We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed-form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model.

https://www.jstor.org/stable/pdfplus/2329288.pdf

A Simple Approach to Valuing Risky Fixed and Floating Rate Debt

Remaining useful life estimation - A review on the statistical data driven approaches

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

The authors evaluated 4 sequential sampling models for 2-choice decisions--the Wiener diffusion, Ornstein-Uhlenbeck (OU) diffusion, accumulator, and Poisson counter models--by fitting them to the response time (RT) distributions and accuracy data from 3 experiments. Each of the models was augmented with assumptions of variability across trials in the rate of accumulation of evidence from stimuli, the values of response criteria, and the value of base RT across trials. Although there was substantial model mimicry, empirical conditions were identified under which the models make discriminably different predictions. The best accounts of the data were provided by the Wiener diffusion model, the OU model with small-to-moderate decay, and the accumulator model with long-tailed (exponential) distributions of criteria, although the last was unable to produce error RTs shorter than correct RTs. The relationship between these models and 3 recent, neurally inspired models was also examined.

A comparison of sequential sampling models for two-choice reaction time

The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker–Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin–Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to – the irregular nature of spiking in cortical neurons and neural gain modulation.

https://link.springer.com/content/pdf/10.1007%2Fs00422-006-0068-6.pdf

A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein-Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

https://www.jstor.org/stable/pdfplus/1427102.pdf

A new integral equation for the evaluation of first-passage-time probability densities

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

On the two-boundary first-crossing-time problem for diffusion processes

Motivated by some unsolved problems of biological interest, such as the description of firing probability densities for Leaky Integrate-and-Fire neuronal models, we consider the first-passage-time problem for Gauss-diffusion processes along the line of Mehr and McFadden (J R Stat Soc B 27:505–522, 1965). This is essentially based on a space-time transformation, originally due to Doob (Ann Math Stat 20:393–403, 1949), by which any Gauss-Markov process can expressed in terms of the standard Wiener process. Starting with an analysis that pinpoints certain properties of mean and autocovariance of a Gauss-Markov process, we are led to the formulation of some numerical and time-asymptotically analytical methods for evaluating first-passage-time probability density functions for Gauss-diffusion processes. Implementations for neuronal models under various parameter choices of biological significance confirm the expected excellent accuracy of our methods.

The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

The leaky integrate-and-fire neuronal model proposed in Stevens and Zador (1998), in which time constant and resting potential are postulated to be time dependent, is revisited within a stochastic framework in which the membrane potential is mathematically described as a gauss-diffusion process. The first-passage-time probability density, miming in such a context the firing probability density, is evaluated by either the Volterra integral equation of Buonocore, Nobile, and Ricciardi (1987) or, when possible, by the asymptotics of Giorno, Nobile, and Ricciardi (1990). The model examined here represents an extension of the classic leaky integrate-and-fire one based on the Ornstein-Uhlenbeck process in that it is in principle compatible with the inclusion of some other physiological characteristics such as relative refractoriness. It also allows finer tuning possibilities in view of its accounting for certain qualitative as well as quantitative features, such as the behavior of the time course of the membrane potential prior to firings and the computation of experimentally measurable statistical descriptors of the firing time: mean, median, coefficient of variation, and skewness. Finally, implementations of this model are provided in connection with certain experimental evidence discussed in the literature.

Aniello Buonocore

Papers

A new integral equation for the evaluation of first-passage-time probability densities

On the two-boundary first-crossing-time problem for diffusion processes

The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

On a stochastic leaky integrate-and-fire neuronal model

Restricted Ornstein-Uhlenbeck process and applications in neuronal models with periodic input signals