Randomized controlled clinical trials provide the gold standard for evidence generation in relation to the efficacy of a new treatment in clinical research. Relevant information from previous studies may be desirable to incorporate in the design and analysis of a new trial, with the Bayesian paradigm providing a coherent framework to formally incorporate prior knowledge. Many established methods involve the use of a discounting factor, sometimes related to a measure of 'similarity' between historical and the new trials. However, it is often the case that the sample size is highly nonlinear in those discounting factors. This hinders communication with subject-matter experts to elicit sensible values for borrowing strength at the trial design stage. Focussing on a method that can incorporate historical data from multiple sources, we highlight a particular issue of nonmonotonicity and explain why this causes issues with interpretability of discounting factors (hereafter referred to as 'weights'). We propose a solution from which an analytical sample size formula is derived. We then propose a linearization technique such that the sample size changes uniformly over the weights. This leads to interpretable weights (as a percentage of information to borrow/discount) which could facilitate easier elicitation of expert opinion on their values.

Optimal Strategies in Controlled Markov Chains Under a Stochastic Discount Factor with Applications to Investment Models

We derive sufficient optimality conditions for an infinite horizon optimal control problem with a vanishing discounting factor and demonstrate the obtained results by examples.

Abstract We utilize Bayesian model averaging to estimate a stochastic discount factor (SDF) for single-stock options. A Bayesian model averaging SDF outperforms reduced-form benchmark models in-sample and out-of-sample in pricing option return anomalies and portfolios. We document that the SDF is dense in characteristics with the implied-realized volatility spread, option return momentum, and jump risk emerging as the most likely included factors. The option SDF exhibits a distinct business cycle pattern and aligns more closely with its counterpart in the stock market than in the bond market.

ABSTRACTThis article is interested in future allocations of scarce resources in an environment where upper bounds and lower bounds are fixed on the stream of consumptions or extractions of the scarce resource. It is shown that the optimal planning of consumptions does not depend on the sequence of discounting factors as soon as they are decreasing at a rate smaller than a bound linked to the concavity of the utility function and the choice of the sequences of lower and upper bounds. The optimal solution is unique and exhibits two regimes with a pivotal period in the middle. Therefore, one gets plans satisfying some kind of intergenerational fairness: the upper bounds prevent the first generations from consuming too much of the resource to the detriment of future generations and the lower bounds ensure minimal consumption for these generations. We also consider the role of the horizon and of a potential regret after a revision for the bounds. The argument is then extended to partially renewable resources.

Abstract We provide an axiomatic approach to the allocation of responsibility for GHG emissions in supply chains. Considering a set of axioms standardly used in networks and decision theory, and consistent with legal principles underlying responsibility, we show that responsibility measures shall be based on exponential discounting of upstream and downstream emissions. From a network theory perspective, the proposed responsibility measure corresponds to a convex combination of the Bonacich centralities for the upstream and downstream weighted adjacency matrices. Scope 1 emissions, consumption-based accounting and income-based accounting are obtained as particular cases of our approach, which also gives a precise meaning to scope 3 emissions while avoiding double-counting. We apply our approach to the assessment of country-level responsibility for global GHG emissions and to sector-level responsibility in the USA. We examine how the responsibility of countries/sectors varies with the discounting of indirect emissions. We identify three groups of countries/sectors: producers of emissions whose responsibility decreases with the discounting factor, consumers of emissions whose responsibility increases with the discounting factor, and an intermediary group whose responsibility mostly depends on the network position and varies non-monotonically with the discounting factor. Overall, our axiomatic approach provides strong normative foundations for the definition of reporting requirements for indirect emissions and for the allocation of responsibility in claims for climate-related loss and damage.

Discounting the influence of future events is a key paradigm in economics and it is widely used in computer-science models, such as games, Markov decision processes (MDPs), reinforcement learning, and automata. While a single game or MDP may allow for several different discount factors, nondeterministic discounted-sum automata (NDAs) were only studied with respect to a single discount factor. It is known that every class of NDAs with an integer as the discount factor has good computational properties: It is closed under determinization and under the algebraic operations min, max, addition, and subtraction, and there are algorithms for its basic decision problems, such as automata equivalence and containment. Extending the integer discount factor to an arbitrary rational number, loses most of these good properties. We define and analyze nondeterministic discounted-sum automata in which each transition can have a different integral discount factor (integral NMDAs). We show that integral NMDAs with an arbitrary choice of discount factors are not closed under determinization and under algebraic operations and that their containment problem is undecidable. We then define and analyze a restricted class of integral NMDAs, which we call tidy NMDAs, in which the choice of discount factors depends on the prefix of the word read so far. Among their special cases are NMDAs that correlate discount factors to actions (alphabet letters) or to the elapsed time. We show that for every function $\theta$ that defines the choice of discount factors, the class of $\theta$-NMDAs enjoys all of the above good properties of NDAs with a single integral discount factor, as well as the same complexity of the required decision problems. Tidy NMDAs are also as expressive as deterministic integral NMDAs with an arbitrary choice of discount factors.Comment: arXiv admin note: text overlap with arXiv:2301.04086

ABSTRACTWe consider European option pricing when the volatility of the underlying stock is stochastic and affected by economic cycles. We further assume that market liquidity risks have a significant impact on the price of the stock that is not negligible, and stock prices should be adjusted according to a liquidity discounting factor. For the purpose of option pricing, we transform the established model dynamics under the physical measure into those under a risk‐neutral measure, which forms a foundation in the subsequent closed‐form derivation of the characteristic function. An analytical option pricing formula is then obtained, and numerical tests together with sensitivity analysis are also performed. Through an empirical analysis, we demonstrate that our model, which incorporates stochastic liquidity, significantly outperforms the version with constant liquidity.

This paper presents an innovative approach to financial market modelling by integrating fuzzy discount factors into the decision-making process, thereby reflecting the complexities of human behaviour. Traditional financial models often fail to account for market dynamics’ psychological factors. The proposed method utilizes fuzzy logic to encapsulate the uncertainty and subjective judgment inherent in financial decisions. By representing financial variables as fuzzy numbers, the model better simulates the way humans assess information and make decisions under uncertainty. The incorporation of fuzzy discount factors marks a significant shift from deterministic to a more realistic representation of financial markets, suitable for practical application. This methodology offers a nuanced investment strategy that balances theoretical rigour with real-world applicability, appealing to a broad spectrum of investors. The aim of the following paper is to introduce an alternative to price modelling with the use of fuzzy return rates, which results in some errors in the mathematical model. The solution has the form of introducing fuzzy discount factors (FDFs) that retain the advantages of the fuzzy approach (e.g., encompassing subjectivity and imprecision) while preserving the shape of the fuzzy number modelling a price.

The grocery stores usually sell the deterioration (perishable) products. One example of a deterioration product is 1-liter packaged milk product. Often when approaching or even reaching the expiration date, those products still cannot be sold out but with the discount offered, it is hope that those products will be sold more. It is necessary to determine the time for giving the discount and the discount rate that give the optimal net profit. This study develops a deterioration product inventory model by considering the expiration factor and the discount factor. By providing the required parameter values ​​in the model, a simulation was carried out, and it can be determined the level of inventory, the time of the discount offered, and the amount of the discount offered, which results in an optimal net profit.

Machine learning private equity returns

Vulnerable options with regime switching and stochastic liquidity

This research aims to analyze the feasibility of adding one Sunward brand SWE210 type excavator unit at PT Mahakam Jaya Perkasa in Sangasanga, East Kalimantan. This research uses quantitative method with field techniques and literary technique as data collection technique. The sampling uses financial reports, such as balance sheets, profit and loss, and cash flow. The data analysis techniques use are Net Present Value (NPV), Internal Rate of Return (IRR), Profitability Index (PI), and Payback Period (PP). The research results use the Net Present Value (NPV) method shows that the total PV Cash Flow is IDR 1.765.401.658. At the same time, the amount of investment proposed by the company is IDR 1.100.000.000 and the resulting Net Present Value is IDR 665.401.658, with Internal Rate of Return (IRR) amounting to 17,853% greater than the Discount Factor value of 17%, the Profitability Index (PI) value is 1,60 which can be said to be greater than 1. The Payback Period (PP) has a result of 7 months with a long economic investment period of 5 years. So, it can be said that the investment is feasible.

Output feedback fault-tolerant Q-learning for discrete-time linear systems with actuator faults

Application of Standardized Tissue Repair Discount Factors to Guide Spinal Cord Dose in Reirradiation

This paper studies constrained Markov decision processes under the total expected discounted cost optimality criterion, with a state-action dependent discount factor that may take any value between zero and one. Both the state and the action space are assumed to be Borel spaces. By using the linear programming approach, consisting in stating the control problem as a linear problem on a set of occupation measures, we show the existence of an optimal stationary Markov policy. Our results are based on the study of both weak-strong topologies in the space of occupation measures and Young measures in the space of Markov policies.

Abstract We propose estimators of the stochastic discount factor using large cross-sections of individual stocks. We introduce a short time-block structure on a large N, T panel to exploit unbalanced panels of individual stock returns and suggest a novel bias correction to achieve the consistency of our estimators. Our estimators can accommodate pre-specified traded and nontraded factors, and latent factors. The estimators perform well in simulations. We apply our estimators to return data for U.S. individual stocks over a 50-year sample period and identify those factors in popular asset pricing models that command significant premia. A number of proposed nontraded factors have insignificant risk premia. Contrary to many studies, we find the market factor has a significant premium, as do profitability, value, and momentum factors.

Testing the boundaries of applicability of standard Stochastic Discount Factor models

Federated learning (FL) in mobile edge computing has emerged as a promising machine-learning paradigm in the Internet of Things, enabling distributed training without exposing private data. It allows multiple mobile devices (MDs) to collaboratively create a global model. FL not only addresses the issue of private data exposure but also alleviates the burden on a centralized server, which is common in conventional centralized learning. However, a critical issue in FL is the imposed computing for local training on multiple MDs, which often have limited computing capabilities. This limitation poses a challenge for MDs to actively contribute to the training process. To tackle this problem, this paper proposes an adaptive dataset management (ADM) scheme, aiming to reduce the burden of local training on MDs. Through an empirical study on the influence of dataset size on accuracy improvement over communication rounds, we confirm that the amount of dataset has a reduced impact on accuracy gain. Based on this finding, we introduce a discount factor that represents the reduced impact of the size of the dataset on the accuracy gain over communication rounds. To address the ADM problem, which involves determining how much the dataset should be reduced over classes while considering both the proposed discounting factor and Kullback-Leibler divergence (KLD), a theoretical framework is presented. The ADM problem is a non-convex optimization problem. To solve it, we propose a greedy-based heuristic algorithm that determines a suboptimal solution with low complexity. Simulation results demonstrate that our proposed scheme effectively alleviates the training burden on MDs while maintaining acceptable training accuracy.

Abstract This paper analyzes the effect of price leadership on collusion among firms with different discount factors. We first find that price leadership relaxes the incentive constraints for collusion. We then derive a dynamic collusion path in which the firms with lower discount factors initially occupy the largest market share and then gradually cede it to the firms with higher discount factors. This collusion path is shaped by the conflicting forces of fairness and efficiency. Additionally, price leadership can restore the efficiency implied by differentiated time preferences in repeated games.