[...]... class of optimal financial control problems where the control takes only two (or three) different discrete values The non-singular optimal control solution of linear-analytic systems in finance with bounded control is commonly known as the bang-bang control The problem of finding the optimal control becomes one of finding the switch- xvi OPTIMAL CONTROL MODELS IN FINANCE ing times in the dynamic financial... (iv) the initial and terminal conditions – various types in different models Optimal control models in finance can take different forms including the following: bang-bang control, deterministic and stochastic models, finite and infinite horizon models, aggregative and disaggregative, closed and open loop models, overtaking or multi-criteria models, time optimal models, overlapping generation models, ... time interval of time say The optimum path for such an interval is called a singular arc; on such an arc, the associated problem only gives A singular arc is very common in real-world trajectory following problems OPTIMAL CONTROL MODELS IN FINANCE 8 Indifference Principle 6 In Blatt [2, 1976], certain financial optimal control models that are concerned with optimal control with a cost of switching control. .. exercises This book reports initial efforts in providing some useful extensions; further work is necessary to complete the research agenda Optimal control models have applications to a wide range of different areas in finance: optimal portfolio choice, optimal corporate finance, financial engineering, stochastic finance, valuation, optimal consumption and investment, financial planning, risk management,... instruments OPTIMAL CONTROL MODELS IN FINANCE 6 3 Pontryagin Theorem The Pontryagin theorem was first introduced in Pontryagin [69, 1962] Consider a minimization problem in finance given as follows: T is planning horizon, subject to a differential equation and some constraint: Here (1.23) represents the differential equation (1.24) represents the constraint on control Let the optimal control problem (1.22)... Models 4 7 Bang-Bang Control In some optimal control problems, when the dynamic equation is linear in the control bang-bang control (the control only jumps on the extreme points in the feasible area of the constraints on the control) is likely to be optimal Here a small example is used to explain this concept Consider the following constraints on the control: In this case the control is restricted... in the deterministic optimization strand by extending the existing literature In this chapter, a typical general financial optimal control model is given in Section 1.1 to explain the formula of the optimal control problems and their accompanying optimal control theories In addition, some classical concepts in operations research and famous standard optimal control theories are introduced in Section... algorithms which can be used for computing optimal control models in finance and can be classified under the algorithms for continuous and discrete optimal control models (Islam [36, 2001]) Algorithms for continuous optimal control models in finance include: (i) gradient search methods; (ii) algorithms based two value boundary problems; (iii) dynamic programming, approximate solution methods (steady-state... use OPTIMAL CONTROL MODELS IN FINANCE 16 7.6 Switching Costs Model An investment model for the natural resource industry was introduced in Richard and Mihall’s paper [73, 2001] with switching cost The problem combines both absolutely continuous and impulse stochastic control In particular, the control strategy involves a sequence of interventions at discrete times However, this component of the control. .. dynamic financial system; (c) mathematical structure; and (d) computational methods and programs While Islam and Craven [38, 2002] have recently made some extensions to these areas, their work does not explicitly focus on bang-bang control models in finance The objective of this book is to present some suggested improvements in modeling bang-bang control in OPTIMAL CONTROL MODELS IN FINANCE 2 finance in . of finding the switch- xvi OPTIMAL CONTROL MODELS IN FINANCE ing times in the dynamic financial system. A cost of switching control is added to usual models. essential tools for most optimal control problems including dynamic optimization models in finance. Optimal control modeling, both deterministic and stochastic,