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 Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles Gerhard-Wilhelm Weber 1*, Büşra Temoçin1, 2,Azer Kerimov 1, Diogo Pinheiro 3, Nuno Azevedo 3 1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Statistics, Ankara University, Ankara, Turkey 3 CEMAPRE, ISEG – Technical University of Lisbon, Lisbon, Portugal * 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

3. Lévy Processes

4. Bubbles, Jumps and Insiders

5. Stochastic Control of StochasticHybrid Systems I

6. Merton’s Optimal Consumption – Investment Problem

7. Hamilton-Jacobi-Bellman Equation

8. Numerical Methods

9. Stochastic Hybrid Systems II

11. StockPrice Dynamics NIG Lévy Asset Price Model Normal Inverse Gaussian Process (NIG) is the subclass of generalized hyperbolic laws and has the following representation: where Ö. Önalan, 2006

12. StockPrice Dynamics NIG Lévy Asset Price Model Ö. Önalan, 2006

13. Lévy Processes Some basic Definition: 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. A process is adapted if and only if, for every realisation and every n, this process is known at time n.

14. Lévy Processes 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:

15. Lévy Processes (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

18. Lévy Processes For every Lévy process, the following property holds: whereis the characteristic exponent of .

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. The Lévy-Khintchine Formulaallows us to study the distributional properties of a Lévy process.

26. is a switching jump-diffusion between jumps.M.K. Ghosh and A. Bagchi,2004

27. Stochastic Control of Hybrid Systems The asset pricedynamics is given by the following system: The price process switches from one jump-diffusion path to another as the discrete component moves from one step to another. M.K. Ghosh and A. Bagchi,2004

28. Stochastic Control of Hybrid Systems is defined as M.K. Ghosh and A. Bagchi,2004

29. Stochastic Control of Hybrid Systems Under boundedness, measurability and Lipschitz conditions a pathwise unique solution of (1) exists. Considering the SDE the unique solution in time interval is found as M.K. Ghosh and A. Bagchi,2004

30. Merton’sConsumptionInvestment Problem 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, so as tomaximizeexpectedlifetimeutility. Assuming a relativeconsumption rate the wealth process evolves with the following SDE:

31. Merton’sConsumptionInvestment Problem The objective is Assuming a logarithmicutilityfunction, which is iso-elastic, theoptimization problem becomes D. David and Y. Yolcu Okur,2009

32. Merton’sConsumptionInvestment Problem Weassume a constantrelative risk aversion (CRRA) utilityfunction where Hence, the objective takes the following form

33. Hamilton-Jacobi-BellmanEquation In applying the dynamic programming approach, we solve theHamilton-Jacobi-Bellman (HJB) equationassociated with the utility maximization problem (2). From (W. Fleming and R. Rishel, 1975) we have that the corresponding HJB equation is given by subject to the terminal condition wherethe value function is given by

34. Hamilton-Jacobi-BellmanEquation In solving the HJB equation (3), the static optimization problem can be handled separately to reduce theHJB equation (3) to a nonlinear partial differential equation of only. Introducing the Lagrange function as whereis the Lagrange multiplier.

35. Hamilton-Jacobi-BellmanEquation The first-order necessaryconditions with respect to and , respectively, of the static optimizationproblem with Lagrangean (4) are given by D. Akumeand G.-W. Weber,2010

36. Hamilton-Jacobi-BellmanEquation Simultaneous resolution of these first-order conditions yields the optimal solutions and. Substituting these into (3) gives thePDE with terminal condition which can then be solved for the optimal value function

37. Hamilton-Jacobi-BellmanEquation Becauseof the nonlinearity in and, the first-order conditions together withthe HJB equation are a nonlinear system. So the PDE equation (5) has no analytic solution and numerical methods such as Newton‘s methodor Sequential Quadratic Programming (SQP) are required to solve for , , and iteratively. D. Akumeand G.-W. Weber,2010

38. Bubbles, Jumps and Insiders Notation: D. David and Y. Yolcu Okur,2009

39. Bubbles, Jumps and Insiders Bubble!

40. Bubbles, Jumps and Insiders Forward Integrals Forward Process D. David and Y. Yolcu Okur,2009

41. Bubbles, Jumps and Insiders It Formula for Forward Integrals D. David and Y. Yolcu Okur,2009

42. Bubbles, Jumps and Insiders D. David and Y. Yolcu Okur,2009

43. Bubbles, Jumps and Insiders D. David and Y. Yolcu Okur,2009

44. Bubbles, Jumps and Insiders D. David and Y. Yolcu Okur,2009

45. Bubbles, Jumps and Insiders Brownian Motion Case

46. Bubbles, Jumps and Insiders Mixed Case

47. Bubbles, Jumps and Insiders Mixed Case

48. Stochastic Control of Hybrid Systems Stochastic Hybrid Model II

49. Stochastic Control of Hybrid Systems Stochastic Hybrid Model II

50. Stochastic Control of Hybrid Systems Stochastic Hybrid Model II

51. Conclusion The emerging financial sector needs more and more advanced methods from mathematics and OR. NIG-Levy process meana better approximation for the stock price process. 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.

52. References 1.J. Nocedal, and S. Wright, Martingale methods in financial modeling, Springer Verlag, Heidelberg, 1999. 2.W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, Berlin, 1975. 3. J. Amendinger, P.Imkellerand M. Schweizer, Additional logarithmic utility of an insider, Stochastic Processes and Their Applications, 75 (1998) 263–286. 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. 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. 6. Ö. Önalan, Martingale Measures for NIG Lévy Processes with Applications to Mathematical Finance, International Research Journal of Finance and Economics ISSN 1450-2887 Issue 34 (2009).

53. References 7.D. Applebaum,Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004. 8.J.Bertoin, Lévy Processes, Cambridge University Press, 1996. 9.O.E.Barndorff-Nielsen, Normal Inverse Gaussian Processes and the Modelling of Stock Returns, Research Report ,Department Theoretical Statistics, Aarhus University. 1995.