# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020003

## On correlated defaults and incomplete information

 1 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 2 Corresponding author. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China 3 Department of Mathematics, Imperial College, London, SW7 2AZ, UK

Received  June 2018 Revised  July 2019 Published  January 2020

In this paper, we study a continuous time structural asset value model for two correlated firms using a two-dimensional Brownian motion. We consider the situation of incomplete information, where the information set available to the market participants includes the default time of each firm and the periodic asset value reports. In this situation, the default time of each firm becomes a totally inaccessible stopping time to the market participants. The original structural model is first transformed to a reduced-form model. Then the conditional distribution of the default time together with the asset value of each name are derived. We prove the existence of the intensity processes of default times and also give the explicit form of the intensity processes. Numerical studies on the intensities of the two correlated names are conducted for some special cases.

Citation: Wai-Ki Ching, Jia-Wen Gu, Harry Zheng. On correlated defaults and incomplete information. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020003
##### References:

show all references

##### References:
Default Intensity process $\lambda_2$ when $\tau_1 = 2$
Default Intensities of firm B where $\rho = -0.5$
Default Intensity Process $\lambda_2$ when $\rho = 0$
Default Intensity Process $\lambda_2$ when $\tau_1 = 2$ and $\rho = 0.1$
Default Intensity Process $\lambda_2$ when $\tau_1 = 2$ and $\rho = -0.1$
 [1] Tzong-Yow Lee. Asymptotic results for super-Brownian motions and semilinear differential equations. Electronic Research Announcements, 1998, 4: 56-62. [2] Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2018, 14 (3) : 953-966. doi: 10.3934/jimo.2017084 [3] Miquel Oliu-Barton. Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights. Journal of Dynamics & Games, 2019, 6 (4) : 259-275. doi: 10.3934/jdg.2019018 [4] Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2239-2265. doi: 10.3934/dcdsb.2013.18.2239 [5] Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077 [6] Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 [7] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553 [8] Chanh Kieu, Quan Wang. On the scale dynamics of the tropical cyclone intensity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3047-3070. doi: 10.3934/dcdsb.2017196 [9] Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. The effect of noise intensity on parabolic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1715-1728. doi: 10.3934/dcdsb.2019248 [10] Martin Swaczyna, Petr Volný. Uniform motions in central fields. Journal of Geometric Mechanics, 2017, 9 (1) : 91-130. doi: 10.3934/jgm.2017004 [11] Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565 [12] Pierre Cardaliaguet, Chloé Jimenez, Marc Quincampoix. Pure and Random strategies in differential game with incomplete informations. Journal of Dynamics & Games, 2014, 1 (3) : 363-375. doi: 10.3934/jdg.2014.1.363 [13] Björn Popilka, Simon Setzer, Gabriele Steidl. Signal recovery from incomplete measurements in the presence of outliers. Inverse Problems & Imaging, 2007, 1 (4) : 661-672. doi: 10.3934/ipi.2007.1.661 [14] Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems & Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685 [15] Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091 [16] Bart Feyaerts, Stijn De Vuyst, Herwig Bruneel, Sabine Wittevrongel. Performance analysis of buffers with train arrivals and correlated output interruptions. Journal of Industrial & Management Optimization, 2015, 11 (3) : 829-848. doi: 10.3934/jimo.2015.11.829 [17] Tai-Chia Lin, Tsung-Fang Wu. Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2165-2187. doi: 10.3934/dcds.2020110 [18] Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931 [19] Stuart S. Antman. Regularity properties of planar motions of incompressible rods. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 481-494. doi: 10.3934/dcdsb.2003.3.481 [20] Ernesto A. Lacomba, Mario Medina. Oscillatory motions in the rectangular four body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 557-587. doi: 10.3934/dcdss.2008.1.557

2018 Impact Factor: 1.025