Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default

In this paper we study the optimal m-states switching problem in finite horizon as well as infinite horizon with risk of default. We allow the switching cost functionals and cost of default to be of polynomial growth and arbitrary. We show uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem with risk of default. This problem is connected with the valuation of a power plant in the energy market.Comment: 25 pages; Real options, Backward stochastic differential equations, Snell envelope, Stopping times, Switching, Viscosity solution of PDEs, Variational inequalities. arXiv admin note: text overlap with arXiv:0805.1306 and arXiv:0904.070

Viscosity Solutions for a System of PDEs and Optimal Switching

In this paper, we study the $m$-states optimal switching problem in finite horizon, when the switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central evaluation in cases where such organizations or state grants or financial assistance to power plants that promotes green energy in their production activity or what uses less polluting modes in their production. We show existence for optimal strategy via a verification theorem then we show existence and uniqueness of the value processes by using an approximation scheme. In the markovian framework we show that the value processes can be characterized in terms of deterministic continuous functions of the state of the process. Those latter functions are the unique viscosity solutions for a system of $m$ variational partial differential inequalities with inter-connected obstacles.Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1102.1256, arXiv:0805.1306, arXiv:0904.0707, arXiv:1202.1108, and arXiv:0707.2663 and arXiv:1104.2689 by other authors. IMA Journal of Mathematical Control and Information (2016

Stochastic Control Representations for Penalized Backward Stochastic Differential Equations

This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of equations, namely multidimensional reflected BSDE and reflected BSDE with a constraint on the hedging part, and gives stochastic control representations for their corresponding penalized equations.Comment: 24 pages in SIAM Journal on Control and Optimization, 201

Uncertainty and Time-to-Build in Bioenergy Crop Production

Over the last years, the cellulosic biofuel mandate has not been enforced by the U.S. Environmental Protection Agency. The uncertainty surrounding the enforcement of the mandate in addition to high production and harvest cost contributes to farmers’ hesitation to plant bioenergy crops such as switchgrass and miscanthus. Previous literature has shown that under uncertainty and sunk cost, the investment threshold is further increased because of the value associated from holding the investment option. This warrants the use of a real option model. In this paper, we extend previous literature by applying a real option model to bioenergy crop production in the United States. We show the spatial allocation of switchgrass under biomass price and agricultural return uncertainty. The empirical model identifies the counties in the contiguous United States that are most likely to change to switchgrass production. Our preliminary results indicate a very small share of land in switchgrass production even at high biomass prices