### Mathematics in Engineering

2020, Issue 3: 557-583. doi: 10.3934/mine.2020025
Research article

# A maximum principle for a stochastic control problem with multiple random terminal times

• Received: 01 April 2020 Accepted: 02 May 2020 Published: 08 May 2020
• In the present paper we derive, via a backward induction technique, an ad hoc maximum principle for an optimal control problem with multiple random terminal times. We thus apply the aforementioned result to the case of a linear quadratic controller, providing solutions for the optimal control in terms of Riccati backward SDE with random terminal time.

Citation: Francesco Cordoni, Luca Di Persio. A maximum principle for a stochastic control problem with multiple random terminal times[J]. Mathematics in Engineering, 2020, 2(3): 557-583. doi: 10.3934/mine.2020025

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• In the present paper we derive, via a backward induction technique, an ad hoc maximum principle for an optimal control problem with multiple random terminal times. We thus apply the aforementioned result to the case of a linear quadratic controller, providing solutions for the optimal control in terms of Riccati backward SDE with random terminal time.

 [1] Barbu V, Cordoni F, Di Persio L (2016) Optimal control of stochastic FitzHugh-Nagumo equation. Int J Control 89: 746-756. doi: 10.1080/00207179.2015.1096023 [2] Bielecki TR, Jeanblanc M, Rutkowski M (2004) Modeling and Valuation of Credit Risk, In: Stochastic Methods in Finance, Berlin: Springer, 27-126. [3] Bielecki TR, Rutkowski M (2013) Credit Risk: Modeling, Valuation and Hedging, Springer Science & Business Media. [4] Capponi A, Chen PC (2015) Systemic risk mitigation in financial networks. J Econ Dynam Control 58: 152-166. doi: 10.1016/j.jedc.2015.06.008 [5] Cordoni F, Di Persio L (2016) A BSDE with delayed generator approach to pricing under counterparty risk and collateralization. Int J Stoch Anal 2016: 1-10. [6] Cordoni F, Di Persio L (2017) Gaussian estimates on networks with dynamic stochastic boundary conditions. Infin Dimens Anal Qu 20: 1750001. doi: 10.1142/S0219025717500011 [7] Cordoni F, Di Persio L (2017) Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions. J Math Anal Appl 451: 583-603. doi: 10.1016/j.jmaa.2017.02.008 [8] Cordoni F, Di Persio L, Prezioso L. A lending scheme for a system of interconnected banks with probabilistic constraints of failure. Available from: https://arxiv.org/abs/1903.06042. [9] Di Persio L, Ziglio G (2011) Gaussian estimates on networks with applications to optimal control. Net Het Media 6: 279-296. [10] Eisenberg L, Noe TH (2001) Systemic risk in financial systems. Manage Sci 47: 236-249. doi: 10.1287/mnsc.47.2.236.9835 [11] El Karoui N, Jeanblanc M, Jiao Y (2010) What happens after a default: The conditional density approach. Stoch Proc Appl 120: 1011-1032. doi: 10.1016/j.spa.2010.02.003 [12] Fleming WH, Soner HM (2006) Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media. [13] Guatteri G, Tessitore G (2008) Backward stochastic Riccati equations and infinite horizon LQ optimal control with infinite dimensional state space and random coefficients. Appl Math Opt 57: 207-235. doi: 10.1007/s00245-007-9020-y [14] Guatteri G, Tessitore G (2005) On the backward stochastic Riccati equation in infinite dimensions. SIAM J Control Opt 44: 159-194. doi: 10.1137/S0363012903425507 [15] Hurd TR (2015) Contagion! The Spread of Systemic Risk in Financial Networks, Springer. [16] Kohlmann M, Zhou XY (2000) Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach. SIAM J Control Opt 38: 1392-1407. doi: 10.1137/S036301299834973X [17] Kohlmann M, Tang S (2002) Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging. Stoch Proc Appl 97: 255-288. doi: 10.1016/S0304-4149(01)00133-8 [18] Ying J, Kharroubi I, Pham H (2013). Optimal investment under multiple defaults risk: A BSDEdecomposition approach. Ann Appl Probab 23: 455-491. doi: 10.1214/11-AAP829 [19] Lipton A (2016) Modern monetary circuit theory, stability of interconnected banking network, and balance sheet optimization for individual banks. Int J Theor Appl Financ 19: 1650034. doi: 10.1142/S0219024916500345 [20] Mansuy R, Yor M (2006) Random Times and Enlargements of Filtrations in a Brownian Setting, Berlin: Springer. [21] Merton RC (1974) On the pricing of corporate debt: The risk structure of interest rates. J financ 29: 449-470. [22] Mou L, Yong J (2007) A variational formula for stochastic controls and some applications. Pure Appl Math Q 3: 539-567. doi: 10.4310/PAMQ.2007.v3.n2.a7 [23] Pham H (2010) Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management. Stoch Proc Appl 120: 1795-1820. doi: 10.1016/j.spa.2010.05.003 [24] Pham H (2009) Continuous-Time Stochastic Control and Optimization with Financial Applications, Springer Science & Business Media. [25] Pham H (2005) On some recent aspects of stochastic control and their applications. Probab Surv 2: 506-549. doi: 10.1214/154957805100000195 [26] Tang S (2003) General linear quadratic optimal stochastic control problems with random coefficients: Linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J Control Opt 42: 53-75. doi: 10.1137/S0363012901387550 [27] Yong J, Zhou XY (1999) Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer Science & Business Media.
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