scholarly journals Sample-Path Large Deviations in Credit Risk

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
V. J. G. Leijdekker ◽  
M. R. H. Mandjes ◽  
P. J. C. Spreij
Keyword(s):  
Credit Risk ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rare Event ◽  
Asymptotic Results ◽  
Loss Process ◽  
Logarithmic Decay ◽  
Sample Path Large Deviations ◽  
Deviation Theory

The event of large losses plays an important role in credit risk. As these large losses are typically rare, and portfolios usually consist of a large number of positions, large deviation theory is the natural tool to analyze the tail asymptotics of the probabilities involved. We first derive a sample-path large deviation principle (LDP) for the portfolio's loss process, which enables the computation of the logarithmic decay rate of the probabilities of interest. In addition, we derive exact asymptotic results for a number of specific rare-event probabilities, such as the probability of the loss process exceeding some given function.

scholarly journals Sample Path Large Deviations for Order Statistics

2011 ◽  
Vol 48 (01) ◽  
pp. 238-257 ◽  
Author(s):  
Ken R. Duffy ◽  
Claudio Macci ◽  
Giovanni Luca Torrisi
Keyword(s):  
Order Statistics ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Sample Paths ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Trimmed Means ◽  
Hill Plots ◽  
Sample Path Large Deviations

We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution function F. If F is strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorokhod M 1 topology. Sanov's theorem is deduced in the Skorokhod M'1 topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

scholarly journals Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

2011 ◽  
Vol 48 (03) ◽  
pp. 688-698 ◽  
Author(s):  
Ken R. Duffy ◽  
Giovanni Luca Torrisi
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rate Function ◽  
Sample Paths ◽  
Heavy Tailed ◽  
Poisson Shot Noise ◽  
Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

scholarly journals Sample Path Large Deviations for Order Statistics

2011 ◽  
Vol 48 (1) ◽  
pp. 238-257 ◽  
Author(s):  
Ken R. Duffy ◽  
Claudio Macci ◽  
Giovanni Luca Torrisi
Keyword(s):  
Order Statistics ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Sample Paths ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Trimmed Means ◽  
Hill Plots ◽  
Sample Path Large Deviations

We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution functionF. IfFis strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the SkorokhodM1topology. Sanov's theorem is deduced in the SkorokhodM'1topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

scholarly journals Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

2011 ◽  
Vol 48 (3) ◽  
pp. 688-698
Author(s):  
Ken R. Duffy ◽  
Giovanni Luca Torrisi
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rate Function ◽  
Sample Paths ◽  
Heavy Tailed ◽  
Poisson Shot Noise ◽  
Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

scholarly journals Transient Asymptotics of Lévy-Driven Queues

2010 ◽  
Vol 47 (1) ◽  
pp. 109-129 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Stable Process ◽  
Sample Path ◽  
Rare Event ◽  
Compound Poisson Process ◽  
Input Process ◽  
Deterministic Function ◽  
Heavy Tailed ◽  
Stationary Workload ◽  
Sample Path Large Deviations

With (Qt)t denoting the stationary workload process in a queue fed by a Lévy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB, QTB > qB) for given positive numbers p and q, and a positive deterministic function TB. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e. TB/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e. TB / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB / B2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

scholarly journals Transient Asymptotics of Lévy-Driven Queues

2010 ◽  
Vol 47 (01) ◽  
pp. 109-129 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Stable Process ◽  
Sample Path ◽  
Rare Event ◽  
Compound Poisson Process ◽  
Input Process ◽  
Deterministic Function ◽  
Heavy Tailed ◽  
Stationary Workload ◽  
Sample Path Large Deviations

With (Q t ) t denoting the stationary workload process in a queue fed by a Lévy input process (X t ) t , this paper focuses on the asymptotics of rare event probabilities of the type P(Q 0 > pB, Q T B > qB) for given positive numbers p and q, and a positive deterministic function T B . We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to T B being sublinear (i.e. T B /B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that T B is superlinear (i.e. T B / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to T B / B 2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T B increasing superlinearly but subquadratically. We pay special attention to the case T B = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

scholarly journals Rare Event Analysis for Minimum Hellinger Distance Estimators via Large Deviation Theory

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 386
Author(s):  
Anand N. Vidyashankar ◽  
Jeffrey F. Collamore
Keyword(s):  
Generating Function ◽  
Large Deviation ◽  
Model Misspecification ◽  
Rare Event ◽  
Hellinger Distance ◽  
Asymptotic Distributions ◽  
Event Analysis ◽  
Large Deviation Theory ◽  
Continuity Properties ◽  
Deviation Theory

Hellinger distance has been widely used to derive objective functions that are alternatives to maximum likelihood methods. While the asymptotic distributions of these estimators have been well investigated, the probabilities of rare events induced by them are largely unknown. In this article, we analyze these rare event probabilities using large deviation theory under a potential model misspecification, in both one and higher dimensions. We show that these probabilities decay exponentially, characterizing their decay via a “rate function” which is expressed as a convex conjugate of a limiting cumulant generating function. In the analysis of the lower bound, in particular, certain geometric considerations arise that facilitate an explicit representation, also in the case when the limiting generating function is nondifferentiable. Our analysis involves the modulus of continuity properties of the affinity, which may be of independent interest.

scholarly journals Preferential attachment when stable

2019 ◽  
Vol 51 (4) ◽  
pp. 1067-1108
Author(s):  
Svante Janson ◽  
Subhabrata Sen ◽  
Joel Spencer
Keyword(s):  
Continuous Time ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Preferential Attachment ◽  
Rare Event ◽  
Alternative Proof ◽  
Functional Limit ◽  
Lower Tail ◽  
Symmetric Random Walk ◽  
Fine Control

AbstractWe study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $\alpha$th power $(\alpha >1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower-tail large deviation principle (LDP) for $L = \sum_{i=1}^{n} ({S_i^2}/{i^2})$, where $\{S_n \colon n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

scholarly journals Applications of large deviation theory in geophysical fluid dynamics and climate science

2021 ◽  
Author(s):  
Vera Melinda Gálfi ◽  
Valerio Lucarini ◽  
Francesco Ragone ◽  
Jeroen Wouters
Keyword(s):  
Fluid Dynamics ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Large Deviation ◽  
Rare Event ◽  
Low Frequency ◽  
Climate Science ◽  
Geophysical Fluid Dynamics ◽  
Large Deviation Theory ◽  
Deviation Theory

AbstractThe climate is a complex, chaotic system with many degrees of freedom. Attaining a deeper level of understanding of climate dynamics is an urgent scientific challenge, given the evolving climate crisis. In statistical physics, many-particle systems are studied using Large Deviation Theory (LDT). A great potential exists for applying LDT to problems in geophysical fluid dynamics and climate science. In particular, LDT allows for understanding the properties of persistent deviations of climatic fields from long-term averages and for associating them to low-frequency, large-scale patterns. Additionally, LDT can be used in conjunction with rare event algorithms to explore rarely visited regions of the phase space. These applications are of key importance to improve our understanding of high-impact weather and climate events. Furthermore, LDT provides tools for evaluating the probability of noise-induced transitions between metastable climate states. This is, in turn, essential for understanding the global stability properties of the system. The goal of this review is manifold. First, we provide an introduction to LDT. We then present the existing literature. Finally, we propose possible lines of future investigations. We hope that this paper will prepare the ground for studies applying LDT to solve problems encountered in climate science and geophysical fluid dynamics.

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