Computational Models for Materials with Shape Memory: Towards a Systematic Description of Coupled Phenomena

Melnik, Roderick V. N.; Roberts, A. J.

doi:10.1007/3-540-46080-2_51

Cited by 5 publications

(2 citation statements)

References 16 publications

(50 reference statements)

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“…Mathematical modelling provides a powerful tool in solving a wide range of problems in industrial applications [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] and interdisciplinary approaches play an indispensable role in formulating state-of-the-art mathematical models in such applications [61]. For some of these applications it is often the case that a degree of uncertainty about the corresponding system, process, or phenomenon under the study needs to be incorporated into the model.…”

Section: Developing Multi-dimensional Models With Jumpsmentioning

confidence: 99%

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

Zhang

Melnik

2008

Appl Stoch Models Bus & Ind

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The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump-diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises.Many applications in science, engineering, finance, economics, and industry require the solution of the first passage problem. Examples range from population genetics and other applications in biology, to neurobiology and neurophysiology, to molecule kinetics and problems in chemistry and physics, to rare events modelling and quantum information, and to aerospace engineering applications (e.g. [1][2][3][4]). This problem also arises in financial applications. Although methodologies developed in this paper are applicable to a number of other areas, we exemplify our discussion here with problems from finance.In the financial world, individual companies are usually linked together via economic conditions, so default correlation, defined as the risk of multiple companies' default together, has been an important area of research in credit analysis with applications to joint defaults, credit derivatives, asset pricing, and risk management.Currently, there are two dominant groups of theoretical models used in default correlation. One is a reduced form model, such as in [5] that uses a Copula function to parameterize the default correlation. In this context, it is worthwhile noting that Chen and Sopranzetti [6] have translated the joint default probability into a bivariate normal probability function, making the analysis and applications of such models potentially more convenient.The second group of models for default correlation is a structural form model. Zhou [7] and Hull and White [8] were the first to incorporate default correlation into the Black-Cox first passage structural model. They obtained similar closed-form solutions for two ...

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Section: Developing Multi-dimensional Models With Jumpsmentioning

confidence: 99%

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

Zhang

Melnik

2008

Appl Stoch Models Bus & Ind

Self Cite

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“…This approach allowed to study both stress-and temperature-induced phase transformations and associated hysteresis phenomena in SMA structures in a unified manner [46] and to extend the developed technique to the Cattaneo-Vernotte law for heat conduction following principles of extended thermodynamics in the context of SMA [15]. Multi-layered structures with embedded thermoelectric SMA actuators were considered as one of the areas of applications of the developed dynamic models [47], and applications in nanotechnology were discussed in [48,49]. The interest to shape memory effects in the context of the latter applications has been increasing and recently a number of materials that can have such effects under appropriate temperature conditions (e.g., Ni-based alloys) have been studied both theoretically [50] and experimentally [51].…”

Section: Dynamic Models and Their Approximationsmentioning

confidence: 99%