scholarly journals On exact sampling of stochastic perpetuities

2011 ◽  
Vol 48 (A) ◽  
pp. 165-182 ◽  
Author(s):  
Jose H. Blanchet ◽  
Karl Sigman

A stochastic perpetuity takes the formD∞=∑n=0∞exp(Y1+⋯+Yn)Bn, whereYn:n≥0) and (Bn:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively byDn+1=AnDn+Bn,n≥0, whereAn=eYn;D∞then satisfies the stochastic fixed-point equationD∞D̳AD∞+B, whereAandBare independent copies of theAnandBn(and independent ofD∞on the right-hand side). In our framework, the quantityBn, which represents a random reward at timen, is assumed to be positive, unbounded with EBnp<∞ for somep>0, and have a suitably regular continuous positive density. The quantityYnis assumed to be light tailed and represents a discount rate from timenton-1. The RVD∞then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples ofD∞. Our method is a variation ofdominated coupling from the pastand it involves constructing a sequence of dominating processes.

2011 ◽  
Vol 48 (A) ◽  
pp. 165-182 ◽  
Author(s):  
Jose H. Blanchet ◽  
Karl Sigman

A stochastic perpetuity takes the form D∞=∑n=0∞ exp(Y1+⋯+Yn)Bn, where Yn:n≥0) and (Bn:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by Dn+1=AnDn+Bn, n≥0, where An=eYn; D∞ then satisfies the stochastic fixed-point equation D∞D̳AD∞+B, where A and B are independent copies of the An and Bn (and independent of D∞ on the right-hand side). In our framework, the quantity Bn, which represents a random reward at time n, is assumed to be positive, unbounded with EBnp <∞ for some p>0, and have a suitably regular continuous positive density. The quantity Yn is assumed to be light tailed and represents a discount rate from time n to n-1. The RV D∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D∞. Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.


2020 ◽  
Vol 72 (12) ◽  
pp. 41-42
Author(s):  
Judy Feder

This article, written by JPT Technology Editor Judy Feder, contains highlights of paper URTEC 198318, “How Not to Squander Billions on Your Next Unconventional Venture,” by Creties Jenkins, SPE, and Mark McLane, SPE, Rose and Associates, prepared for the 2019 SPE/AAPG/SEG Asia Pacific Unconventional Resources Technology Conference, Brisbane, Australia, 18-19 November. The paper has not been peer reviewed. During the past decade, hundreds of unconventional oil and gas projects have failed to deliver the value sought by shareholders. Two common mistakes have been focusing on production attainment instead of value creation, and incorrectly thinking that enough was understood about a given reservoir to proceed with development. Companies must implement a staged approach that exposes capital incrementally in a responsible fashion and an assurance process that provides a framework for conducting and reviewing work so that mistakes may be analyzed to influence future decisions. The complete paper provides a work flow for making better decisions about investing in unconventional projects. Introduction In 2019, an analysis of 16,000 unconventional wells operated by 29 of the largest producers in Texas and North Dakota revealed that these companies spent $112 billion more in cash over the past 10 years than they generated from operations. A primary contributor to this shortfall was optimistic production forecasts based on a small number of early wells. These types of projections lead companies to commit to development projects before they understand the true variability in well performance and, most importantly, whether the average well will be commercial (i.e., able to pay for the cost to drill, complete, and tie in). Commercial is defined here as attaining a present value greater than zero at the corporate discount rate. If this is 10%, a net present value (NPV) of zero equates to a 10% rate of return. The Challenge More than 50 shale plays across North America have been tested for their production potential. Of these, only a dozen or so (approximately 25%) have been commercially developed. Thus, the first order of business is to focus on the right play in the right basin. But even within a productive basin, operators need to be in the commercial fairway, which is commonly a fraction of the total basin area regardless of play type. The probability of commercializing a new unconventional play in a frontier basin is low. Although a well can be drilled practically anywhere in the basin and encounter mobile hydrocarbons, this does not reduce the commercial risk relative to conventional plays. Instead, it transfers the risk (threat of fiscal loss) to later stages, in which it must be shown that unconventional wells can produce at sufficient rates, costs can be reduced to make these wells commercially viable, and results are repeatable.


Author(s):  
John K. McSweeney

This chapter quantifies the dynamics of a crossword puzzle by using a network structure to model it. Specifically, the chapter determines how the interaction between the structure of cells in the puzzle and the difficulty of the clues affects the puzzle's solvability. It first builds an iterative stochastic process that exactly describes the solution and obtains its deterministic approximation, which gives a very simple fixed-point equation to solve for the final solution proportion. The chapter then shows via simulation on actual crosswords from the Sunday edition of The New York Times that certain network properties inherent to actual crossword networks are important predictors of the final solution size of the puzzle.


2011 ◽  
Vol 54 (3) ◽  
pp. 464-471
Author(s):  
Tea-Yuan Hwang ◽  
Chin-Yuan Hu

AbstractIn this paper, a fixed point equation of the compound-exponential type distributions is derived, and under some regular conditions, both the existence and uniqueness of this fixed point equation are investigated. A question posed by Pitman and Yor can be partially answered by using our approach.


2019 ◽  
Vol 34 (05) ◽  
pp. 1950027 ◽  
Author(s):  
Oliver J. Rosten

A Legendre transform of the recently discovered conformal fixed-point equation is constructed, providing an unintegrated equation encoding full conformal invariance within the framework of the effective average action.


2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Cui-Xia Li ◽  
Shi-Liang Wu

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like method are presented. The computational efficiency of the modified SOR-like method is better than that of the SOR-like method by some numerical experiments.


2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Haitao Che ◽  
Haibin Chen

In this article, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. Firstly, we transfer the original problem to the constrained multiple-sets split equality problem and a fixed point equation system is established. Then, we show the equivalence of the constrained multiple-sets split equality problem and the fixed point equation system. Secondly, we present a relaxed self-adaptive projection algorithm for the fixed point equation system. The advantage of the self-adaptive step size is that it could be obtained directly from the iterative procedure. Furthermore, we prove the convergence of the proposed algorithm. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.


2019 ◽  
Vol 51 (4) ◽  
pp. 967-993
Author(s):  
Jorge I. González Cázares ◽  
Aleksandar Mijatović ◽  
Gerónimo Uribe Bravo

AbstractWe exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes [27]) and use it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse the performance of the algorithm numerically (the code, written in Julia 1.0, is available on GitHub).


2003 ◽  
Vol 35 (02) ◽  
pp. 377-394 ◽  
Author(s):  
Amke Caliebe

LetT= (T1,T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|&gt;0&lt; ∞ almost surely. We consider the following equation for distributions μ:W≅ ∑j=1∞TjWj, whereW,W1,W2,… have distribution μ andT,W1,W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions forTj≥ 0: essentially under the condition that E ∑j=1∞Tj2log+Tj2&lt; ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.


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