7.6 Switching Costs Model

An investment model for the natural resource industry was introduced in Richard and Mihalls 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 strategy does not have all the features of impulse control because the sizes of the jumps associated with each interventions strategy are not part of the control strategy but are constrained to the pattern...,1, -1,1, -1,..., jumping between two levels. This kind of control patterns is also considered in this research.

Perthame [68, 1984] first introduced the combination of impulse and absolutely continuous stochastic control problems which have been further studied by Brekke and B.0ksendal [3, 1991]. Mundaca and 0ksendal [62, 1998] and Cadenillas and Zapatero [6, 2000] work on the applications to the control of currency exchange rates.

7.7 CPET

Time optimal control problems (which are not considered in this research) with bang-bang control associated with or without singular arc solutions can make the calculation difficult. A novel problem transformation called the Control Parameterization Enhancing Transform (CPET) was introduced in reference [51, 1997] to provide a computationally simple and numerically accurate solution without assuming that the optimal control is pure bang-bang control for time optimal control problems. A standard parameterization algorithm can calculate the exact switching times and singular control values of the original problem with CPET. Also, with the CPET, switching points for the control match the control parameterization knot points naturally, and hence piece-wise integration can be conveniently and accurately carried out in the usual control parameterization manner.

Several models used this technique and gave numerical results with extremely high accuracy. They are F-8 flight aircraft (first introduced in reference [31, 1977]) [71, 1994], the well-known dodgem car problem [29, 1975], and a stirred tank mixer [34, 1976]. The generalizations of CPET technique to a large range of problems were also introduced in reference [72, 1999].

7.8 STC

A new control method, the switching time computation (STC) method, which finds a suitable concatenation of constant-input arcs (or, equivalently, the places of switchings) that would take a given single-input non-linear system from a given initial point to the target, was first introduced in Kaya and Noakes paper [43, 1994]. It was also described in detail of the mathematical

reasoning in paper [44, 1996]. The method is applicable to single-input nonlinear systems. It finds the switching times for a piecewise-constant input with a given number of switchings. It can also be used for solving the time-optimal bang-bang control problem. The TOBC algorithm, which is based on the STC method, is given for this purpose. Since the STC method is basically designed for a non-linear control system, the problem of the initial guess is equally difficult when it is applied to a linear system. For the optimization procedure, an improper guess for the arc times may cause the method to fail in linear systems. However, the initial guess can be improved by experience from the failures. In non-linear systems, there does not exist a scheme for guessing a proper starting point in general optimization procedures. In general, the STC method handles a linear or a non-linear system without much discrimination. The reason is that the optimization is carried out in arc time space and even a linear system has a solution that is complicated in arc time space. The STC method has been applied as part of the TOBC algorithm to two ideal systems and a physical system (F-8 aircraft). They have been shown to be fast and accurate. The comparisons with results obtained through MISER3 software have demonstrated the efficiency of the STC method, both in its own right in finding bang-bang controls and in finding time-optimal bang-bang controls when incorporated in the TOBC algorithm. There is also a possibility to generalize the system from a single-input system to the multi-input system, which needs more computer programming involving.

7.9 Leap-frog Algorithm

Pontryagins Maximum Principle gives the necessary conditions for opti-mality of the behavior of a control system, which requires the solution of a two-point boundary-value problem (TPBVP). Noakes [64, 1997] has developed a global algorithm for finding a geodesic joining two given points on a Riemannian manifold. A geodesic problem is a special type of TPBVP. The algorithm can be viewed as a solution method for a special type of TPBVP. It is simple to implement and works well in practice. The algorithm is called the Leap-Frog Algorithm because of the nature of its geometry. Application of the Leap-Frog Algorithm to optimal control was first announced in Kaya and Noakes [45, 1997]. This algorithm gave promising results when it was applied to find an optimal control for a class of systems with unbounded control. In Kaya and Noakes paper [46, 1998], a direct and intuitive implementation of the algorithm for general non-linear systems with unbounded controls has been discussed. This work gave a more detailed and extended account of the announcement. A theoretical analysis of the Leap-Frog Algorithm for a class of optimal control problems with bounded controls in the plane was given in Kaya and Noakes paper [47, 1998]. The Leap-Frog Algorithm assumes that the problem is already solved locally. This requirement translates to the case

of optimal control as the availability of a local solution of the problem. This is related to the structure of the small-time reachable sets.

7.10 An obstruction optimal control problem

In Craven [17, 1999], an optimal control problem relating to flow around an obstacle (original proposed by Giannesi [32, 1996]) can be treated as a minimization problem which leads to a necessary condition or an optimal control problem. In this paper, Craven gave the discretization augment, and proved that, if an optimal path exists, the Pontryagin principle can be used to calculate the optimum. The optimum was verified to be reached by a discretization of the problem, and was also proved to be a global minimum.

7.11 Computational approaches to stochastic optimal control models in finance

Computational approaches specific to stochastic financial optimal control models are relatively well developed in the literature. However, computational approaches to deterministic financial optimal control models are not well documented in the literature, the standard general computational approaches to optimal control discussed above are applied to financial models as well. Some discussion of the computational approaches with specific applications to finance may be seen in Islam and Craven [38, 2002].

7.12 Comparisons of the methods

While a discussion of the comparisons of the general computational approaches is provided below, such comparisons are also relevant when the general approaches are applied to financial models. In financial optimal control models, the control function is approximated by a vector on some vector space of finite dimension in all algorithms for numerical computation of such an optimal control model 1.38-1.40. There are some examples with different chosen approximations. The RIOTS 95 package [77, 1997] which uses MATLAB, uses various spline approximations, solves the optimization problems by projected descent methods; MISER3 [33, 1987], uses a step-function to approximate the control function, solves the optimization problems by sequential quadratic programming; OCIM [15, 1998], uses conjugate gradient methods. Different implementations behave differently especially on the functions defined on a restricted domain, since some optimization methods might want to search the area outside the domain. Although a step-function is obviously a crude approximation, it produces accurate results shown in many instants in reference [84, 1991]. Since integrating the dynamic equation to obtain is a smooth operation, the high-frequency oscillations are attenuated. In Craven [14, 1995], if this attenuation is suffi-