scholarly journals Importance sampling of heavy-tailed iterated random functions

2018 ◽  
Vol 50 (3) ◽  
pp. 805-832
Author(s):  
Bohan Chen ◽  
Chang-Han Rhee ◽  
Bert Zwart
Keyword(s):  
Markov Chain ◽  
Stationary Solution ◽  
Importance Sampling ◽  
Heavy Tails ◽  
Lipschitz Functions ◽  
Natural Conditions ◽  
Random Functions ◽  
State Dependent ◽  
Large Jump ◽  
Heavy Tailed

AbstractWe consider the stationary solutionZof the Markov chain {Zn}n∈ℕdefined byZn+1=ψn+1(Zn), where {ψn}n∈ℕis a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} whenxis large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψnsuch that, for largex, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (04) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (4) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (Xk:k≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (Sn:n≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail ofSnin a large deviation regime asn↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

2010 ◽  
Vol 47 (2) ◽  
pp. 301-322 ◽  
Author(s):  
Jose Blanchet ◽  
Jingchen Liu
Keyword(s):  
Random Walk ◽  
Importance Sampling ◽  
Mean Squared Error ◽  
First Passage Time ◽  
Passage Time ◽  
Natural Generalization ◽  
Efficient Estimation ◽  
State Dependent ◽  
Negative Drift ◽  
Heavy Tailed

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

scholarly journals Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

2010 ◽  
Vol 47 (02) ◽  
pp. 301-322 ◽  
Author(s):  
Jose Blanchet ◽  
Jingchen Liu
Keyword(s):  
Random Walk ◽  
Importance Sampling ◽  
Mean Squared Error ◽  
First Passage Time ◽  
Passage Time ◽  
Natural Generalization ◽  
Efficient Estimation ◽  
State Dependent ◽  
Negative Drift ◽  
Heavy Tailed

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

Growth Optimal Investment Strategy: The Impact of Reallocation Frequency and Heavy Tails

2012 ◽  
Vol 13 (2) ◽  
pp. 228-240 ◽  
Author(s):  
G. Bamberg ◽  
A. Neuhierl
Keyword(s):  
Heavy Tails ◽  
Geometric Mean ◽  
Asymptotic Optimality ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Risky Asset ◽  
Optimal Investment Strategy ◽  
Optimality Properties ◽  
Heavy Tailed ◽  
The Impact

Abstract The strategy to maximize the long-term growth rate of final wealth (maximum expected log strategy, maximum geometric mean strategy, Kelly criterion) is based on probability theoretic underpinnings and has asymptotic optimality properties. This article reviews the allocation of wealth in a two-asset economy with one risky asset and a risk-free asset. It is also shown that the optimal fraction to be invested in the risky asset (i) depends on the length of the basic return period and (ii) is lower for heavy-tailed log returns than for light-tailed log returns.

scholarly journals A New Class of Heavy-Tailed Distribution and the Stock Market Returns in Germany

2017 ◽  
Vol 4 (2) ◽  
pp. 13 ◽  
Author(s):  
John Oden ◽  
Kevin Hurt ◽  
Susan Gentry
Keyword(s):  
Stock Market ◽  
Global Economy ◽  
Heavy Tails ◽  
Financial Sector ◽  
Superior Performance ◽  
Market Returns ◽  
Stock Market Returns ◽  
Volatility Clustering ◽  
Empirical Performance ◽  
Heavy Tailed

As the fourth largest economy over the world, Germany’s financial sector plays a key role in the global economy. As one of the most important components of the financial sector, the equity market played a more and more important role. Thus, risk management of its stock market is crucial for welfare of its market participants. To account for the two stylized facts, volatility clustering and conditional heavy tails, we take advantage of the framework in Guo (2016) and consider empirical performance of the GARCH model with normal reciprocal inverse Gaussian distribution in fitting the German stock return series. Our results indicate the NRIG distribution has superior performance in fitting the stock market returns.

scholarly journals Inhomogeneous phase-type distributions and heavy tails

2019 ◽  
Vol 56 (4) ◽  
pp. 1044-1064 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Mogens Bladt
Keyword(s):  
Real World ◽  
Heavy Tails ◽  
Heavy Tail ◽  
Transition Rates ◽  
Modelling Framework ◽  
Inhomogeneous Phase ◽  
Fire Insurance ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Heavy Tailed

AbstractWe extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.

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