Dynamics under Various Assumptions on Time and Uncertainty

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

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

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  • 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
  • 5. System Dynamics Primary Education
    • The first stage of compulsory education is primary or elementary education.
    • 6. In most countries, it is compulsory for children to receive primary education,
    though in many jurisdictions it is permissible for parents to provide it.
    • The transition from elementary school to secondary school or high school is
    somewhat arbitrary, but it generally occurs at about eleven or twelve yearsof age.
    • The major goals of primary education are achieving basic literacy and
    numeracy amongst all pupils, as well as establishing foundations
    in science,geography, history and other social sciences.
    • The relative priority of various fields, and the methods used to teach them,
    are an area of considerable political debate.
    • Some of the expected benefits from primary education are the reduction of
    the infant mortality rate, the population growth rate, of the
    crude birth and death rate, and so on.
  • 7. System Dynamics Primary Education
    • Because of the importance of primary education, there are several models
    proposed to study the factors influencing the primary school enrollment and
    progressions.
    • Various models developed to analyze issues in basic education are as follows:
    • 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.
  • 13. System Dynamics Primary Education
    • To design a good policy, issues to be considered / understood are:
    - reasons behind enrollment, drop-outs and repeaters in school,
    - perceived quality of teaching by students and parents,
    - educational level and income level of parents,
    - expectations from school by parents,
    - perceived quality of teaching by the District EducationalOfficer(DEO),
    - needs of different infrastructure facilities(space, ventilation, sanitation, etc.)
    in the school,
    - and much more….
  • 14. System Dynamics Primary Education
    • to analyzeimportance of infrastructural facilities on the quality of the
    primary education system and its correlation to attributes related to the
    primary education system and environment including parents, teachers,
    infrastructure and so on (cross impact matrix),
    • to predict the effects of infrastructural facilities on the quality of primary
    education system so that a discussion can be started on
    how to developpolicies,
    • to developintervention policies using societal problem handling methods
    such as COMPRAM(DeTombe1994, 2003),
    once the outputs of the model become known with available data.
  • 15. System Dynamics Primary Education
    • Modeling and forecasting the behaviour of complex systems are necessary
    if we are to exert some degree of control over them.
    • Propertiesof variables and interactions in our large-scale system:
    • 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)
  • 18. System Dynamics Primary Education
    • A variable’s response to a given impact decreases to 0as
    that variable approachesits upper or lower bound.
    • It is generally found that bounded growth and decay processes exhibit
    this sigmoidal character.
    • All other issues being constant, a variable (attribute) produces
    agreater impact onthe system as it grows larger(ceteris paribus).
    • Complex interactions are described by a looped network of
    binary interactions
    (this is the basis of cross-impact analysis).
  • 19. System Dynamics
    • The choice of relevant attributes has to be made carefully, keeping in mind both
    short-termandlong-termconsequences of solutions (decisions).
    • All attributes can be associated with given levels that may indicate
    quantitativeor qualitative possession.
    • When entities interact through their attributes, the levelsof the attributes might change, i.e., the system behaves in certain directions.
    • 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.
    • 24. However, interrelationsamong variables should be analyzedcarefully beforeintroducing new policies.
  • System Dynamics 4 Steps of Implementation
    Step 1.Set the initial values to identified attributes obtained from published sources and surveys conducted.
    Step 2.Build a cross-impact matrix with the identified relevant attributes.
    a. Summing the effectsof column attributes on rows indicates the effect of each attribute in the matrix.
    b. The parameters ijcan be determined by creating a pairwise correlation matrixafter collecting the data,and adjusted by subjective assessment.
    c. Qualitative impacts
    are quantifiedsubjectively as
    shown in the table.
    Qualitative impacts can be
    extracted from a published
    sources and survey data set.
  • 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:
  • 26. System Dynamics Problems with Migrant Primary School Students in Turkey
    Problems with Migrant Primary School Students in Turkey
    • Lack of Turkish language
    • 27. Lack of parental interest
    • 28. Lack of adequate shelter, food
    • 29. Not able to assimilate in urban life and society
  • System Dynamics Problems with Migrant Primary School Students in Turkey
    Entities in the System Dynamics Model
    student,
    teacher,
    family,
    school environment,
    Ministry of Education.
    Entity Relationshipdiagram
  • 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).
  • 34. System Dynamics Problems with Primary Education in India
    • Level of enrollment(loe)
    Beforeimplementation of policy variables (red line):
    - sharp increaseat the beginning phase of the simulation
    (first 12 iterations), and then there is steady decrease
    after a certain period of time (first 12 iterations).
    After implementation of policy variables (blue line):
    • steadyincreasefrom initial value (0.71) to unity.
    Analysis:
    • increase in level of enrollment in first 12 iterations happens
    because of the response lag in population
    to bad quality school system,
    • instant impact on the level enrollment after implementation
    of policy variables because students and parents are
    more eager to have the children attend a nice looking
    healthy school.
  • 35. System Dynamics Problems with Primary Education in India
    • Level of repeaters (lr)
    Before implementation of policy variables (red line):
    - steady increasefrom 0.05 to unity in 50 iterations.
    After implementation of policy variables (blue line):
    • steadyincreasefrom initial value 0.05 to 0.12 in first 14 iterations and then declinedto zero in 50 iterations.
    Analysis:
    • level of repeaters has instant impact on level of repeaters
    ifthe infrastructural facilities are bad
    (assumption: teachers are doing their best in teaching –
    this variable is static -sovariable impacted here is teaching aids),
    • increase in level of repeaters in first 14 iterations is because improvement in the infrastructure doesnot have an instant impact on the level of repeaters.
  • Bio-Systems
    Medicine
    Environment
    ... Finance
    Health Care
    prediction of gene patterns based on
    DNA microarraychip experiments
    with
    M.U. Akhmet, H. Öktem
    S.W. Pickl, E. Quek Ming Poh
    T. Ergenç, B. Karasözen
    J. Gebert, N. Radde
    Ö. Uğur, R. Wünschiers
    M. Taştan, A. Tezel, P. Taylan
    F.B. Yilmaz, B. Akteke-Öztürk
    S. Özöğür, Z. Alparslan-Gök
    A. Soyler, B. Soyler, M. Çetin
    S. Özöğür-Akyüz, Ö. Defterli
    N. Gökgöz, E. Kropat
  • 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:
  • 56.
  • 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:
  • 72. HybridStochastic Control
    • standard Brownian motion
    • 73. continuous state Solves an SDE whose jumps are governed by the discrete state.
    • 74. discrete state Continuous time Markov chain.
    • 75. control
    Chess Review, Nov. 21, 2005
    Control of Stochastic Hybrid Systems, Robin Raffard
  • 76. Applications
    hybrid
    • Engineering:Maintain dynamical system in safe domain for maximum time.
    • 77. Systems biology: Parameter identification.
    • 78. Finance: Optimal portfolio selection.
  • Method:1st step
    hybrid
    Derive a PDE satisfied by the objective function in terms of the generator:
    • Example 1:
    If
    then
    • Example 2:
    If
    then
  • 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
  • 81. Appendix
    System Dynamics Epidemics of HIV
    • The epidemic of AIDS has been steadily spreading for the past two decades, and now affects every country in the world.
    Estimated prevalence of HIV among young adults (15-49) per country at the end of 2005 (source: http://en.wikipedia.org/wiki/AIDS)
    • Each year, more people die, and the number of HIV+ people continues to rise despite national and international HIV prevention policies and dedicated public healthcare strategies.
    • 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.
  • 84. Appendix
    System Dynamics Epidemics of HIV
    Different issues to consider:
    • To design good policy, issues to be considered / understood are:
    - psyche and behavioral pattern of HIV+,
    - interactions between the society and HIV+,
    - socio and economic situation in which HIV+ are living,
    - awareness level of society about HIV and its transmission channels,
    - circumstances that make HIV people to do intentional transmission,
    - and much more.
  • 85. Appendix
    System Dynamics Epidemics of HIV
    Goals:
    • to analyze the phenomenon of intentional transmission of HIV/AIDS and its correlation to attributes related to the virus carrier and his/her community including family members, work colleagues and healthcare infrastructure (cross impact matrix),
    • 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,
    • 87. to develop intervention policies using societal problem handling methods such asCOMPRAM (DeTombe 1994, 2003) once the outputs of the model become known with available data.
    Possible Solutions:
    Some popular approaches are:
    • system thinking approach (cross impact analysis),
    • 88. system dynamics,
    • 89. mathematical modeling,
    • 90. etc..
  • Appendix
    System Dynamics Epidemics of HIV
    Proposed approach:
    A cross impact analysis method has been proposed here to study the behaviour of HIV+ persons with respect to the disclosure of their status:
    • basic entities and their relationships with the behavior of HIV+ persons are described, and a list of attributes are provided,
    • 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,
    • 95. significance of intentional infection transmission rates due to various sources can then be identified and analyzed.
  • Appendix
    System Dynamics Epidemics of HIV
    Conclusions:
    • A cross-impact model is developed here to study the intentional transmission of HIV bynon-disclosure of status in various risky situations.
    • 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.
    • 98. Two policy variables may be proposed as intervention to non-disclosure:
    - The first could be investing funds in improving hygiene and preventative measures in healthcare institutions.
    - However, such a policy should be accompanied by supporting the HIV+ individuals with economic aid if they are unemployed, free access to special AIDS clinics and access to housing units where they will not be subjected to any harassment.
    • The proposed cross impact model enables the identification of important factors that result in non-disclosure and could invoke new intervention policies and regulations to prevent the intentional transmission of HIV/AIDS in different countries and societal environments.
  • Stochastic Control of Hybrid Systems
    Appendix
    We consider a financial market consisting of:
    • one risk-free asset with price satisfying:
    • 99. riskyassets with prices following:
  • Stochastic Control of Hybrid Systems
    Appendix
    We assume:
  • 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 .
  • 124. Simulating NIG Distributed Random Variables
    Appendix
    We consider the simulation algorithm for sampling from an
    NIG (α, β, μ, δ) distributed random variableL.
    An 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
  • 130. Asset Price Dynamics
    Appendix
    Dynamics of Jump Models:
    • Jump-DiffusionModels,
    • 131. Infinite Activity Lévy Models.
  • Jump Diffusion Models
    Appendix
    • Special cases of exponential-Lévy models.
    • 132. Frequency of jumps is finite.
    • 133. Stock price follows a geometric Brownian motion between jumps.
  • Jump Diffusion Models
    Appendix
    • Merton’s Model
    • 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.
  • 136. Jump Diffusion Models
    Appendix
    • Kou’s Model
    • 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.
  • 139. Jump Diffusion Models
    Appendix
    • Bates’s Model
    • 140. Combines compound Poisson jumps and stochastic volatility.
    • 141. Easy to simulate and efficient MC methods can be employedfor pricing path-dependent options.
    • 142. Dynamics of the stock price is given by the following SDEs:
    with , : Brownian motionswith non-zero correlation, and
    : a compound Poisson process.
    • Can also be considered as a SV extension of the Merton’s model.
    • 143. Jumps of the log-price process do not have to be Gaussian.
  • Infinite Activity Lévy Models
    Appendix
    • Infinitely many jumps in any finite time interval.
    • 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.
    • 148. High activityis accounted for by a large/infinite number of small jumps.
  • Variance Gamma Process
    Appendix
    • The characteristic function is given by
    • 149. Infinitely divisibledistribution.
    • 150. Can also be defined by considering as time-changed Brownian motionwith drift
  • Normal Inverse Gaussian Process
    Appendix
    • The characteristic function is given by
    • 151. Can also be defined by Inverse Gaussian time-changed Brownian motion.
    • 152. Negative and positive values of result in negative and positive skewness, respectively.
  • Meixner Process
    Appendix
    • The characteristic function is given by
    • 153. Special case of the Generalized z(GZ) distributions.
    • 154. Moments of all orders exist.
    • 155. Has semi-heavy tails.
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