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July  2016, 21(5): 1483-1505. doi: 10.3934/dcdsb.2016008

Optimal switching at Poisson random intervention times

Gechun Liang 1, and  Wei Wei 2,

1. 

Department of Mathematics, King's College London, Strand, London, WC2R 2LS, United Kingdom
2. 

Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United Kingdom

Received  October 2013 Revised  March 2014 Published  April 2016

This paper introduces a new class of optimal switching problems, where the player is allowed to switch at a sequence of exogenous Poisson arrival times, and the underlying switching system is governed by an infinite horizon backward stochastic differential equation system. The value function and the optimal switching strategy are characterized by the solution of the underlying switching system. In a Markovian setting, the paper gives a complete description of the structure of switching regions by means of the comparison principle.
Keywords: ordinary differential equation system., infinite horizon BSDE system, Optimal switching, Poisson process, optimal stopping.
Mathematics Subject Classification: 60H10, 60G40, 93E2.
Citation: Gechun Liang, Wei Wei. Optimal switching at Poisson random intervention times. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1483-1505. doi: 10.3934/dcdsb.2016008
References:
[1]

E. Bayraktar and M. Egami, On the one-dimensional optimal switching problem, Mathematics of Operations Research, 35 (2010), 140-159. doi: 10.1287/moor.1090.0432.

[2]

E. Bayraktar and M. Ludkovski, A sequential tracking of a hidden Markov chain using point process observations, Stochastic Processes and Their Applications, 119 (2009), 1792-1822. doi: 10.1016/j.spa.2008.09.003.

[3]

A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Gauthier-Villars, Paris, 1984.

[4]

K. Brekke and B. Oksendal, Optimal switching in an economic activity under uncertainty, SIAM J. Control Optim., 32 (1994), 1021-1036. doi: 10.1137/S0363012992229835.

[5]

P. Briand and H. Ying, Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, Journal of Functional Analysis, 155 (1998), 455-494. doi: 10.1006/jfan.1997.3229.

[6]

R. Carmona and M. Ludkovski, Pricing asset scheduling flexibility using optimal switching, Applied Mathematical Finance, 15 (2008), 405-447. doi: 10.1080/13504860802170507.

[7]

M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model, SIAM Journal on Financial Mathematics, 1 (2010), 780-810. doi: 10.1137/090770552.

[8]

R. W. R. Darling and E. Pardoux, Backwards SDE with random terminal time and applications to semilinear elliptic PDE, The Annals of Probability, 25 (1997), 1135-1159. doi: 10.1214/aop/1024404508.

[9]

K. Duckworth and M. Zervos, A model for investment decisions with switching costs, The Annals of Applied probability, 11 (2001), 239-260. doi: 10.1214/aoap/998926992.

[10]

P. Dupuis and H. Wang, Optimal stopping with random intervention times, Adv. in Appl. Probab., 34 (2002), 141-157. doi: 10.1239/aap/1019160954.

[11]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.

[12]

S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192. doi: 10.1287/moor.1060.0228.

[13]

S. Hamadène and J. Zhang, Switching problem and related system of reflected backward SDEs, Stochastic Processes and Their Applications, 120 (2010), 403-426. doi: 10.1016/j.spa.2010.01.003.

[14]

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121. doi: 10.1007/s00440-009-0202-1.

[15]

J. Lempa, Optimal stopping with information constraint, Applied Mathematics and Optimization, 66 (2012), 147-173. doi: 10.1007/s00245-012-9166-0.

[16]

G. Liang, Stochastic control representations for penalized backward stochastic differential equations, SIAM Journal on Control and Optimization, 53 (2015), 1440-1463. doi: 10.1137/130942681.

[17]

G. Liang, E. Lütkebohmert and W. Wei, Funding liquidity, debt tenor structure, and creditor's belief: An exogenous dynamic debt run model, Mathematics and Financial Economics, 9 (2015), 271-302. doi: 10.1007/s11579-015-0144-6.

[18]

V. Ly Vath and H. Pham, Explicit solution to an optimal switching problem in the two-regime case, SIAM Journal on Control and Optimization, 46 (2007), 395-426. doi: 10.1137/050638783.

[19]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.

[20]

É. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[21]

H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8.

[22]

H. Pham, V. Ly Vath and X. Y. Zhou, Optimal switching over multiple regimes, SIAM Journal on Control and Optimization, 48 (2009), 2217-2253. doi: 10.1137/070709372.

[23]

A. Porchet, N. Touzi and X. Warin, Valuation of power plants by utility indifference and numerical computation, Math. Methods Oper. Res., 70 (2009), {47-75}. doi: 10.1007/s00186-008-0231-z.

[24]

S. Tang and J. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach, Stochastics: An International Journal of Probability and Stochastic Processes, 45 (1993), 145-176. doi: 10.1080/17442509308833860.

[25]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

show all references

References:
[1]

E. Bayraktar and M. Egami, On the one-dimensional optimal switching problem, Mathematics of Operations Research, 35 (2010), 140-159. doi: 10.1287/moor.1090.0432.

[2]

E. Bayraktar and M. Ludkovski, A sequential tracking of a hidden Markov chain using point process observations, Stochastic Processes and Their Applications, 119 (2009), 1792-1822. doi: 10.1016/j.spa.2008.09.003.

[3]

A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Gauthier-Villars, Paris, 1984.

[4]

K. Brekke and B. Oksendal, Optimal switching in an economic activity under uncertainty, SIAM J. Control Optim., 32 (1994), 1021-1036. doi: 10.1137/S0363012992229835.

[5]

P. Briand and H. Ying, Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, Journal of Functional Analysis, 155 (1998), 455-494. doi: 10.1006/jfan.1997.3229.

[6]

R. Carmona and M. Ludkovski, Pricing asset scheduling flexibility using optimal switching, Applied Mathematical Finance, 15 (2008), 405-447. doi: 10.1080/13504860802170507.

[7]

M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model, SIAM Journal on Financial Mathematics, 1 (2010), 780-810. doi: 10.1137/090770552.

[8]

R. W. R. Darling and E. Pardoux, Backwards SDE with random terminal time and applications to semilinear elliptic PDE, The Annals of Probability, 25 (1997), 1135-1159. doi: 10.1214/aop/1024404508.

[9]

K. Duckworth and M. Zervos, A model for investment decisions with switching costs, The Annals of Applied probability, 11 (2001), 239-260. doi: 10.1214/aoap/998926992.

[10]

P. Dupuis and H. Wang, Optimal stopping with random intervention times, Adv. in Appl. Probab., 34 (2002), 141-157. doi: 10.1239/aap/1019160954.

[11]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.

[12]

S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192. doi: 10.1287/moor.1060.0228.

[13]

S. Hamadène and J. Zhang, Switching problem and related system of reflected backward SDEs, Stochastic Processes and Their Applications, 120 (2010), 403-426. doi: 10.1016/j.spa.2010.01.003.

[14]

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121. doi: 10.1007/s00440-009-0202-1.

[15]

J. Lempa, Optimal stopping with information constraint, Applied Mathematics and Optimization, 66 (2012), 147-173. doi: 10.1007/s00245-012-9166-0.

[16]

G. Liang, Stochastic control representations for penalized backward stochastic differential equations, SIAM Journal on Control and Optimization, 53 (2015), 1440-1463. doi: 10.1137/130942681.

[17]

G. Liang, E. Lütkebohmert and W. Wei, Funding liquidity, debt tenor structure, and creditor's belief: An exogenous dynamic debt run model, Mathematics and Financial Economics, 9 (2015), 271-302. doi: 10.1007/s11579-015-0144-6.

[18]

V. Ly Vath and H. Pham, Explicit solution to an optimal switching problem in the two-regime case, SIAM Journal on Control and Optimization, 46 (2007), 395-426. doi: 10.1137/050638783.

[19]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.

[20]

É. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[21]

H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8.

[22]

H. Pham, V. Ly Vath and X. Y. Zhou, Optimal switching over multiple regimes, SIAM Journal on Control and Optimization, 48 (2009), 2217-2253. doi: 10.1137/070709372.

[23]

A. Porchet, N. Touzi and X. Warin, Valuation of power plants by utility indifference and numerical computation, Math. Methods Oper. Res., 70 (2009), {47-75}. doi: 10.1007/s00186-008-0231-z.

[24]

S. Tang and J. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach, Stochastics: An International Journal of Probability and Stochastic Processes, 45 (1993), 145-176. doi: 10.1080/17442509308833860.

[25]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

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