 k= 1 α TT ≤ deg(C )TT s.t.       j j α TE ≤ deg(C )TE j j boundson outdegrees α ET ≤ deg(D )ET i i α EE ≤ deg(D )EE i i

68. Regulatory NetworksunderUncertainty The Continuous Regression Problem: R S T Σ Σ Σ − − ^ (k) (k) (k)  (k) +  ^ E E X Maximize X ∩ ∩ s r r s r = 1 s = 1 k= 1 R Σ α TT TT ≤ P TT ( )TT s.t. , ξ A       j j r jr j0 r =1 R α TE Σ ≤ TE P TE ( )TE , ξ A j j r j0 jr r =1 boundson outdegrees R ET Σ α ET P ET ( )ET ≤ , ζ A i i s i0 is ContinuousConstraints /Probabilities s =1 R Σ EE α EE P EE ( )EE ≤ , ζ A i i s is i0 s =1

69. Stock Markets

70. Financial Dynamics drift and diffusion term Ex.: price, wealth, interest rate, volatility processes

71. Financial Dynamics Milstein Scheme : and, based on our finitely many data:

73. continuous state Solves an SDE whose jumps are governed by the discrete state.

74. discrete state Continuous time Markov chain.

75. controlChess Review, Nov. 21, 2005 Control of Stochastic Hybrid Systems, Robin Raffard

77. Systems biology: Parameter identification.

79. Method:2ndand 3rd step hybrid Rewrite original problem as deterministic PDE optimization program Solve PDE optimization program using adjoint method Simple and robust…

80. References http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf Thank you very much for your attention! gweber@metu.edu.tr

82. Well-known modes of transmission of HIV are sexual contact, direct contact with HIV-infectedblood or fluids and perinatal transmission from mother to child.

83. Transmission can be classified into intentional and unintentional transmission - intentional if the infected person knows that he/she is HIV+ and he/she does not disclose it when there is a risk of transmission, - otherwise, transmission is not intentional.

86. to predict the effects of intentional transmission on the spread of HIV so that a discussion can be started on how to develop policies that induce openness about the HIV+ status of individuals,

88. system dynamics,

89. mathematical modeling,

91. the data requirements for the model are summarized along with a questionnaire to collect the data,

92. the cross impact among attributes can be measured by pairwise correlation analysis,

93. partial cross impact matrix that conveys information on the influence of one variable over the other is illustrated using qualitative judgement,

94. parameters are then fed into mathematical equations that change the level of variables throughout simulation iterations,

96. Here, we identify certain entities and attributes that might affect an HIV+ patient’s attitude towards disclosure.

97. We describe the simulation method and the data to be collected if such a model is to be executed.

100. Stochastic Control of Hybrid Systems Appendix Wealth Process We assume:

101. Stochastic Control of Hybrid Systems Appendix Wealth Process Be

102. Stochastic Control of Hybrid Systems Appendix Wealth Process Be

103. Stochastic Control of Hybrid Systems Appendix Wealth Process The investor is faced with the problem of finding strategies that maximize the utilityof (i) his consumption for all, and (ii) his terminal wealth at time T.

104. Stochastic Control of Hybrid Systems Appendix Wealth Process

105. Stock Prices and Indices Appendix d-fine, 2002 Ö. Önalan, 2006

106. Root Monte Carlo Simulation Appendix B. Dupire, 2007

107. Lévy Processes Appendix Definition: A cadlag adapted processes defined on a probability space (Ώ,F,P) is said to be a Lévy processes,if it possesses the following properties:

108. Lévy Processes Appendix ί ) ίί )For is independent of , i.e., has independentincrements. ίίί)For any is equal in distribution to (the distribution of does not depend on t ); has stationary increments. ίv) For every, , i.e.,is stochastically continuous. In the presence of ( ί ), ( ίί ), ( ίίί ), this is equivalent to the condition

109. Lévy Processes Appendix There is strong interplay between Lévy processes and infinitely divisibledistributions. Proposition:If is a Lévy processes, thenis infinitely divisible foreach. Proof : For any and any : Together with the stationarity and independence of increments we concludethat the random variableis infinitely divisible.

110. Lévy Processes Appendix Moreover, for alland all we define hence, for rational : .

111. Lévy Processes Appendix For every Lévy process, the following property holds: where is the characteristic exponent of .

112. Lévy Processes Appendix The triplet is called the Lévyor characteristic triplet; is called the Lévyor characteristic exponent. Here,: drift term, : diffusion coefficient, and :Lévy measure.

113. Lévy Processes Appendix The Lévy measure is a measure on which satisfies . This means that a Lévy measure has no mass at the origin, but infinitely many jumps can occur around the origin. The Lévy measure describes the expected number of jumps of a certain height in a time in interval length 1.

114. Lévy Processes Appendix The sum of all jumps smaller than some does not converge. However, the sum of the jumps compensated by their mean does converge. This pecularity leads to the necessity of the compensator term . If the Lévy measure is of the form , then f (x) is called the Lévy density. In the same way as the instant volatility describes the local uncertainty of a diffusion, the Lévy density describes the local uncertainity of a pure jump process. The Lévy-Khintchine Formulaallows us to study the distributional properties of a Lévy process. Another key concept, the Lévy-Ito Decomposition Theorem, allows one to describe the structure of a Lévy process sample path.

115. Lévy Processes Appendix Every Lévy process is a combination of a Brownian motion with drift and a possibly infinite sum of independent compound Poisson processes. This also means that every Lévy process can be approximated with arbitrary precision by a jump-diffusion process. In particular, the Lévy measure v describes arrival rates for jumps of every size for each component of Lt . Jumps of sizes in the set A occur according to a Poisson processwith intensity parameter , A being a any interval bounded away from 0. The Lévy measure of the process L may also be defined as A pure jump Lévy process can have a finite activity (aggregate jump arrival rate: finite) or infinite activity (infinitely many jumps possibly occuring in any finite time interval).

116. Lévy Processes Appendix A sample path of an NIG Lévy process is drawn with jumps being of irregular size which implies the very jagged shape of the picture. Source: Barndorff Nielsen and N.Shephard (2002) , pp. 22.

117. NormalInverse Gaussian Distribution Appendix The normal inverse Gaussian (NIG) distributions were introduced by Barndorff-Nielsen (1995) as a subclass of generalized hyperbolic laws with , so that . The normal inverse Gaussian (NIG) distribution is defined by the following probability density function: ,

118. NormalInverse Gaussian Distribution Appendix where,0 ≤│β│≤ α and K1is the modified Bessel function of third kind with index 1: The tail behavior of the NIG density is characterized by the following asymptotic relation:

119. NIG-density for different values of ; here, and . NormalInverse Gaussian Distribution Appendix

120. NIG-density for different values of ; here, and . NormalInverse Gaussian Distribution Appendix

121. NIG Lévy Asset Price Model Appendix Under the probability measureP, we consider the modeling of the asset price process as the exponential of anLévy process whereL is an NIG Lévy process. The NIG distribution is infinitely divisible and, hence, it generates a Lévy process. The log-returns of the model have independent and stationary increments.

122. NIG Lévy Asset Price Model Appendix Any infinitely divisible distributionX generates a Lévy process. According to the construction, increments of length 1 have distributionX, i.e., . But, in general, none of the increments of length different from 1 has a distribution of the same class. The price process is the solution of the stochastic differential equation where is an NIG Lévy process and is the jump of L at time t. The solution of above SDE is given by

123. NIG Lévy Asset Price Model Appendix is the unique solution of the following equation: . Under the corresponding probability , the process is again a Lévy process which is called the Esscher transform. This Esscher equivalent martingale measureis given by .

125. Sample Y from N (0,1) .

126. ReturnAn IGvariateZcan be sampled as follows, firstly drawing a random variable ,which is distributed, defining a random variable: T.H. Rydberg (1997)

127. Simulating NIG Distributed Random Variables Appendix and then letting being uniform distributed and. We see that to sample, a standard normal Y, a distributed and a uniform appear.

128. Simulating NIG Distributed Random Variables Appendix The probability density function of the inverse Gaussian distribution: The parameters aand bare functions of α and β : , .

129. NIG Lévy Asset Price Model Appendix

132. Frequency of jumps is finite.

134. Tries to capture skewnessand excess kurtosisof the log return density.

135. Stock price has the following SDE: with : a Brownian motion, and : a compound Poisson process with i.i.d. lognormally distributed jump sizes.

137. Employed to produce analytical solutions for path-dependent options, (barrier, lookback) because of the memoryless property of exponential density.

138. Stock price has the following SDE: with : a Brownian motion, and : a Poisson process with asymmetric double exponentially distributed logarithmic jump sizes.

140. Combines compound Poisson jumps and stochastic volatility.

141. Easy to simulate and efficient MC methods can be employedfor pricing path-dependent options.

144. Empirical performance of these models generally is not improved by adding a diffusion component.

145. Easier to calibrate than JD models since

146. the globalerror declines rapidly,

147. all the parameters vary simultaneously from the initial ones during the calibration.

149. Infinitely divisibledistribution.

151. Can also be defined by Inverse Gaussian time-changed Brownian motion.

153. Special case of the Generalized z(GZ) distributions.

154. Moments of all orders exist.

156. References Part 1 Pedamallu, C.S.,Özdamar, L.,Akar, H., Weber, G.-W., and Özsoy, A., Investigating academic performanceof migrant students: a system dynamics perspectivewith an application to Turkey, to appear in International Journal ofProduction Economics, the special issue on Models for CompassionateOperations,Sarkis, J., guest editor. Pedamallu, C.S., Ozdamar, L., Kropat , E., and Weber, G.-W., A system dynamics model for intentional transmission of HIV/AIDS using cross impact analysis, to appear in CEJOR (Central European Journal of Operations Research). Pedamallu, C.S.,Özdamar, L., Weber, G.-W., and Kropat,E., A system dynamics model to studythe importance of infrastructure facilities on quality of primary education system in developing countries, in the proceedings of PCO 2010, 3rd Global Conference on Power Control and Optimization,February 2-4, 2010, Gold Coast, Queensland, Australia ISBN: 978-0-7354- 0785-5 (AIP: American Institute of Physics) 321-325. Weimer-Jehle, W., Cross-impact balances: A system-theoretical approach to cross-impact analysis, Technological Forecasting and Social Change, 73:4, May 2006, pp. 334-361.

157. References Part 2 Achterberg, T., Constraint integer programming, PhD. Thesis, Technische Universitat Berlin, Berlin, 2007. Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems. Academic Press, San Diego; 2004. Chen, T., He, H.L., and Church, G.M., Modeling gene expression with differential equations, Proceedings of Pacific Symposium on Biocomputing 1999, 29-40. Ergenc, T., and Weber, G.-W., Modeling and prediction of gene-expression patterns reconsidered with Runge-Kutta discretization, Journal of Computational Technologies 9, 6 (2004) 40-48. Gebert, J., Laetsch, M., Pickl, S.W., Weber, G.-W., and Wünschiers ,R., Genetic networks and anticipation of gene expression patterns, Computing Anticipatory Systems: CASYS(92)03 - Sixth International Conference,AIP Conference Proceedings 718 (2004) 474-485. Hoon, M.D., Imoto, S., Kobayashi, K., Ogasawara, N., andMiyano, S., Inferring gene regulatory networks from time-ordered gene expression data of Bacillus subtilis using dierential equations, Proceedings of Pacific Symposium on Biocomputing (2003) 17-28. Pickl, S.W., and Weber, G.-W., Optimization of a time-discrete nonlinear dynamical system from a problem of ecology - an analytical and numerical approach, Journal of Computational Technologies 6, 1 (2001) 43-52. Sakamoto, E., and Iba, H., Inferring a system of differential equations for a gene regulatory network by using genetic programming, Proc. Congress on Evolutionary Computation 2001, 720-726. Tastan, M., Analysis and Prediction of Gene Expression Patterns by Dynamical Systems, and by a Combinatorial Algorithm, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2005.

158. References Part 2 Tastan , M., Pickl, S.W., and Weber, G.-W., Mathematical modeling and stability analysis of gene-expression patterns in an extended space and with Runge-Kutta discretization, Proceedings of Operations Research, Bremen, 2006, 443-450. Wunderling, R., Paralleler und objektorientierter Simplex Algorithmus, PhD Thesis. Technical Report ZIB-TR 96-09. Technische Universitat Berlin, Berlin, 1996. Weber, G.-W., Alparslan -Gök, S.Z., and Dikmen, N., Environmental and life sciences: Gene-environment networks-optimization, games and control - a survey on recent achievements, deTombe, D. (guest ed.), special issue of Journal of Organizational Transformation and Social Change 5, 3 (2008) 197-233. Weber, G.-W., Taylan, P., Alparslan-Gök, S.Z., Özögur, S., and Akteke-Öztürk, B., Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation, TOP 16, 2 (2008) 284-318. Weber, G.-W., Alparslan-Gök, S.Z ., and Söyler, B., A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics,Environmental Modeling & Assessment 14, 2 (2009) 267-288. Weber, G.-W., and Ugur, O., Optimizing gene-environment networks: generalized semi-infinite programming approach with intervals,Proceedings of International Symposium on Health Informatics and Bioinformatics Turkey '07, HIBIT, Antalya, Turkey, April 30 - May 2 (2007). Yılmaz, F.B., A Mathematical Modeling and Approximation of Gene Expression Patterns by Linear and Quadratic Regulatory Relations and Analysis of Gene Networks, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2004.

159. References Part 3 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. 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 ThroughComputer Experiments, Springer, 1994. Korn, R., and Korn, E., Options Pricing and Portfolio Optimization: Modern Methods ofFinancial 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).

160. References Part 3 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 Lévyprocesses with applications to mathematicalfinance, presentation at Advanced Mathematical Methods for Finance, Side, Antalya, Turkey, April 26-29, 2006. Taylan, P., Weber, G.-W.,and Kropat, E.,Approximation of stochastic differential equationsby additive modelsusing splines and conic programming, International Journal of Computing Anticipatory Systems 21(2008) 341-352. Taylan, P., Weber, G.-W., and Beck, A.,New approaches to regression by generalized additive modelsand continuous optimization for modernapplications in finance, science and techology, Optimization 56, 5-6 (2007) 1-24. Taylan, P., Weber, G.-W.,andYerlikaya, F., A new approach to multivariate adaptive regression splineby using Tikhonov regularization and continuous optimization, TOP 18, 2 (December 2010) 377-395. Rydberg, T.H., The normal inverse gaussianlévy process: simulation and approximation, Stochastic Models, 13 (1997) 4, 887 — 910. Seydel, R., Tools for ComputationalFinance, Springer, Universitext, 2004. Weber, G.-W., Taylan, P., Akteke-Öztürk, B., and Uğur, Ö., Mathematical and datamining contributions dynamics and optimization of gene-environment networks,inthe special issue Organization in Matter fromQuarks to Proteins of Electronic Journalof Theoretical Physics. Weber, G.-W.,Taylan, P., Yıldırak, K.,and Görgülü, Z.K., Financial regression and organization, DCDIS-B (Dynamics of Continuous, Discrete andImpulsive Systems (Series B)) 17, 1b (2010) 149-174.