A new risk measure, the Lambda Value-at-Risk (VaR), was proposed from a theoretical point of view as a generalization of the ordinary VaR in the literature. Motivated by the recent developments in risk sharing problems for the VaR and other risk measures, we study the optimization of risk sharing for the Lambda VaR. Explicit formulas of the inf-convolution and sum-optimal allocations are obtained with respect to the left Lambda VaRs, the right Lambda VaRs, or a mixed collection of the left and right Lambda VaRs. The inf-convolution of Lambda VaRs constrained to comonotonic allocations is investigated. Explicit formula for worst-case Lambda VaRs under model uncertainty induced by likelihood ratios is also given.

The Global Financial Crisis of 2007–2008 has elicited various debates, ranging from ethics over the stability of the banking system to subtle technical issues regarding the Gaussian and other copulas. We want to look at the crisis from a particular perspective. Credit derivatives have much in common with treaty reinsurance, including risk transfer via pooling and layering, scarce data, skewed distributions, and a limited number of specialised players in the market. This leads to a special mixture of mathematical/statistical and behavioural challenges. Reinsurers have been struggling to cope with these, not always successfully, but they have learned some lessons over the course of more than one century in business. This has led to certain rules being adopted by the reinsurance market and to a certain mindset being adopted by the individuals working in the industry. Some cultures established in the reinsurance world could possibly inspire markets like the credit derivatives market, but the subtle differences between the two worlds matter. We will see that traditional reinsurance has built-in incentives for (some) fairness, while securitisation can foster opportunism.

Climate risks are systemic risks and may be clustered according to so-called volatilities, uncertainties, complexities, and ambiguities (VUCA) criteria. We analyze climate risk in the VUCA concept and provide a framework that allows to interpret systemic risks as model risk. As climate risks are characterized by deep uncertainties (unknown unknowns), we argue that precautionary and resilient principles should be applied instead of capital-based risk measures (reasonable for known unknows). A prominent example of the proposed principles is the precommitment approach (PCA). Within the PCA, subjective probabilities allow to discriminate between tolerable risks and acceptable ones. The amount of determined solvency capital for acceptable risks and estimations of model risk may be aggregated by means of a multiplier approach. This framework is in line with the three-pillar approach of Solvency II, especially with the recovery and resolution plan. Furthermore, it fits smoothly to a hybrid approach of micro- and macroprudential supervision.

This article presents an innovative adaptive-observer-based scaled four-channel (4-CH) control approach applying damping injection for nonlinear teleoperation systems, which unify the study of robotic dynamic uncertainties, operator/environment force acquirements and asymmetric time-varying delays in the same framework. First, a scaled 4-CH scheme with damping injection is developed to handle time-varying delay while guaranteeing the passivity of communication channels. Then, the improved extended active observer (IEAOB) is deployed to derive the operator/environment force while addressing the issues of measurement noise and model uncertainties. Furthermore, the system stability is analyzed by choosing Lyapunov functional. Finally, the proposed method is validated through simulation.

Patient-specific surgical simulations require the patient-specific identification of the constitutive parameters. The sparsity of the experimental data and the substantial noise in the data (e.g., recovered during surgery) cause considerable uncertainty in the identification. In this exploratory work, parameter uncertainty for incompressible hyperelasticity, often used for soft tissues, is addressed by a probabilistic identification approach based on Bayesian inference. Our study particularly focuses on the uncertainty of the model: we investigate how the identified uncertainties of the constitutive parameters behave when different forms of model uncertainty are considered. The model uncertainty formulations range from uninformative ones to more accurate ones that incorporate more detailed extensions of incompressible hyperelasticity. The study shows that incorporating model uncertainty may improve the results, but this is not guaranteed.

This paper reflects on the notion of partial ambiguity. Using a framework decomposing ambiguity into distinct layers of analysis, among which are risk and model uncertainty, and allowing for different attitudes toward these layers, I show that partial ambiguity may prove less desirable than full ambiguity, even under ambiguity aversion. This observation poses difficulties for interpreting the notion of partial ambiguity in relation to the partial information available to determine the potential compositions of an ambiguous urn. Two Ellsberg-style thought experiments are described to challenge the meaning of partial ambiguity further, and an alternative interpretation, based on a more ambiguous relation, is discussed.

This paper addresses robust stability and position tracking problems in teleoperation systems subject to varying delay in the communication medium, uncertainties in the models of manipulators, and non-passive interaction forces in the terminations. Fixed-structure nonlinear control law is developed based on the notion of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) scheme. Then, utilizing the Lyapunov–Krasovskii theorem, sufficient conditions are derived in terms of Linear Matrix Inequalities (LMIs) to tune the controller parameters. Differently from literature, the objectives are achieved without requirement for any passive parts in the model of interaction forces. Comparative simulations and experimental results demonstrate the applicability and superiority of the proposed method.

This paper presents analytical representations for an optimal insurance contract under distortion risk measure and in the presence of model uncertainty. We incorporate ambiguity aversion and distortion risk measure through the model of Robert and Therond [(2014) ASTIN Bulletin: The Journal of the IAA, 44(2), 277–302.] as per the framework of Klibanoff et al. [(2005) A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.]. Explicit optimal insurance indemnity functions are derived when the decision maker (DM) applies Value-at-Risk as risk measure and is ambiguous about the loss distribution. Our results show that: (1) under model uncertainty, ambiguity aversion results in a distorted probability distribution over the set of possible models with a bias in favor of the model which yields a larger risk; (2) a more ambiguity-averse DM would demand more insurance coverage; (3) for a given budget, uncertainties about the loss distribution result in higher risk level for the DM.

In this paper, a neural network (NN)-based tracking controller is proposed for a general class of type (m,s) wheeled mobile manipulators (WMMs) subjected to model uncertainties with prescribed transient and steady-state performance specifications. First, an input–output model of WMMs is derived by introducing proper output equations. Then, the prescribed performance technique is employed to propose a proportional integral derivative trajectory tracking controller for WMMs to ensure that the tracking errors converge to a smaller, arbitrary ultimate bound with a predefined maximum overshoot/undershoot and convergence speed. The learning capabilities of multilayer NNs are incorporated into the controller to approximate the uncertain nonlinear dynamics of the robot. An adaptive saturation-type controller is utilized to compensate NN estimation errors and external disturbances. A Lyapunov-based stability analysis is used to demonstrate that the tracking errors are uniformly ultimately bounded and converge to a small neighborhood of zero with a guaranteed prescribed performance. Numerical computer simulations are presented to show the effectiveness of the proposed controller.

We considered a pension fund that needs to hedge uncertain long-term liabilities. We modeled the pension fund as a robust investor facing an incomplete market and fearing model uncertainty for the evolution of its liabilities. The robust agent is assumed to minimize the shortfall between the assets and liabilities under an endogenous worst-case scenario by means of solving a min–max robust optimization problem. When the funding ratio is low, robustness reduces the demand for risky assets. However, cherishing the hope of covering the liabilities, a substantial risk exposure is still optimal. A longer investment horizon or a higher funding ratio weakens the investor's fear of model misspecification. If the expected equity return is overestimated, the initial capital requirement for hedging can be decreased by following the robust strategy.

This paper presents latest thinking from the Institute and Faculty of Actuaries’ Model Risk Working Party and follows on from their Phase I work, Model Risk: Daring to Open the Black Box. This is a more practical paper and presents the contributors’ experiences of model risk gained from a wide range of financial and non-financial organisations with suggestions for good practice and proven methods to reduce model risk. After a recap of the Phase I work, examples of model risk communication are given covering communication: to the Board; to the regulator; and to external stakeholders. We present a practical framework for model risk management and quantification with examples of the key actors, processes and cultural challenge. Lessons learned are then presented from other industries that make extensive use of models and include the weather forecasting, software and aerospace industries. Finally, a series of case studies in practical model risk management and mitigation are presented from the contributors’ own experiences covering primarily financial services.

Do quasi-physically based models with more detail perform better than regression or other empirical models? This is a question that was raised many years ago and still remains. In an effort to respond to this question, the author reviews the needs and concerns of users and then divides the large number of models into three classes: (1) screening, (2) research, and (3) planning, monitoring, and assessment. Empirical and causal (physically based) models are contrasted and the advantages and disadvantages of each described. Sources of model uncertainty (properties of data bases, model structure, parameter estimation methods, algorithmic implementation, verification and validation, and future users) that lead to skepticism about models' performance are investigated. Simulation scale and spatial variability are also important considerations. The leading question is then discussed from the perspective of screening, research, and planning, monitoring, and regulatory models.

Taming the beast of uncertainty has been the grand project to which actuaries have dedicated much of their energy and skill over at least the last 50 years – roughly the time since, in Hans Bühlmann's (1989) famous term, “Actuaries of the Second Kind” emerged.

With the increasing use of complex quantitative models in applications throughout the financial world, model risk has become a major concern. Such risk is generated by the potential inaccuracy and inappropriate use of models in business applications, which can lead to substantial financial losses and reputational damage. In this paper, we deal with the management and measurement of model risk. First, a model risk framework is developed, adapting concepts such as risk appetite, monitoring, and mitigation to the particular case of model risk. The usefulness of such a framework for preventing losses associated with model risk is demonstrated through case studies. Second, we investigate the ways in which different ways of using and perceiving models within an organisation both lead to different model risks. We identify four distinct model cultures and argue that in conditions of deep model uncertainty, each of those cultures makes a valuable contribution to model risk governance. Thus, the space of legitimate challenges to models is expanded, such that, in addition to a technical critique, operational and commercial concerns are also addressed. Third, we discuss through the examples of proxy modelling, longevity risk, and investment advice, common methods and challenges for quantifying model risk. Difficulties arise in mapping model errors to actual financial impact. In the case of irreducible model uncertainty, it is necessary to employ a variety of measurement approaches, based on statistical inference, fitting multiple models, and stress and scenario analysis.

We use robust control to study how a central bank in an economy with imperfect interest rate pass-through conducts monetary policy if it fears that its model could be misspecified. We find that, first, whether robust optimal monetary policy under commitment responds more cautiously or more aggressively depends crucially on the source of shock. Imperfect pass-through amplifies the robust policy. Second, if the central bank is concerned about uncertainty, it dampens volatility in the inflation rate preemptively but accepts higher volatility in the output gap and loan rate. However, for highly sticky loan rates, insurance against model misspecification becomes particularly pricy. Third, if the central bank fears uncertainty only in the IS equation or the loan rate equation, the robust policy shifts its concern for stabilization away from inflation.

Using a macroeconomic model with asset prices, we analyze how optimal monetary policy and macroeconomic dynamics and performance are affected by a central bank's desire to be robust against model misspecifications. We show that a higher central bank preference for robustness implies a more aggressive reaction of the nominal interest rate to the expected future inflation rate and inflation shocks. The dynamic stability of the equilibrium is not modified for a sufficiently high preference for robustness. However, the speed of dynamic convergence is lower under robust control compared to a benchmark case without it and implies supplementary economic costs. Finally, an increase in the preference for robustness comes at the cost of higher macroeconomic and financial volatility in the presence of inflation shocks. It has no effect on the reaction of inflation, output gap, and asset price gap to shocks affecting goods and financial markets.

The use of generalized linear models (GLM) to estimate claims reserves has become a standard method in insurance. Most frequently, the exponential dispersion family (EDF) is used; see e.g. England, Verrall. We study the so-called Tweedie EDF and test the sensitivity of the claims reserves and their mean square error of predictions (MSEP) over this family. Furthermore, we develop second order Taylor approximations for the claims reserves and the MSEPs for members of the Tweedie family that are difficult to obtain in practice, but are close enough to models for which claims reserves and MSEP estimations are easy to determine. As a result of multiple case studies, we find that claims reserves estimation is relatively insensitive to which distribution is chosen amongst the Tweedie family, in contrast to the MSEP, which varies widely.

In this paper we examine the claims reserving problem using Tweedie's compound Poisson model. We develop the maximum likelihood and Bayesian Markov chain Monte Carlo simulation approaches to fit the model and then compare the estimated models under different scenarios. The key point we demonstrate relates to the comparison of reserving quantities with and without model uncertainty incorporated into the prediction. We consider both the model selection problem and the model averaging solutions for the predicted reserves. As a part of this process we also consider the sub problem of variable selection to obtain a parsimonious representation of the model being fitted.