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In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of three monotone operators are analyzed. The framework modifies the forward–backward–forward dynamical system by adding a cocoercive operator to the inclusion. The existence, uniqueness, and weak asymptotic convergence of the generated trajectories are discussed. A variable metric forward–backward–half forward dynamical system with the essence of non-self-adjoint linear operators is introduced. The proposed notion, in turn, extends the forward–backward–forward dynamical system and forward–backward dynamical system in the framework of variable metric by relaxing some conditions on the metrics. The distributed dynamical system is further explored to compute a generalized Nash equilibrium in a monotone game as an application. A numerical example is provided to illustrate the convergence of trajectories.

In this work, we introduce two Bregman projection algorithms with self-adaptive stepsize for solving pseudo-monotone variational inequality problem in a Hilbert space. The weak and strong convergence theorems are established without the prior knowledge of Lipschitz constant of the cost operator. The convergence behavior of the proposed algorithms with various functions of the Bregman distance are presented. More so, the performance and efficiency of our methods are compared to other related methods in the literature.

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a positive weight. We give a variational principle for limit shapes in this setting, and show that the resulting Euler-Lagrange equation can be integrated, giving explicit limit shapes parameterized by analytic functions., 37 pages, 21 figures

We consider the propagation dynamics of a general heterogeneous reaction-diffusion system under a shifting environment. By developing the fixed-point theory for second order non-autonomous differential system and constructing appropriate upper and lower solutions, we show there exists a nondecreasing wave front with the speed consistent with the habitat shifting speed. We further show the uniqueness of forced waves by the sliding method and some analytical skills, and we obtain the global stability of forced waves by applying the dynamical systems approach. Moreover, we establish the spreading speed of the system by appealing to the abstract theory of monotone semiflows. Applications and numerical simulations are also given to illustrate the analytical results.

Assume that the function $f:[0,\infty )\rightarrow \mathbb {R}$ is operator monotone in $[0,\infty )$ . We can define the perspective $\mathcal {P}_{f}\left (B,A\right ) $ by setting $$ \mathcal{P}_{f}\left( B,A\right) :=A^{1/2}f\left( A^{-1/2}BA^{-1/2}\right) A^{1/2}, $$ where A, B > 0. In this paper, we show among others that, if σ ≥ C ≥ ρ > 0, D > 0, ς ≥ Q ≥ τ > 0 and 0 < n ≤ D − C ≤ N for some constants ρ, σ, ς, τ, n, N, then $$ \begin{array}{@{}rcl@{}} 0& \le& \frac{n}{N{\varsigma}^{2}}\left[ \mathcal{P}_{f}\left( {\varsigma} ,N+\sigma \right) -\mathcal{P}_{f}\left( {\varsigma} ,\sigma \right) \right] Q^{2} \\ & \leq& \mathcal{P}_{f}\left( Q,D\right) -\mathcal{P}_{f}\left( Q,C\right) \\ & \leq& \frac{N}{n\tau^{2}}\left[ \mathcal{P}_{f}\left( \tau ,n+\rho \right) -\mathcal{P}_{f}\left( \tau ,\rho \right) \right] Q^{2}. \end{array} $$ Applications for the weighted operator geometric mean and the perspective $$ \mathcal{P}_{\ln \left( \cdot +1\right) }\left( B,A\right) :=A^{1/2}\ln \left( A^{-1/2}BA^{-1/2}+1\right) A^{1/2},~ A,B>0 $$ are also provided.

We generalize the disutility of effort function in the linear-Constant Absolute Risk Aversion (CARA) pure moral hazard model. We assume that agents are heterogeneous in ability. Each agent’s ability is observable and treated as a parameter that indexes the disutility of effort associated with the task performed. In opposition to the literature (the “traditional” scenario), we find a new, “novel” scenario, in which a high-ability agent may be offered a weaker incentive contract than a low-ability one, but works harder. We characterize the conditions for the existence of these two scenarios: formally, the “traditional” (“novel”) scenario occurs if and only if the marginal rate of substitution of the marginal disutility of effort function is increasing (decreasing) in effort when evaluated at the second-best effort. If, further, this condition holds for all parameter values and matching is endogenous, less (more) talented agents work for principals with riskier projects in equilibrium. This implies that the indirect and total effects of risk on incentives are negative under monotone assortative matching.

The paper presents two inertial viscosity-type extragradient algorithms for finding a common solution of the variational inequality problem involving a monotone and Lipschitz continuous operator and of the fixed point problem with a demicontractive mapping in real Hilbert spaces. Our algorithms use a simple step size rule which is generated by some calculations at each iteration. Two strong convergence theorems are obtained without the prior knowledge of the Lipschitz constant of the operator. The numerical behaviors of the proposed algorithms in some numerical experiments are reported and compared with previously known ones.

In this paper, hybrid best proximity points for a class of proximally mixed monotone set-valued mappings are discussed. In order to get rid of the fetter of continuity, we introduce the proximally ...

In many practical applications, a multivariate outcome is measured on two or more groups of subjects. When these variables are analyzed separately, the information is not fully exhausted, as possib...

We introduce a flexible lifetime distribution called Burr III-Inverse Weibull (BIII-IW). The new proposed distribution has well-known sub-models. The BIII-IW density function includes exponential, left-skewed, right-skewed and symmetrical shapes. The BIII-IW model's failure rate can be monotone and non-monotone depending on the parameter values. To show the importance of the BIII-IW distribution, we establish various mathematical properties such as random number generator, ordinary moments, conditional moments, residual life functions, reliability measures and characterizations. We address the maximum likelihood estimates (MLE) for the BIII-IW parameters and estimate the precision of the maximum likelihood estimators via a simulation study. We consider applications to two COVID-19 data sets to illustrate the potential of the BIII-IW model.

Given a set V , the problem of unconstrained submodular maximization with modular costs (USM-MC) asks for a subset S ⊆ V that maximizes f ( S ) - c ( S ), where f is a non-negative, monotone, and submodular function that gauges the utility of S , and c is a non-negative and modular function that measures the cost of S. This problem finds applications in numerous practical scenarios, such as profit maximization in viral marketing on social media. This paper presents ROI-Greedy, a polynomial time algorithm for USM-MC that returns a solution S satisfying [EQUATION], where S * is the optimal solution to USM-MC. To our knowledge, ROI-Greedy is the first algorithm that provides such a strong approximation guarantee. In addition, we show that this worst-case guarantee is tight , in the sense that no polynomial time algorithm can ensure [EQUATION], for any ϵ > 0. Further, we devise a non-trivial extension of ROI-Greedy to solve the profit maximization problem, where the precise value of f ( S ) for any set S is unknown and can only be approximated via sampling. Extensive experiments on benchmark datasets demonstrate that ROI-Greedy significantly outperforms competing methods in terms of the tradeoff between efficiency and solution quality.

In this note, we propose a modification as an alternative to remove the assumption of uniformly monotone as assumed in Aliyu et al. (2019). Numerical experiment shows that the modification improves the numerical performance of the method.

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality., submitted, 34 pages

In this paper, a new derivative-free approach for solving nonlinear monotone system of equations with convex constraints is proposed. The search direction of the proposed algorithm is derived based on the modified scaled Davidon-Fletcher-Powell (DFP) updating formula in such a way that it is sufficiently descent. Under some mild assumptions, the search direction is shown to be bounded. Subsequently, the convergence result of the proposed method is established. The performance of the proposed algorithm on a collection of some test problems as well as signal recovery problems is demonstrated in comparison with some existing algorithms with similar characteristics. The results of the numerical experiments confirm the efficiency as well as the robustness of the proposed algorithm by comparing it with some existing methods in the literature.

In this paper, in the setting of Hadamard spaces, a iterative scheme is proposed for approximating a solution of the inclusion problem for a finite family of monotone operators which is a unique solution of a variational inequality. Some applications in convex minimization and fixed point theory are also presented to support the main result.

In this paper, we propose three types of absolute continuity for monotone measures and present some of their basic properties. By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. The Egoroff theorem, the Riesz theorem and the Lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case. These three generalized convergence theorems include as special cases several previous versions of Egoroff-like theorem, Riesz-like theorem and Lebesgue-like theorem for monotone measures, respectively.

In this paper, we study an impulsive reaction-advection-diffusion model given by a reaction-advection-diffusion equation with the Allee effect in high-dimensional space composed with a discrete-time map. We establish the existence, uniqueness up to translation, and global stability of bistable traveling waves for impulsive reaction-advection-diffusion models. The methods involve the monotone semiflow approach, the upper and lower solutions, and spreading speeds of monostable systems. Numerical simulations verify the correctness of our conclusions.

We further investigate (M)-property, an important structure characteristic of monotone measures. Some necessary and/or sufficient conditions of (M)-property are shown and some characteristics of (M)-property are described. It is shown that the restriction of a monotone measure to an atom of its own possesses (M)-property on this atom. By means of the (M)-property we characterize the equivalence among the Choquet integral, the pan-integral, the concave integral, and the Shilkret integral on atoms of monotone measures and obtain some special results. We also show a necessary and sufficient condition for the concave integral and the pan-integral to coincide on atoms of monotone measures.

This paper investigates the existence of a nondegenerate pure-strategy Bayesian-Nash equilibrium in a double auction between one seller and several heterogeneous buyers in the independent private value framework. It begins with three examples describing some of the model’s particular features. After studying a number of continuity-related properties of the interim payoff functions, we construct a sequence of strategy profiles that converges to a nondegenerate monotone Bayesian-Nash equilibrium of the game under some strict monotonicity conditions. The equilibrium existence result is applied to a double auction with risk-averse bidders.

We consider a weighted initial value problem for first-order functional differential equations with monotone non-linearities which may have a non-integrable singularity with respect to the independent variable. Conditions for the existence of solutions in a suitable weighted space of locally absolutely continuous functions are given.

In this paper, we introduce two algorithms for finding a common solution of the monotone inclusion problem and the fixed point problem for a relatively nonexpansive mapping in reflexive Banach spaces. The weak convergence results for both algorithms are established without the prior knowledge of the Lipschitz constant of the mapping. An application to the variational inequality problem is considered. Finally, some numerical experiments of the proposed algorithms including comparisons with other algorithms are provided.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

For solving pseudo-monotone variational inequality problems in Hilbert spaces, we introduce a new algorithm with self-adaptive technique. The algorithm is based on the projection method and inertia...

In this paper, we consider reflected three-operator splitting methods for monotone inclusion problems in real Hilbert spaces. To do this, we first obtain weak convergence analysis and nonasymptotic...

In this article, the notions of partially continuous mappings, partially bounded, partially compact and partially closed sets of an ordered E-metric space are defined. These notions are used to introduce partial E-measure of noncompactness and prove the fixed point results of monotone mappings and sum of two monotone mappings. In this way we generalize many results including the well known results of Schauder, Darbo and Krasnoselskii, in the settings of ordered E-metric and ordered Banach spaces. We also provide nontrivial examples and existence results for a class of integral equations to validate the significance of our theory and results.

We propose a method of constructing convergent high order schemes for Hamilton-Jacobi equations on triangular meshes, which is based on combining a high order scheme with a first order monotone scheme. According to this methodology, we construct adaptive schemes of weighted essentially non-oscillatory type on triangular meshes for non-convex Hamilton-Jacobi equations in which the first order monotone approximations are occasionally applied near singular points of the solution (discontinuities of the derivative) instead of weighted essentially non-oscillatory approximations. Through detailed numerical experiments, the convergence and effectiveness of the proposed adaptive schemes are demonstrated.

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences $$\{f_{n}\}_{n=1}^{\infty}$$ and $$\{g_{n}\}_{n=1}^{\infty}$$ such that (1) $$f_{n}\leq f_{n+1}\leq f$$ and $$f\leq g_{n+1}\leq g_{n}$$ for all integers $$n \geq 1$$ ; (2) $$f_{n}$$ and $$g_{n}$$ are continuous and Gâteaux differentiable on $$X$$ ; (3) $$f_n \to f$$ and $${g_n \to f}$$ uniformly on $$X$$ ; (4) $$f_n$$ and $$g_n$$ are coercive and b-Lipschitzian. Moreover, we also prove that if $$X^{*}$$ is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists $$m>0$$ such that $$|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|$$ and $$|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|$$ whenever $$x, y \in X$$ and $$n\in N$$ .

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

The aim of this work is to study a stochastic obstacle problem governed by a T-monotone operator, a random force and a multiplicative stochastic reaction in the frame of Sobolev spaces. After proving a result of existence and uniqueness of the variational solution, by using an ad hoc perturbation of the stochastic reaction and a penalization of the constraint, we prove Lewy–Stampacchia’s inequalities associated with the problem finally.

We consider the first boundary value problem for quasilinear parabolic differential–difference equations. The uniqueness and existence of generalized solutions are proved. The proofs are based on the theory of linear differential–difference equations and the theory of monotone (pseudomonotone) operators.

Properties of Menger-monotone sets are studied. It is proved that all boundedly weakly compact Menger-connected $$(\omega\rhd n)$$ -approximatively compact sets are suns. The existence of a continuous selection of the Chebyshev near-center map (relative to $$V$$ ) is proved for the case in which $$V\subset C(Q)$$ is a $$B^2$$ -infinitely connected set in the space $$C(Q)$$ .

We characterize the three-dimensional Banach spaces in which any Chebyshev set is monotone path-connected. Namely, we show that in a three-dimensional space each Chebyshev set is monotone path- connected if and only if one of the following two conditions is satisfied: any exposed point of the unit sphere of is a smooth point or (that is, the unit sphere of is a cylinder). Bibliography: 17 titles.

In this paper, we study N -player Colonel Blotto games with incomplete information about battlefield valuations. Such games arise in job markets, research and development, electoral competition, security analysis, and conflict resolution. For M ≥ N + 1 battlefields, we identify a Bayes-Nash equilibrium in which the resource allocation to a given battlefield is strictly monotone in the valuation of that battlefield. We also explore extensions such as heterogeneous budgets, the case M ≤ N , full-support type distributions, and network games.

Continuing the first part of this work, we study an implicit evolution inclusion with time-dependent maximal monotone operator in a separable Hilbert space. This inclusion involves some perturbation given by a multivalued history-dependent operator. We establish the existence of a solution. The solution is unique provided that the values of the multivalued history-dependent operator are singletons. We derive an explicit evolution inclusion resolved for the derivative with the same solution set. The results are applied to the implicit sweeping process generated by a moving closed convex set. All results are new since the implicit evolution inclusions with maximal monotone operators of general form have not been studied yet. The perturbing multivalued history-dependent operator, addressed in the first part, adds novelty too. As for the implicit sweeping processes, the main available results are particular cases of ours.

We define a topological characteristic, the fixed point index with respect to a convex closed subset of a Banach space for a class of completely fundamentally restrictible multivalued maps which can be represented as a composition of maps with aspheric values. This class includes, in particular, maps which are condensing with respect to a monotone nonsingular measure of noncompactness. Maps of this type naturally arise in the study of nonlinear systems with impulse effects. Applications of the index to some fixed point theorems are considered.

The paper refines the currently known estimates of sums of sine series with monotone coefficients, as well as the upper bound for the sum of the cosine series with convex coefficients.

The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral operators on the cones, associate norms over some cones in Lebesgue spaces, the norms of the dilation operator and embedding operators on weighted Lorentz spaces with general weights. Under some more general conditions, we present order sharp estimates for Hardy-type operators on the cones.

Many problems in engineering and social sciences can be transformed into system of nonlinear equations. As a result, a lot of methods have been proposed for solving the system. Some of the classical methods include Newton and Quasi Newton methods which have rapid convergence from good initial points but unable to deal with large scale problems due to the computation of Jacobian matrix or its approximation. Spectral and conjugate gradient methods proposed for unconstrained optimization, and later on extended to solve nonlinear equations do not require any computation of Jacobian matrix or its approximation, thus, are suitable to handle large scale problems. In this paper, we proposed a spectral conjugate gradient algorithm for solving system of nonlinear equations where the operator under consideration is monotone. The search direction of the proposed algorithm is constructed by taking the convex combination of the Dai-Yuan (DY) parameter and a modified conjugate descent (CD) parameter. The proposed search direction is sufficiently descent and under some suitable assumptions, the global convergence of the proposed algorithm is proved. Numerical experiments on some test problems are presented to show the efficiency of the proposed algorithm in comparison with an existing one. Finally, the algorithm is successfully applied in signal recovery problem arising from compressive sensing.

The study of alternation in two-way finite automata ( 2 fa s) has been quite non-systematic. Since the 1970's, various authors with a variety of motivations have studied 2 fa s with various types of alternation and a variety of names, creating a fairly long list of sporadic contributions with little internal consistency. This article attempts to organize the subject into a single unifying framework. We start with a detailed account of all contributions to date, that reveals the large variety of approaches and the lack of consistency between them. We then identify and name four types of automata that these contributions have really studied over the years: general two-way Boolean finite automata ( 2 b fa s); monotone 2 b fa s; basic 2 b fa s; and (monotone basic, or) alternating 2 b fa s ( 2 a fa s). Next, we identify four different ways by which authors have described how such automata compute, each offering a distinct view onto their operation: the circuit view, where computation is modelled by a circuit of Boolean gates; the formula view, where computation is modelled by a Boolean formula over configuration-variables; the run view, where decisions are determined by the existence of appropriate trees of configuration-goal pairs, called “runs”; and the (most classic) tree view, where computation is modelled by a tree of configurations. After carefully defining each 2 b fa type and each view, we prove the following. First, that within each type, every two of the four views are equivalent to each other, in the strong sense that each of them closely mimics the other at every step of the computation. Second, that not all types of 2 b fa s are equivalent: although general 2 b fa s are as powerful as monotone 2 b fa s, and basic 2 b fa s are as powerful as 2 a fa s (up to polynomial differences in the number of states), general 2 b fa s may need exponentially fewer states than 2 a fa s.

In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Gruss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we a...

Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.

In this paper, we consider the existence of extremal solutions for a type of fractional compartment models. By use of a new comparison result, some new sufficient conditions for the existence of solutions are established by combining the monotone iterative technique and the methods of lower and upper solutions. Finally, an example is presented to illustrate the validity of our main results.

In a recent paper, Bauschke et al. study $$\rho $$ ρ -comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent $$J_A.$$ J A . In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for $$\rho $$ ρ -comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. $$zer\, A$$ z e r A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for $$\rho $$ ρ -comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

In this paper, the consequences of missing observations on data-based multicollinearity were analyzed. Different missing values has a different effect on multicollinearity in the system of multiple regression model. Therefore, to ascertain the clear relationship between both multicollinearity and skipping values on monotone and arbitrary missing values, the collinear effects were potentially studied on two types of missing values. Similarly, the comparison was done to investigate each response of multicollinearity on each pattern of the missing values with the same informatics data. It was found that tolerance and variance inflation factor fluctuates due to the missing of information from the sample analyzed at a different percentages of the missing values.It was observed that the more missing values available in the sample obtain from either population statistics or survey than multicollinearity will be found in the system of multiple regression, this is because as the number of Missingness increase it shows a drastic decrease from the tolerance level on both monotone and arbitrary types as observed from the analysis.

The paper concerns with two new schemes for approximating solutions of equilibrium problems involving monotone and Lipschitz-type bifunctions in Hilbert spaces. We describe how to incorporate the t...

Setting a proper minimum capital requirement is one of the most fundamental problems in the insurance industry. Ruin theory proposes a solution to this problem by identifying the minimum capital that a company needs to hold in order to stay solvent with a high probability. In this note we discuss the ruin theory risk consistency. More precisely we show that the ruin-consistent Value-at-Risk (VaR) is not continuous in probability, in $$L^{p},0\le p