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Register a new userOn Numerical Solution of Structural model for the Probability of Default under a Regime-Switching Synchronous-Jump Tempered Stable Levy Model with Desingularized Meshfree Collocation method
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Added by
Davood Damircheli
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In the paper [Hainaut, D. and Colwell, D.B., A structural model for credit risk with switching processes and synchronous jumps, The European Journal of Finance44(33) (4238):3262-3284], the authors exploit a synchronous-jump regime-switching model to compute the default probability of a publicly-traded company. Here, we first generalize the proposed Levy model to a more general setting of tempered stable processes recently introduced into the finance literature. Based on the singularity of the resulting partial integro-differential operator, we propose a general framework based on strictly positive-definite functions to de-singularize the operator. We then analyze an efficient meshfree collocation method based on radial basis functions to approximate the solution of the corresponding system of partial integro-differential equations arising from the structural credit risk model. We show that under some regularity assumptions, our proposed method naturally de-sinularizes the problem in the tempered stable case. Numerical results of applying the method on some standard examples from the literature confirm the accuracy of our theoretical results and numerical algorithm.
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We propose a Markov regime switching GARCH model with multivariate normal tempered stable innovation to accommodate fat tails and other stylized facts in returns of financial assets. The model is used to simulate sample paths as input for portfolio optimization with risk measures, namely, conditional value at risk and conditional drawdown. The motivation is to have a portfolio that avoids left tail events by combining models that incorporates fat tail with optimization that focuses on tail risk. In-sample test is conducted to demonstrate goodness of fit. Out-of-sample test shows that our approach yields higher performance measured by Sharpe-like ratios than the market and equally weighted portfolio in recent years which includes some of the most volatile periods in history. We also find that suboptimal portfolios with higher return constraints tend to outperform optimal portfolios.
Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).
In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilatation, then back-substitute dilatation, and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. We then apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lam{e} parameters satisfy $lambda geq mu$ to avoid adding extra constraints on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.
In this paper we present numerical simulations of a macroscopic vision-based model [1] derived from microscopic situation rules described in [2]. This model describes an approach to collision avoidance between pedestrians by taking decisions of turning or slowing down based on basic interaction rules, where the dangerousness level of an interaction with another pedestrian is measured in terms of the derivative of the bearing angle and of the time-to-interaction. A meshfree particle method is used to solve the equations of the model. Several numerical cases are considered to compare this model with models established in the field, for example, social force model coupled to an Eikonal equation [3]. Particular emphasis is put on the comparison of evacuation and computation times. References 1. Degond P., Appert-Rolland C., Pettere J., Theraulaz G., Vision-based macroscopic pedestrian models, Kinetic and Related models, AIMs 6(4), 809-839 (2013) 2. Ondrej J., Pettere J., Olivier A.H., Donikian S., A synthetic-vision based steering approach for crowd simulation, ACM Transactions on Graphics, 29(4), Article 123 (2010) 3. Etikyala R., Gottlich S., Klar A., Tiwari S., Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models, Mathematical Models and Methods in Applied Sciences, 20(12), 2503-2523 (2014)
The inclusion of domain (point) sources into a three dimensional boundary element method while solving the Helmholtz equation is described. The method is fully desingularized which allows for the use of higher order quadratic elements on the surfaces of the problem with ease. The effect of the monopole sources ends up on the right hand side of the resulting matrix system. Several carefully chosen examples are shown, such as sources near and within a concentric spherical core-shell scatterer as a verification case, a curved focusing surface and a multi-scale acoustic lens.
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