6th International Summer School<br />                            National Universi...
Outline<br /><ul><li>  Stock Price Dynamics
  Lévy Processes
  Bubbles, Jumps and Insiders
  Stochastic Control of StochasticHybrid Systems I
  Merton’s Optimal Consumption – Investment Problem
  Hamilton-Jacobi-Bellman Equation
  Numerical Methods
Stochastic Hybrid Systems II
  Conclusion</li></li></ul><li>StockPrice Dynamics<br />d-fine, 2002<br />
StockPrice Dynamics<br />NIG Lévy Asset Price Model<br />Normal Inverse Gaussian Process  (NIG) is the subclass of general...
StockPrice Dynamics<br />NIG Lévy Asset Price Model<br />Ö. Önalan, 2006<br />
Lévy Processes<br />Some basic Definition: <br />A process is said to be càdlàg (“continu à droite, limite à gauche”) or R...
Lévy Processes<br />Definition: A cadlag adapted processes <br />defined on a probability space (Ώ,F,P) is said to be a <b...
Lévy Processes<br />(ί )<br />(ίί )For   is independent of     , <br />i.e., has independentincrements.<br />(ίίί) For any...
Lévy Processes<br /><ul><li>There is strong interplay between </li></ul>Lévy processesand infinitely divisibledistribution...
Lévy Processes<br /><ul><li>Moreover, for alland all            we define</li></ul>hence, for rationalt > 0:<br />        ...
Lévy Processes<br />For every Lévy process, the following property holds:<br />whereis the characteristic exponent of     ...
Lévy Processes<br /><ul><li>The triplet is called the Lévyor characteristic triplet and</li></ul>is called the Lévyor char...
Lévy Processes<br />The Lévy measure is a measure on  which satisfies<br />                                               ...
Lévy Processes<br /><ul><li>The sums of all jumps smaller than some              does not converge. </li></ul>However, the...
In the same way as the instant volatility describes the local uncertainty of a diffusion, the Lévy density describes the l...
The Lévy-Khintchine Formulaallows us to study the distributional properties </li></ul>of a Lévy process. <br />
Lévy Processes<br /><ul><li>Another key concept, the Lévy-Ito Decomposition Theorem, allows one to describe the structure ...
Stochastic Control of Hybrid Systems<br />Stochastic Hybrid Model I<br /><ul><li>is a pure jump process taking values in  ...
                          is a switching jump-diffusion between jumps.</li></ul>M.K. Ghosh and A. Bagchi,2004 <br />
Stochastic Control of Hybrid Systems<br />The asset pricedynamics is given by the following system:<br />The price process...
Stochastic Control of Hybrid Systems<br />is defined as  <br />M.K. Ghosh and A. Bagchi,2004 <br />
Stochastic Control of Hybrid Systems<br />Under boundedness, measurability and Lipschitz conditions                       ...
Merton’sConsumptionInvestment Problem<br />An investor with a finite lifetime must determine the amounts       that will b...
Merton’sConsumptionInvestment Problem<br />The objective is<br />Assuming a logarithmicutilityfunction, which is iso-elast...
Merton’sConsumptionInvestment Problem<br />Weassume a constantrelative risk aversion (CRRA) utilityfunction<br />where<br ...
Hamilton-Jacobi-BellmanEquation<br />In applying the dynamic programming approach, we solve theHamilton-Jacobi-Bellman (HJ...
Hamilton-Jacobi-BellmanEquation<br />In solving the HJB equation (3), the static optimization problem<br />can be handled ...
Hamilton-Jacobi-BellmanEquation<br />The first-order necessaryconditions with respect to            and      ,<br />respec...
Hamilton-Jacobi-BellmanEquation<br />Simultaneous resolution of these first-order conditions yields the optimal solutions ...
Hamilton-Jacobi-BellmanEquation<br />Becauseof the nonlinearity in and, the first-order conditions together withthe HJB eq...
Bubbles, Jumps and Insiders<br />Notation:<br />D. David and Y. Yolcu Okur,2009 <br />
Bubbles, Jumps and Insiders<br />Bubble!<br />
Bubbles, Jumps and Insiders<br />Forward Integrals<br />Forward Process<br />D. David and Y. Yolcu Okur,2009 <br />
Bubbles, Jumps and Insiders<br />It     Formula for Forward Integrals<br />D. David and Y. Yolcu Okur,2009 <br />
Bubbles, Jumps and Insiders<br />D. David and Y. Yolcu Okur,2009 <br />
Bubbles, Jumps and Insiders<br />D. David and Y. Yolcu Okur,2009 <br />
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Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles

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

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Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles

  1. 1. 6th International Summer School<br /> National University of Technology of the Ukraine<br /> Kiev, Ukraine, August 8-20, 2011<br />Impulsive and Stochastic Hybrid Systems <br />in the Financial Sector<br />with Bubbles<br />Gerhard-Wilhelm Weber 1*,<br />Büşra Temoçin1, 2,Azer Kerimov 1, Diogo Pinheiro 3, Nuno Azevedo 3<br /> 1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey<br /> 2 Department of Statistics, Ankara University, Ankara, Turkey<br /> 3 CEMAPRE, ISEG – Technical University of Lisbon, Lisbon, Portugal<br /> * Faculty of Economics, Management and Law, University of Siegen, Germany<br />Center for Research on Optimization and Control, University of Aveiro, Portugal<br />Universiti Teknologi Malaysia, Skudai, Malaysia<br /> Advisor to EURO Conferences<br />
  2. 2. Outline<br /><ul><li> Stock Price Dynamics
  3. 3. Lévy Processes
  4. 4. Bubbles, Jumps and Insiders
  5. 5. Stochastic Control of StochasticHybrid Systems I
  6. 6. Merton’s Optimal Consumption – Investment Problem
  7. 7. Hamilton-Jacobi-Bellman Equation
  8. 8. Numerical Methods
  9. 9. Stochastic Hybrid Systems II
  10. 10. Conclusion</li></li></ul><li>StockPrice Dynamics<br />d-fine, 2002<br />
  11. 11. StockPrice Dynamics<br />NIG Lévy Asset Price Model<br />Normal Inverse Gaussian Process (NIG) is the subclass of generalized hyperbolic laws and has the following representation:<br />where<br />Ö. Önalan, 2006<br />
  12. 12. StockPrice Dynamics<br />NIG Lévy Asset Price Model<br />Ö. Önalan, 2006<br />
  13. 13. Lévy Processes<br />Some basic Definition: <br />A process is said to be càdlàg (“continu à droite, limite à gauche”) or RCLL (“right continuous with left limits”) is a function that is right-continuous and has left limits in every point. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. <br /> A process is adapted if and only if, for every realisation and every n,<br />this process is known at time n.<br />
  14. 14. Lévy Processes<br />Definition: A cadlag adapted processes <br />defined on a probability space (Ώ,F,P) is said to be a <br />Lévy processes,if it possesses the following properties:<br />
  15. 15. Lévy Processes<br />(ί )<br />(ίί )For is independent of , <br />i.e., has independentincrements.<br />(ίίί) For any is equal in distribution to<br />(the distribution of does not depend on t );<br />has stationary increments.<br />(ίv) For every, <br /> , <br />i.e.,is stochastically continuous.<br />In the presence of ( ί ), ( ίί ), ( ίίί ), this is equivalent to the condition<br />
  16. 16. Lévy Processes<br /><ul><li>There is strong interplay between </li></ul>Lévy processesand infinitely divisibledistributions.<br />Proposition:If is a Lévy processes, thenis infinitely divisible foreach.<br />Proof : For any and any : <br /><ul><li>Together with the stationarity and independence of increments we conclude</li></ul>that the random variableis infinitely divisible. <br />
  17. 17. Lévy Processes<br /><ul><li>Moreover, for alland all we define</li></ul>hence, for rationalt > 0:<br /> . <br />
  18. 18. Lévy Processes<br />For every Lévy process, the following property holds:<br />whereis the characteristic exponent of . <br />
  19. 19. Lévy Processes<br /><ul><li>The triplet is called the Lévyor characteristic triplet and</li></ul>is called the Lévyor characteristic exponent,<br /><ul><li> where: drift term,</li></ul>:diffusion coefficient and<br />:Lévy measure.<br />
  20. 20. Lévy Processes<br />The Lévy measure is a measure on which satisfies<br /> .<br /><ul><li>This means that a Lévy measure has no mass at the origin,</li></ul> but infinitely many jumps can occur around the origin.<br /><ul><li>The Lévy measure describes the expected number of jumps of a certain size </li></ul>ina time in interval length 1. <br />
  21. 21. Lévy Processes<br /><ul><li>The sums of all jumps smaller than some does not converge. </li></ul>However, the sum of the jumps compensated by their mean does converge. <br />This pecularity leads to the necessity of the compansator term . <br /><ul><li>If the Lévy measure is of the form , then f (x) is called the Lévy density.
  22. 22. 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.
  23. 23. The Lévy-Khintchine Formulaallows us to study the distributional properties </li></ul>of a Lévy process. <br />
  24. 24. Lévy Processes<br /><ul><li>Another key concept, the Lévy-Ito Decomposition Theorem, allows one to describe the structure of a Lévy process sample path.</li></ul>Lévy-Ito Decomposition Theorem:<br /> Let be a Lévy process, the distribution of parametrized by<br /> Then decomposes as , where is a Brownian motion, and is an independent Poisson point process with intensity measure , and is a martingale with jumps <br />
  25. 25. Stochastic Control of Hybrid Systems<br />Stochastic Hybrid Model I<br /><ul><li>is a pure jump process taking values in ,
  26. 26. is a switching jump-diffusion between jumps.</li></ul>M.K. Ghosh and A. Bagchi,2004 <br />
  27. 27. Stochastic Control of Hybrid Systems<br />The asset pricedynamics is given by the following system:<br />The price process switches from one jump-diffusion path to another as the discrete component moves from one step to another.<br />M.K. Ghosh and A. Bagchi,2004 <br />
  28. 28. Stochastic Control of Hybrid Systems<br />is defined as <br />M.K. Ghosh and A. Bagchi,2004 <br />
  29. 29. Stochastic Control of Hybrid Systems<br />Under boundedness, measurability and Lipschitz conditions a pathwise unique solution of (1) exists.<br /> Considering the SDE <br />the unique solution in time interval is found as <br />M.K. Ghosh and A. Bagchi,2004 <br />
  30. 30. Merton’sConsumptionInvestment Problem<br />An investor with a finite lifetime must determine the amounts that will be consumed and the fraction of wealth that will be invested in a stock portfolio,<br />so as tomaximizeexpectedlifetimeutility. <br />Assuming a relativeconsumption rate<br />the wealth process evolves with the following SDE:<br />
  31. 31. Merton’sConsumptionInvestment Problem<br />The objective is<br />Assuming a logarithmicutilityfunction, which is iso-elastic,<br />theoptimization problem becomes<br />D. David and Y. Yolcu Okur,2009 <br />
  32. 32. Merton’sConsumptionInvestment Problem<br />Weassume a constantrelative risk aversion (CRRA) utilityfunction<br />where<br />Hence, the objective takes the following form<br />
  33. 33. Hamilton-Jacobi-BellmanEquation<br />In applying the dynamic programming approach, we solve theHamilton-Jacobi-Bellman (HJB) equationassociated with the utility maximization problem (2).<br />From (W. Fleming and R. Rishel, 1975) we have that the corresponding HJB equation is given by<br />subject to the terminal condition<br />wherethe value function is given by<br />
  34. 34. Hamilton-Jacobi-BellmanEquation<br />In solving the HJB equation (3), the static optimization problem<br />can be handled separately to reduce theHJB equation (3) to a nonlinear partial differential equation of only.<br />Introducing the Lagrange function as<br />whereis the Lagrange multiplier.<br />
  35. 35. Hamilton-Jacobi-BellmanEquation<br />The first-order necessaryconditions with respect to and ,<br />respectively, of the static optimizationproblem with Lagrangean (4) are given by<br />D. Akumeand G.-W. Weber,2010 <br />
  36. 36. Hamilton-Jacobi-BellmanEquation<br />Simultaneous resolution of these first-order conditions yields the optimal solutions and. <br />Substituting these into (3) gives thePDE<br />with terminal condition<br />which can then be solved for the optimal value function<br />
  37. 37. Hamilton-Jacobi-BellmanEquation<br />Becauseof the nonlinearity in and, the first-order conditions together withthe HJB equation are a nonlinear system.<br />So the PDE equation (5) has no analytic solution and numerical methods such as Newton‘s methodor Sequential Quadratic Programming (SQP) <br />are required to solve for <br /> , , and<br />iteratively.<br />D. Akumeand G.-W. Weber,2010 <br />
  38. 38. Bubbles, Jumps and Insiders<br />Notation:<br />D. David and Y. Yolcu Okur,2009 <br />
  39. 39. Bubbles, Jumps and Insiders<br />Bubble!<br />
  40. 40. Bubbles, Jumps and Insiders<br />Forward Integrals<br />Forward Process<br />D. David and Y. Yolcu Okur,2009 <br />
  41. 41. Bubbles, Jumps and Insiders<br />It Formula for Forward Integrals<br />D. David and Y. Yolcu Okur,2009 <br />
  42. 42. Bubbles, Jumps and Insiders<br />D. David and Y. Yolcu Okur,2009 <br />
  43. 43. Bubbles, Jumps and Insiders<br />D. David and Y. Yolcu Okur,2009 <br />
  44. 44. Bubbles, Jumps and Insiders<br />D. David and Y. Yolcu Okur,2009 <br />
  45. 45. Bubbles, Jumps and Insiders<br />Brownian Motion Case<br />
  46. 46. Bubbles, Jumps and Insiders<br />Mixed Case<br />
  47. 47. Bubbles, Jumps and Insiders<br />Mixed Case<br />
  48. 48. Stochastic Control of Hybrid Systems<br />Stochastic Hybrid Model II<br />
  49. 49. Stochastic Control of Hybrid Systems<br />Stochastic Hybrid Model II<br />
  50. 50. Stochastic Control of Hybrid Systems<br />Stochastic Hybrid Model II<br />
  51. 51. Conclusion<br />The emerging financial sector needs more and more advanced methods from mathematics and OR.<br />NIG-Levy process meana better approximation for the stock price process. <br />Impulsive and hybrid behaviour is turning out to become very import in modelling and decision making in finance, but also in ecomomics, natural and social sciences and engineering.<br />
  52. 52. References<br />1.J. Nocedal, and S. Wright, Martingale methods in financial modeling, Springer Verlag, Heidelberg, 1999.<br />2.W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, Berlin, 1975.<br />3. J. Amendinger, P.Imkellerand M. Schweizer, Additional logarithmic utility of an insider, Stochastic Processes and Their Applications, 75 (1998) 263–286.<br />4. D. Akume and G.-W. Weber, Risk-constrained dynamic portfolio Management, Dynamics of Continuous, Discrete and Impulsive SystemsSeries B: Applications & Algorithms 17 (2010) 113-129.<br />5.D. David andY. Okur, Optimal consumption and portfolio for an Insider in a market with jumps, Communications on Stochastic Analysis Vol. 3, No. 1 (2009) 101-117.<br />6. Ö. Önalan, Martingale Measures for NIG Lévy Processes with Applications to Mathematical Finance, International Research Journal of Finance and Economics<br />ISSN 1450-2887 Issue 34 (2009).<br />
  53. 53. References<br />7.D. Applebaum,Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004.<br />8.J.Bertoin, Lévy Processes, Cambridge University Press, 1996.<br />9.O.E.Barndorff-Nielsen, Normal Inverse Gaussian Processes and the Modelling of Stock Returns, Research Report ,Department Theoretical Statistics, Aarhus University. 1995. <br />

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