Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles

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

Stochastic Control of StochasticHybrid Systems I

Merton’s Optimal Consumption – Investment Problem

Hamilton-Jacobi-Bellman Equation

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

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

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.

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:

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

Lévy Processes ,[object Object],Lévy processesand infinitely divisibledistributions. Proposition:If is a Lévy processes, thenis infinitely divisible foreach. Proof : For any and any : ,[object Object],that the random variableis infinitely divisible.

Lévy Processes ,[object Object],hence, for rationalt > 0: .

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

Lévy Processes ,[object Object],is called the Lévyor characteristic exponent, ,[object Object],:diffusion coefficient and :Lévy measure.

Lévy Processes The Lévy measure is a measure on which satisfies . ,[object Object], but infinitely many jumps can occur around the origin. ,[object Object],ina time in interval length 1.

Lévy Processes ,[object Object],However, the sum of the jumps compensated by their mean does converge. This pecularity leads to the necessity of the compansator term . ,[object Object]

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.

Lévy Processes ,[object Object],Lévy-Ito Decomposition Theorem: Let be a Lévy process, the distribution of parametrized by Then decomposes as , where is a Brownian motion, and is an independent Poisson point process with intensity measure , and is a martingale with jumps

Stochastic Control of Hybrid Systems Stochastic Hybrid Model I ,[object Object]

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

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

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

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

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:

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

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

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

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.

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

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

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

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

Bubbles, Jumps and Insiders Bubble!

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

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

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

Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles

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Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles