Runge-Kutta scheme for stochastic optimal control problems

Download

index.pdf

Date

2017

Author

Öz Bakan, Hacer

Metadata

Show full item record

Item Usage Stats

233
views

115
downloads

In this thesis, we analyze Runge-Kutta scheme for the numerical solutions of stochastic optimal control problems by using discretize-then-optimize approach. Firstly, we dis- cretize the cost functional and the state equation with the help of Runge-Kutta schemes. Then, we state the discrete Lagrangian and take the partial derivative of it with respect to its variables to get the discrete optimality system. By comparing the continuous and discrete optimality conditions, we ﬁnd a relationship between the Runge-Kutta coef- ﬁcients of the state and adjoint equation, so that we present Runge-Kutta scheme for the adjoint pair (p(t),q(t)). Similar to the deterministic setting, the issue of conver- gence is important when dealing with a numerical scheme. In stochastic case, this can be achieved either by using the strong-order convergence or weak-order convergence criteria. We match the stochastic Taylor expansion on the exact solution of continuous optimality system with the stochastic Taylor expansion of approximate solution of our discrete optimality system, term by term, in order to get both strong and weak-order conditions. The thesis ends with a conclusion and a future outlook to forthcoming research and application.

Subject Keywords

Runge-Kutta formulas., Stochastic differential equations., Stochastic partial differential equations., Stochastic differential equations

URI

http://etd.lib.metu.edu.tr/upload/12621378/index.pdf
https://hdl.handle.net/11511/26953

Collections

Graduate School of Applied Mathematics, Thesis

Suggestions

OpenMETU
Core

Advances and applications of stochastic Ito-Taylor approximation and change of time method in the financial sector Öz, Hacer; Weber, Gerhard Wilhelm; Department of Financial Mathematics (2013) In this thesis, we discuss two different approaches for the solution of stochastic differential equations (SDEs): Ito-Taylor method (IT-M) and change of time method (CT-M). First approach is an approximation in space-domain and the second one is a probabilistic transformation in time-domain. Both approaches may be considered to substitute SDEs for more “practical” representations and solutions. IT-M was most studied for one-dimensional SDEs. The main aim of this work is to extend the theory of one-dimension...
Sturm comparison theory for impulsive differential equations Özbekler, Abdullah; Ağacık, Zafer; Department of Mathematics (2005) In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super...
Periodic solutions and stability of differential equations with piecewise constant argument of generalized type Büyükadalı, Cemil; Akhmet, Marat; Department of Mathematics (2009) In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincaré, the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant...
Optimal control and reduced order modelling of Fitzhugh–Nagumo equation Küçükseyhan , Tuğba; Karasözen, Bülent; Uzunca, Murat; Department of Scientific Computing (2017) In this thesis, we investigate model order reduction and optimal control of FitzHugh-Nagumo equation (FHNE). FHNE is coupled partial differential equations (PDEs) of activator-inhibitor types. Diffusive FHNE is a model for the transmission of electrical impulses in a nerve axon, whereas the convective FHNE is a model for blood coagulation in a moving excitable media. We discretize these state FHNEs using a symmetric interior penalty Galerkin (SIPG) method in space and an average vector field (AVF) method in...
A Stochastic Maximum Principle for a Markov Regime-Switching Jump-Diffusion Model with Delay and an Application to Finance Savku, Emel; Weber, Gerhard Wilhelm (2018-11-01) We study a stochastic optimal control problem for a delayed Markov regime-switching jump-diffusion model. We establish necessary and sufficient maximum principles under full and partial information for such a system. We prove the existence-uniqueness theorem for the adjoint equations, which are represented by an anticipated backward stochastic differential equation with jumps and regimes. We illustrate our results by a problem of optimal consumption problem from a cash flow with delay and regimes.

Citation Formats

H. Öz Bakan, “Runge-Kutta scheme for stochastic optimal control problems,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.