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People back Faculty Graduate Students Staff. Regular Faculty Baxendale, Peter : Stochastic dynamical systems; equilibrium, stability and bifurcation for solutions of stochastic differential equations; applications to stochastic neuronal models. Lototsky, Sergey : Stochastic partial differential equations, optimal nonlinear filtering of diffusion processes, statistical inference for continuous-time processes. Ma, Jin : Stochastic analysis, stochastic differential equations, stochastic control theory, mathematical finance and insurance.
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Mikulevicius, Remigijus Remi : Stochastic differential equations, stochastic analysis. Minsker, Stanislav Stas : Statistical learning theory, non-parametric statistics, concentration inequalities, mathematical finance. W y is a piecewise constant function where. We approximate W y using a hyperbolic tangent function. For larger values of k we obtain a smoother approximation of the function W y see Fig.
There are several necessary conditions that need to be satisfied, such as the state equation 9 and the costate equation 16 along with the stationarity condition Also the initial condition of y 0 is given and we select a guessed initial value of p 0. We also need to make sure the boundary condition of 18 will be satisfied at the final time, T. This is a two point boundary value problem.
A non-standard optimal control problem arising in an economics application
We need to ensure that the iteration value of z used in 17 will be equal to y T at the final time T. This is satisfied in our algorithm only when the p T boundary condition converges.
Then we will have the optimal solution. The minimising free value y T is calculated in an outer loop iteration using the Golden Section or Brent maximising line search algorithm. An optimal solution was obtained with high accuracy.
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The results are. Figures shows the optimal curves of the state variable y t , adjoint variable p t and the integral eta t and the control variable u t. As a comparative approach, we used a different nonlinear programming discrete-time technique to solve the same problem [1,7]. We solved the problem using Euler and also Runge-Kutta discretisation, and an optimization algorithm in order to solve the unknown control variables u k at each time t k . Figures show the plot of optimal results for both the shooting and nonlinear programming NP approaches for the state variable y t , costate and control variable u t.
The results are essentially the same. The NP results are good but the Hamiltonian shooting approach is a much more accurate approach. In this paper we have shown how to solve a non-standard optimal control problem.
We have presented the necessary conditions and the computational procedures in order to obtain optimal solutions. The optimal solution of a test problem has been presented. A shooting algorithm together with a maximising approach was used to obtain a highly accurate solution, and compared with a discrete-time nonlinear programming solution.
Our techniques can be applied to the actual rather more complicated economics problem where the Lagrangian integrand is piecewise constant in many stages and depends upon y T which is a priori unknown.
Financial Mathematics and Control Theory Research
Higher-order calculus of variations on time scales. Mathematical control theory and finance, pp. Springer, Berlin. Hamiltonian approach to optimality conditions in control theory and nonsmooth analysis. Optimal control theory and static optimization in economics, Cambridge University Press, Cambridge. Natural boundary conditions in the calculus of variations.
Mathematical Methods in the Applied Sciences , 33 14 : Wiley, doi: A non-classical class of variational problemswith application in economics. InderScience, doi: Numerical recipes: The art of scientific computing.
Third edition. Cambridge Univ.
Press, Cambridge. Optimal control theory - Applications on management science and economics. Second edition. Optimal production subject to piecewise continuous royalty payment obligations, internal report. Optimal control theory: An introduction. Optimal control and the calculus of variations. Oxford University Press, Oxford.