scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals ON THE ORDER OF MAGNITUDE OF SUMS OF NEGATIVE POWERS OF INTEGRATED PROCESSES

2012 ◽  
Vol 29 (3) ◽  
pp. 642-658 ◽  
Author(s):  
Benedikt M. Pötscher
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Integrated Process ◽  
Order Of Magnitude ◽  
Image Position ◽  
Integrated Processes

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

Composite Thermoelectrics - Exact Results and Calculational Methods

1991 ◽  
Vol 234 ◽  
Author(s):  
David J. Bergman ◽  
Ohad Levy
Keyword(s):  
Quality Factor ◽  
Computer Simulations ◽  
Lower Bounds ◽  
Theoretical Study ◽  
Transport Coefficients ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Thermoelectric Transport

ABSTRACTA theoretical study of composite thermoelectric media has resulted in the development of a number of simple approximations, as well as some exact results. The latter include exact upper and lower bounds on the bulk effective thermoelectric transport coefficients of the composite and upper bounds on the bulk effective thermoelectric quality factor Ze. In particular, as a result of some exact theorems and computer simulations we conclude that Ze can never be greater than the largest value of Z in the different components that make up the composite.

scholarly journals Computational Approaches for Stochastic Shortest Path on Succinct MDPs

2018 ◽  
Author(s):  
Krishnendu Chatterjee ◽  
Hongfei Fu ◽  
Amir Goharshady ◽  
Nastaran Okati
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Shortest Path ◽  
Upper Bounds ◽  
Decision Processes ◽  
Upper And Lower Bounds ◽  
Computational Approaches ◽  
Stochastic Shortest Path ◽  
Markov Decision ◽  
Infinite State

We consider the stochastic shortest path (SSP) problem for succinct Markov decision processes (MDPs), where the MDP consists of a set of variables, and a set of nondeterministic rules that update the variables. First, we show that several examples from the AI literature can be modeled as succinct MDPs. Then we present computational approaches for upper and lower bounds for the SSP problem: (a) for computing upper bounds, our method is polynomial-time in the implicit description of the MDP; (b) for lower bounds, we present a polynomial-time (in the size of the implicit description) reduction to quadratic programming. Our approach is applicable even to infinite-state MDPs. Finally, we present experimental results to demonstrate the effectiveness of our approach on several classical examples from the AI literature.

Longest common subsequences of two random sequences

1975 ◽  
Vol 12 (02) ◽  
pp. 306-315 ◽  
Author(s):  
Vacláv Chvátal ◽  
David Sankoff
Keyword(s):  
Monte Carlo ◽  
Molecular Evolution ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Common Subsequence ◽  
Limiting Behaviour

Summary Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n –1 f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

Bounds for the Natural Frequencies of a Simply Supported beam Carrying a Uniformly Distributed Axial Load

1967 ◽  
Vol 9 (2) ◽  
pp. 149-156 ◽  
Author(s):  
G. Fauconneau ◽  
W. M. Laird
Keyword(s):  
Lower Bounds ◽  
Axial Load ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Distributed Load ◽  
Made In

Upper and lower bounds for the eigenvalues of uniform simply supported beams carrying uniformly distributed axial load and constant end load are obtained. The upper bounds were calculated by the Rayleigh-Ritz method, and the lower bounds by a method due to Bazley and Fox. Some results are given in terms of two loading parameters. In most cases the gap between the bounds over their average is less than 1 per cent, except for values of the loading parameters corresponding to the beam near buckling. The results are compared with the eigenvalues of the same beam carrying half of the distributed load lumped at each end. The errors made in the lumping process are very large when the distributed load and the end load are of opposite signs. The results also indicate that the Rayleigh-Ritz upper bounds computed with the eigenfunctions of the unloaded beam as co-ordinate functions are quite accurate.

scholarly journals Some Refinements and Generalizations of I. Schur Type Inequalities

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xian-Ming Gu ◽  
Ting-Zhu Huang ◽  
Wei-Ru Xu ◽  
Hou-Biao Li ◽  
Liang Li ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Recent Literature ◽  
Structural Characteristics ◽  
Variable Parameter ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Schur Inequality

Recently, extensive researches on estimating the value ofehave been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help ofMapleand automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.

Longest common subsequences of two random sequences

1975 ◽  
Vol 12 (2) ◽  
pp. 306-315 ◽  
Author(s):  
Vacláv Chvátal ◽  
David Sankoff
Keyword(s):  
Monte Carlo ◽  
Molecular Evolution ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Common Subsequence ◽  
Limiting Behaviour

SummaryGiven two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n–1f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

On Proportions of Fit Individuals in Population of Mutation-Based Evolutionary Algorithm with Tournament Selection

2018 ◽  
Vol 26 (2) ◽  
pp. 269-297 ◽  
Author(s):  
Anton V. Eremeev
Keyword(s):  
Local Search ◽  
Evolutionary Algorithm ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Set Cover ◽  
Fitness Level ◽  
Level Model ◽  
Tournament Selection ◽  
Tail Bound

In this article, we consider a fitness-level model of a non-elitist mutation-only evolutionary algorithm (EA) with tournament selection. The model provides upper and lower bounds for the expected proportion of the individuals with fitness above given thresholds. In the case of so-called monotone mutation, the obtained bounds imply that increasing the tournament size improves the EA performance. As corollaries, we obtain an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and polynomial upper bounds on the runtime of EAs on the 2-SAT problem and on a family of Set Cover problems proposed by E. Balas.

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