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Financing policies via stochastic control: a dynamic programming approach

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Abstract

This paper deals with a theoretical stochastic dynamic optimization model for the external financing of firms. We aim at searching for the best intensity of payment that a financier has to apply to a company in order to have a loan repaid. The techniques involved are related to the optimal control theory with exit time. We follow a dynamic programming approach. Our model also presents a distinction between the legal and the illegal financier, and a theoretical comparison analysis of the results is presented. Some numerical examples provide further validation of the theoretical results.

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References

  • AitSahlia F., Pardalos P.M., Sheu Y.C.: Optimal Execution of Time-constrained Portfolio Transactions. In: Konthoghiorges, E.J., Rustem, B., Winker, P. (eds) Computational Methods in Financial Engineering, pp. 95–102. Springer, (2008)

  • Alvarez O., Lasry J.M., Lions P.L.: Convex viscosity solutions and state constraints. J. Math. Pure Appl. 762, 65–288 (1997)

    Google Scholar 

  • Bardi M., Capuzzo-Dolcetta I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equation. Birkhäuser Boston, Boston (1997)

    Book  Google Scholar 

  • Barles G.: Viscosity Solutions of Hamilton–Jacobi Equations. Springer, Paris (1994)

    Google Scholar 

  • Barles G., Burdeau J.: The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems. Commun. Partial Differ. Equ. 20(1–2), 129–178 (1995)

    Article  Google Scholar 

  • Barles G., Perthame B.: Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26, 1133–1148 (1988)

    Article  Google Scholar 

  • Barles G., Rouy E.: A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications. Commun. Partial Differ. Equ. 23(11–12), 1995–2033 (1998)

    Google Scholar 

  • Barone, R., Cerqueti, R., Quaranta, A.G.: Illegal financier and usurer behavior. Eur. J. Law Econ. (2011). doi:10.1007/s10657-010-9183-x

  • Bellman R.: Dynamic programming and stochastic control process. Inf. Control 1, 228–239 (1958)

    Article  Google Scholar 

  • Blanc A.P.: Deterministic exit time control problems with discontinuous exit costs. SIAM J. Control Optim. 35, 399–434 (1997)

    Article  Google Scholar 

  • Borkar V.S.: Optimal Control of Diffusion Processes. Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow (1989)

    Google Scholar 

  • Brealey R.A., Myers S.C., Allen F.: Principles of Corporate Finance. McGraw-Hill/Irwin, New York (2006)

    Google Scholar 

  • Brennan M., Schwartz E.: Corporate income taxes, valuation, and the problem of optimal capital structure. J. Bus. 51, 103–114 (1978)

    Article  Google Scholar 

  • Caballero R.J., Pindyck R.S.: Uncertainty, investment, and industry evolution. Int. Econ. Rev. 37, 641–662 (1996)

    Article  Google Scholar 

  • Cerqueti R.: Dynamic programming via measurable selection. Pac. J. Optim. 5(1), 169–181 (2009)

    Google Scholar 

  • Cerqueti, R., Quaranta, A.G.: The perspective of a bank in granting credits: an optimization model. Optim. Lett. (2011). doi:10.1007/s11590-011-0310-6

  • Chang C.P., Lin J.H.: Bank as a liquidity provider and interest rate discovery: an option-based optimization. Expert Syst. Appl. 31, 360–369 (2006)

    Article  Google Scholar 

  • Christodoulos, A.F., Pardalos, P.M. (eds): Encyclopedia of Optimization. Springer, Berlin (2009)

    Google Scholar 

  • Cifarelli M.D., Masciandaro D., Peccati L., Salsa S., Tagliani A.: Success or failure of a firm under different financing policies: a dynamic stochastic model. Eur. J. Oper. Res. 136(3), 471–482 (2002)

    Article  Google Scholar 

  • Crandall M.G., Lions P.L.: Condition d’unicité pour les solutions généralisées des quations de Hamilton-Jacobi du premier ordre. Les Comptes Rendus de l’Académie des Sciences Paris-Séries I: Mathematique 292(3), 183–186 (1981)

    Google Scholar 

  • Crandall M.G., Lions P.L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

    Article  Google Scholar 

  • Crandall M.G., Lions P.L.: Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton–Jacobi equations. Ill. J. Math. 31(4), 665–688 (1987)

    Google Scholar 

  • Crandall M.G., Evans L.C., Lions P.L.: Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)

    Article  Google Scholar 

  • Crandall M.G., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  Google Scholar 

  • Damodaran A.: Applied Corporate Finance: A Users Manual, 2nd edn. Wiley, New York (2006)

    Google Scholar 

  • Dixit A.: Hysteresis, import penetration, and exchange-rate pass-through. Q. J. Econ. 104, 205–228 (1989)

    Article  Google Scholar 

  • Fleming W.H., Soner H.M.: Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics (New York). Springer, New York (1993)

    Google Scholar 

  • Florentin J.J.: Optimal control of continuous time, Markov, stochastic systems. J. Electron. Control 10, 473–488 (1961)

    Google Scholar 

  • Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1977)

    Google Scholar 

  • Kahn C., Pennacchi G., Sopranzetti B.: Bank consolidation and the dynamics of consumer loan interest rates. J. Bus. 78(1), 99–199 (2005)

    Article  Google Scholar 

  • Krouse C.G., Lee W.Y.: Optimal equity financing of the corporation. J. Finan. Quant. Anal. 8, 539–563 (1973)

    Article  Google Scholar 

  • Krylov N.V.: Controlled Diffusion Processes. Springer, New York (1980)

    Google Scholar 

  • Ladyzhenskaya O.A., Uraltseva N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    Google Scholar 

  • Leahy, J.: The Optimality of Myopic Behavior in a Competitive Model of Entry and Exit. Discussion Paper No. 1566, Harvard University, August (1991)

  • Leland H.: Corporate debt value, bond covenants, and optimal capital structure. J. Finance 49(4), 1213–1252 (1994)

    Google Scholar 

  • Leland H., Toft K.B.: Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. J. Finance 51(3), 987–1019 (1996)

    Google Scholar 

  • Li Y., MacLean L.C., Ziemba W.T.: Security and Wealth Aspects of Optimal Capital Growth Models with Minimum Expected Time Criteria. University of British Columbia, Mimeo (1996)

    Google Scholar 

  • Lions P.L.: Control of diffusion processes in R N. Commun. Pure Appl. Math. 34(1), 121–147 (1981)

    Article  Google Scholar 

  • Lions, P.L.: Generalized solutions of Hamilton–Jacobi equations. In: Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston (1982)

  • Lions P.L.: Équations de Hamilton–Jacobi et Solutions de Viscosité, pp. 83–97. Ennio De Giorgi Colloquium, Paris (1983a)

    Google Scholar 

  • Lions P.L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part I: the dynamic programming principle and applications. Commun. Partial Differ. Equ. 8, 1101–1174 (1983b)

    Article  Google Scholar 

  • Lions P.L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part II: viscosity solutions and uniqueness. Commun. Partial Differ. Equ. 8, 1229–1276 (1983c)

    Article  Google Scholar 

  • Lions, P.L.: Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations, Part III. In: Nonlinear PDE and Applications, Séminaire du Collége de France, vol. V. Pitman (1985)

  • Malisoff M.: Viscosity solutions of the Bellman equation for exit time optimal control problems with non-Lipschitz dynamics. ESAIM Control Optim. Calc. Var. 6, 415–441 (2001)

    Article  Google Scholar 

  • Malisoff M.: Viscosity solutions of the Bellman equation for exit time optimal control problems with vanishing Lagrangians. SIAM J. Control Optim. 40(5), 1358–1383 (2002)

    Article  Google Scholar 

  • Malisoff M.: Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians. Syst. Control Lett. 50(1), 65–79 (2003)

    Article  Google Scholar 

  • Malisoff, M., Sussmann, H.J.: Further results on the Bellman equation for exit time optimal control problems with nonnegative Lagrangians: the case of Fuller’s Problem. In: Proceedings of the 39th IEEE Conference on Decision and Control (Sydney, Australia), vol. 3, pp. 2308–2310 (2000)

  • Masciandaro, D., Peccati, L., Tagliani, A.: Why usury can be cheaper? Atti del XXI Convegno A.M.A.S.E.S, Rome, Italy 453–468 (1997)

  • Modigliani F., Miller M.: The cost of capital corporation finance, and the theory of investment. Am. Econ. Rev. 48, 261–277 (1958)

    Google Scholar 

  • Pardalos, P.M., Tsitsiringos, V. (eds): Financial Engineering, Supply Chain and E-commerce. Kluwer, Dordrecht (2002)

    Google Scholar 

  • Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton (1970)

    Google Scholar 

  • Sethi S.P.: Optimal equity financing model of Krouse and Lee: corrections and extensions. J. Finan. Quant. Anal. 13(3), 487–505 (1978)

    Article  Google Scholar 

  • Sethi S.P., Taksar M.I.: Optimal financing of a corporation subject to random returns. Math. Finance 12(2), 155–172 (2002)

    Article  Google Scholar 

  • Stanhouse B., Stock D.: Managing the risk of loan prepayments and the optimal structure of short term lending rates. Ann. Finance 4, 197–215 (2008)

    Article  Google Scholar 

  • Tirole J.: The Theory of Corporate Finance. Princeton University Press, Princeton (2006)

    Google Scholar 

  • Ye J.J.: Discontinuous solutions of the Hamilton–Jacobi equation for exit time problems. SIAM J. Control Optim. 38(4), 1067–1085 (2000)

    Article  Google Scholar 

  • Yong J., Zhou X.Y.: Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics. Springer, New York (1999)

    Google Scholar 

Download references

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Correspondence to Roy Cerqueti.

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Cerqueti, R. Financing policies via stochastic control: a dynamic programming approach. J Glob Optim 53, 539–561 (2012). https://doi.org/10.1007/s10898-011-9725-y

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