A Bellman approach for two-domains optimal control problems in RN\R^N

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space $\R^N$. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness

Value function for regional control problems via dynamic programming and Pontryagin maximum principle

In this paper we focus on regional deterministic optimal control problems, i.e., problems where the dynamics and the cost functional may be different in several regions of the state space and present discontinuities at their interface. Under the assumption that optimal trajectories have a locally finite number of switchings (no Zeno phenomenon), we use the duplication technique to show that the value function of the regional optimal control problem is the minimum over all possible structures of trajectories of value functions associated with classical optimal control problems settled over fixed structures, each of them being the restriction to some submanifold of the value function of a classical optimal control problem in higher dimension.The lifting duplication technique is thus seen as a kind of desingularization of the value function of the regional optimal control problem. In turn, we extend to regional optimal control problems the classical sensitivity relations and we prove that the regularity of this value function is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem

A hybrid approach for the implementation of the Heston model

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm

A hybrid tree/finite-difference approach for Heston-Hull-White type models

We study a hybrid tree-finite difference method which permits to obtain efficient and accurate European and American option prices in the Heston Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we provide a new simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed method

On the dynamics of capital accumulation across space

We solve an optimal growth model in continuous space, continuous and bounded time. The optimizer chooses the optimal trajectories of capital and consumption across space and time by maximizing an objective function with both space and time discounting. We extract the corresponding Pontryagin conditions and prove their sufficiency. We end up with a system of two parabolic differential equations with the corresponding boundary conditions. Then, we study the roles of initial capital and technology distributions over space in various scenarios.economics, economic geography, parabolic differential equations, optimal control

Effects of Anti-Nutritional Compounds in Pongamia Seedcake on Intake, Digestion, and Ruminal Fermentation in Beef Cattle

Effects of increasing levels of the anti-nutritional compounds karanjin and pongamol on intake, nutrient utilization and ruminal fermentation were evaluated with a growing diet (trial I) and a forage diet (trial II). Steers had ad libitum access to growing diet and Bermuda grass hay during trial 1 and trial 2, respectively. At the same time diets were fed, steers received their allocated doses of karanjin and pongamol via ruminal cannula to determine acceptable inclusion level of pongamia seedcakes containing varying levels of karanjin and pongamol, which are contained in the residual oil of commercially available pongamia seedcakes. Twelve steers in both studies were assigned to completely randomized block design consisting of a control (no karanjin and no pongamol) and one of 3 levels of karanjin and pongamol. Dosing increasing levels karanjin and pongamol linearly (P < 0.01) decreased intake in Trial 1. No significant effects (P ≥ 0.12) were observed for total DMD, OMD, NDFD, ADFD and CPD. Ruminal total VFA concentration and pH were not affected by inclusion levels (P = 0.19 and P = 0.51, respectfully). The lowest dose of karanjin and pongamol resulted in a greater (P < 0.01) molar proportion of propionate than other treatments, resulting in a reduction in acetate:propionate ratio (P = 0.02) for that treatment. Among all VFA molar concentration measurements, only isobutyrate exhibited a treatment × hour effect (P = 0.02). When a forage diet was fed (Trial 2) increasing levels of karanjin and pongamol dosing resulted in a quadratic decrease (P < 0.05) in intake of total DM, OM, NDF and ADF with the. Crude protein intake did not differ (P = 0.14) among treatments. There was no effect (P ≥ 0.22) among treatments on DMD, OMD, NDFD and ADFD. Total VFA concentration and ruminal pH were not affected by inclusion level (P = 0.59 and 0.72, respectively. Therefore, acetate:propionate ratio also had no treatment effect (P = 0.84). Increasing levels of karanjin and pongamol decrease intake, although digestibility is not impacted. Ruminal fermentation of steers fed with complete ration is slightly altered with karanjin and pongamol intake, however, none alteration was observed when steers were fed only Bermuda grass hay