Search

We develop a comprehensive methodology for the fluctuation theory of continuous-time, skip-free Markov chains, extending and improving the recent work of Choi and Patie for discrete-time, skip-free Markov chains. As a significant application, we use it to derive a full set of fluctuation identities regarding exiting a finite or infinite interval for Markov branching processes with immigration, thereby uncovering many new results for this classic family of continuous-time Markov chains. The theory also allows us to recover in a simple manner fluctuation identities for skip-free downward compound Poisson processes. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the arrival statistics of unbiased planar random walkers to complex target configurations. In contrast to problems posed in finite domains, simple moments of the first passage time distribution, such as its mean and variance, are not defined. Therefore, it is necessary to obtain the full arrival statistics. We describe several methods to obtain these distributions and other associated quantities such as splitting probabilities. One approach combines a Laplace transform of the underlying parabolic equation with matched asymptotic analysis followed by numerical transform inversion. The second approach is similar but uses a boundary integral equation method to solve for the transformed variable. To validate the results of this theory, and to obtain the arrival time statistics in very general configurations of absorbers, we introduce an efficient kinetic Monte Carlo (KMC) method that describes trajectories as a combination of large but exactly solvable projection steps. The effectiveness of these methodologies is demonstrated on a variety of challenging examples highlighting the applicability of these methods to a variety of practical scenarios, such as source inference. A particularly useful finding arising from these results is that homogenization theories, in which complex configurations are replaced by equivalent simple ones, are remarkably effective at describing arrival time statistics. [ABSTRACT FROM AUTHOR]

Let \mathbb {X}=(\mathbb {X}_t)_{t\geq 0} be the subdiffusive process defined, for any t\geq 0, by \mathbb {X}_t = X_{\ell _t} where X=(X_t)_{t\geq 0} is a Lévy process and \ell _t=\inf \{s>0; \mathcal {K}_s>t \} with \mathcal {K}=(\mathcal {K}_t)_{t\geq 0} a subordinator independent of X. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair (\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}}) where \begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t>0; \mathbb {X}_t > a+ {b}_t \} \end{equation*} with a \in \mathbb {R} and {b}=({b}_t)_{t\geq 0} a (possibly degenerate) subordinator independent of X and \mathcal {K}. We proceed by providing a detailed analysis of the cases where either \mathbb {X} is a self-similar or is spectrally negative. For the later, we show the fact that the process (\mathbb {T}_a^{({b})})_{a\geq 0} is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable \mathbb {T}_a^{({b})} has the same law as the first passage time of a semi-regenerative process of Lévy type , a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set. [ABSTRACT FROM AUTHOR]

To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. On the other hand, in larger networks a single decision can trigger a cascade of agreements and disagreements that depend on the private information agents have gathered. Our approach provides a bridge between social decision making models in the economics literature, which largely ignore the temporal dynamics of decisions, and the single-observer evidence accumulator models used widely in neuroscience and psychology. [ABSTRACT FROM AUTHOR]

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. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a novel method to price discretely monitored single- and double-barrier options in Lévy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix–vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates. [ABSTRACT FROM AUTHOR]