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Counterparty risk valuation on credit-linked notes under a Markov Chain framework
JIANG Ting-ting, QIAN Xiao-song, ? George Xian-zhi Yuan,
Applied Mathematics-A Journal of Chinese Universities , 2021, 36(1): 31-50.
A credit-linked note (CLN) is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity. In this paper, we study the counterparty risk on CLNs under a Markov chain framework, and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer. Assuming that the respective default intensities are directly and inversely proportional to the interest rate, which follows a CIR process, we obtain the explicit formulae for CLN values through a PDE approach. Finally, credit valuation adjustment (CVA) formula is derived to price counterparty credit risk.
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Numerical solutions of two-dimensional nonlinear integral equations via Laguerre Wavelet method with convergence analysis
K. Maleknejad, M. Soleiman Dehkordi
Applied Mathematics-A Journal of Chinese Universities , 2021, 36(1): 83-98.
In this paper, the approximate solutions for two different type of two-dimensional
nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations
and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre
wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed
into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.
Some numerical examples are presented and results are compared with the analytical solution
to demonstrate the validity and applicability of the proposed method.
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Analysis method and algorithm design of biological sequence problem based on generalized k-mer vector
LIU Wen-li, WU Qing-biao
Applied Mathematics-A Journal of Chinese Universities , 2021, 36(1): 114-127.
K-mer can be used for the description of biological sequences and k-mer distribution is a tool for solving sequences analysis problems in bioinformatics. We can use k-mer vector as a representation method of the k-mer distribution of the biological sequence. Problems, such as similarity calculations or sequence assembly, can be described in the k-mer vector space. It helps us to identify new features of an old sequence-based problem in bioinformatics and develop new algorithms using the concepts and methods from linear space theory. In this study, we defined the k-mer vector space for the generalized biological sequences. The meaning of corresponding vector operations is explained in the biological context. We presented the vector/matrix form of several widely seen sequence-based problems, including read quantification, sequence assembly, and pattern detection problem. Its advantages and disadvantages are discussed. Also, we implement a tool for the sequence assembly problem based on the concepts of k-mer vector methods. It shows the practicability and convenience of this algorithm design strategy.
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Tracy-Widom distribution, Airy2 process and its sample path properties
SU Zhong-gen, LEI Yu-huan, SHEN Tian
Applied Mathematics-A Journal of Chinese Universities , 2021, 36(1): 128-158.
Tracy-Widom distribution was first discovered in the study of largest eigenvalues of high dimensional Gaussian unitary ensembles (GUE), and since then it has appeared in a number of apparently distinct research fields. It is believed that Tracy-Widom distribution have a universal feature like classic normal distribution. Airy2 process is defined through finite dimensional distributions with Tracy-Widom distribution as its marginal distributions. In this introductory survey, we will briefly review some basic notions, intuitive background and fundamental properties concerning Tracy-Widom distribution and Airy2 process. For sake of reading, the paper starts with some simple and well-known facts about normal distributions, Gaussian processes and their sample path properties.
10 articles