scholarly journals On an Integral Equation for the Free Boundary of Stochastic, Irreversible Investment Problems

2012 ◽  
Author(s):  
Giorgio Ferrari
Keyword(s):  
Integral Equation ◽  
Free Boundary ◽  
Irreversible Investment

scholarly journals An Integral Equation Approach to the Irreversible Investment Problem with a Finite Horizon

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2084
Author(s):  
Junkee Jeon ◽  
Geonwoo Kim
Keyword(s):  
Integral Equation ◽  
Free Boundary ◽  
Production Capacity ◽  
Finite Horizon ◽  
Irreversible Investment ◽  
Investment Problem ◽  
Integral Equation Approach ◽  
Increase Productivity ◽  
Hamilton Jacobi Bellman ◽  
Irreversible Investments

This paper studies an irreversible investment problem under a finite horizon. The firm expands its production capacity in irreversible investments by purchasing capital to increase productivity. This problem is a singular stochastic control problem and its associated Hamilton–Jacobi–Bellman equation is derived. By using a Mellin transform, we obtain the integral equation satisfied by the free boundary of this investment problem. Furthermore, we solve the integral equation numerically using the recursive integration method and present the graph for the free boundary.

scholarly journals Irreversible investment under Lévy uncertainty: an equation for the optimal boundary

2016 ◽  
Vol 48 (1) ◽  
pp. 298-314 ◽  
Author(s):  
Giorgio Ferrari ◽  
Paavo Salminen
Keyword(s):  
Integral Equation ◽  
Optimal Investment ◽  
Positive Probability ◽  
Singular Stochastic Control ◽  
Irreversible Investment ◽  
Infinite Time ◽  
Optimal Boundary ◽  
For Profit ◽  
Profit Functions ◽  
New Equation

AbstractWe derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank–El Karoui representation problem. Such a relation and the Wiener–Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point inRwith positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb–Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.

scholarly journals Surface integrals approach to solution of some free boundary problems - II

1989 ◽  
Vol 2 (2) ◽  
pp. 91-100
Author(s):  
Igor Malyshev
Keyword(s):  
Integral Equation ◽  
Free Boundary ◽  
Stefan Problem ◽  
Numerical Experiments ◽  
Free Boundary Problems ◽  
Volterra Equations ◽  
Boundary Problems ◽  
New Algorithms ◽  
Surface Integrals

This paper is a continuation of the publication [1] where integral equation techniques were applied to the solution of a generalized Stefan problem. The regularization of the corresponding system of nonlinear integral Volterra equations offered here is quite different from that in [1], hence - several new algorithms and numerical experiments. For consistency and easy reference we start this paper with sec.6.

LAPLACE TRANSFORMS AND INSTALLMENT OPTIONS

2004 ◽  
Vol 14 (08) ◽  
pp. 1167-1189 ◽  
Author(s):  
GHADA ALOBAIDI ◽  
ROLAND MALLIER ◽  
A. STANLEY DEAKIN
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Free Boundary ◽  
Asymptotic Analysis ◽  
Laplace Transform ◽  
Financial Security ◽  
Present Value ◽  
Lump Sum ◽  
A Free Boundary Problem

An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.

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