Optimal strategies in a risk selection investment model

We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

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Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

Discrete Dynamics in Nature and Society ◽

10.1155/2012/802518 ◽

2012 ◽

Vol 2012 ◽

pp. 1-26 ◽

Cited By ~ 1

Author(s):

Yan Li ◽

Guoxin Liu

Keyword(s):

Ruin Probability ◽

Risk Model ◽

Optimal Strategies ◽

Proportional Reinsurance ◽

Two Dimensional ◽

Compound Poisson ◽

Compound Poisson Risk Model ◽

Optimal Value ◽

Hamilton Jacobi Bellman ◽

Poisson Risk Model

We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.

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Best Proximity Point Results in Complex Valued Metric Spaces

International Journal of Analysis ◽

10.1155/2014/827862 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Binayak S. Choudhury ◽

Nikhilesh Metiya ◽

Pranati Maity

Keyword(s):

Minimum Distance ◽

Metric Spaces ◽

Optimal Solution ◽

Global Optimal Solution ◽

Complex Valued Metric Space ◽

Fixed Point Equation ◽

Point Equation ◽

Global Optimal ◽

Metric Function ◽

Complex Valued

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples.

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Effect of recess depth on lubrication performance of annular recess hydrostatic thrust bearing by constant rate flow

Industrial Lubrication and Tribology ◽

10.1108/ilt-03-2017-0066 ◽

2018 ◽

Vol 70 (1) ◽

pp. 68-75 ◽

Cited By ~ 4

Author(s):

Jun-peng Shao ◽

Guang-dong Liu ◽

Xiao-dong Yu ◽

Yan-qin Zhang ◽

Xiu-li Meng ◽

...

Keyword(s):

Thrust Bearing ◽

Simulation Method ◽

Content Type ◽

Oil Film ◽

Constant Rate ◽

Lubrication Performance ◽

Optimal Value ◽

Improved Performance ◽

Lubrication Characteristics ◽

Volume Method

Purpose The purpose of this paper is to describe a simulation and experimental research concerning the effect of recess depth on the lubrication performance of a hydrostatic thrust bearing by constant rate flow. Design/methodology/approach The computational fluid dynamics and finite volume method have been used to compute the lubrication characteristics of an annular recess hydrostatic thrust bearing with different recess depths. The performances are oil recess pressure, oil recess temperature and oil film velocity. The recess depth has been optimized. A test rig is established for testing the pressure field of the structure of hydrostatic thrust bearing after recess depth optimization, and experimental results show that experimental data are basically identical with the simulation results, which demonstrates the validity of the proposed numerical simulation method. Findings The results demonstrate that the oil film temperature decreases and the oil film pressure first increases and then decreases with an increase in the recess depth, but oil film velocity is constant. To sum up comprehensive lubrication performance, the recess depth of 3.5 mm is its optimal value for the annular recess hydrostatic thrust bearing. Originality/value The computed results indicate that to get an improved performance from a constant flow hydrostatic thrust bearing, a proper selection of the recess depth is essential.

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A Model for Integrated Inventory and Assortment Planning

Management Science ◽

10.1287/mnsc.2021.4149 ◽

2021 ◽

Author(s):

Victor Martínez-de-Albéniz ◽

Sumit Kunnumkal

Keyword(s):

Closed Form Solution ◽

Optimal Solution ◽

Integer Program ◽

Form Solution ◽

Assortment Planning ◽

Dimensional Parameter ◽

Planning Decisions ◽

Fixed Point Equation ◽

Point Equation ◽

Random Instances

Integrating inventory and assortment planning decisions is a challenging task that requires comparing the value of demand expansion through broader choice for consumers with the value of higher in-stock availability. We develop a stockout-based substitution model for trading off these values in a setting with inventory replenishment, a feature missing in the literature. Using the closed form solution for the single-product case, we develop an accurate approximation for the multiproduct case. This approximated formulation allows us to optimize inventory decisions by solving a fractional integer program with a fixed point equation constraint. When products have equal margins, we solve the integer program exactly by bisection over a one-dimensional parameter. In contrast, when products have different margins, we propose a fractional relaxation that we can also solve by bisection and that results in near-optimal solutions. Overall, our approach provides solutions within 0.1% of the optimal policy and finds the optimal solution in 80% of the random instances we generate. This paper was accepted by David Simchi-Levi, optimization.

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Analysis of Crossword Puzzle Difficulty Using a Random Graph Process

The Mathematics of Various Entertaining Subjects ◽

10.23943/princeton/9780691164038.003.0008 ◽

2015 ◽

Author(s):

John K. McSweeney

Keyword(s):

New York ◽

Stochastic Process ◽

Fixed Point ◽

Network Structure ◽

New York Times ◽

Final Solution ◽

Crossword Puzzle ◽

Fixed Point Equation ◽

Point Equation ◽

Network Properties

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.

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A Characterization of the Compound-Exponential Type Distributions

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-086-7 ◽

2011 ◽

Vol 54 (3) ◽

pp. 464-471

Author(s):

Tea-Yuan Hwang ◽

Chin-Yuan Hu

Keyword(s):

Fixed Point ◽

Exponential Type ◽

Existence And Uniqueness ◽

Fixed Point Equation ◽

Point Equation

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.

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A conformal fixed-point equation for the effective average action

International Journal of Modern Physics A ◽

10.1142/s0217751x19500271 ◽

2019 ◽

Vol 34 (05) ◽

pp. 1950027 ◽

Cited By ~ 5

Author(s):

Oliver J. Rosten

Keyword(s):

Fixed Point ◽

Conformal Invariance ◽

Legendre Transform ◽

Fixed Point Equation ◽

Average Action ◽

Point Equation

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.

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Modified SOR-Like Method for Absolute Value Equations

Mathematical Problems in Engineering ◽

10.1155/2020/9231639 ◽

2020 ◽

Vol 2020 ◽

pp. 1-6

Author(s):

Cui-Xia Li ◽

Shi-Liang Wu

Keyword(s):

Fixed Point ◽

Nonlinear Equation ◽

Computational Efficiency ◽

Numerical Experiments ◽

Absolute Value Equations ◽

Convergence Conditions ◽

Absolute Value ◽

Fixed Point Equation ◽

Point Equation ◽

Better Than

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.

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A Relaxed Self-Adaptive Projection Algorithm for Solving the Multiple-Sets Split Equality Problem

Journal of Function Spaces ◽

10.1155/2020/6183214 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Haitao Che ◽

Haibin Chen

Keyword(s):

Fixed Point ◽

Equation System ◽

Projection Algorithm ◽

Step Size ◽

Split Equality Problem ◽

Fixed Point Equation ◽

Point Equation ◽

Equality Problem ◽

Adaptive Step ◽

Self Adaptive

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.

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