Dual moments replace primal moments for measuring risk aversion.
problem Traditional risk aversion measures using mean and variance are insufficient in non-EU models.
method Introduced dual moments as a new measure for absolute risk aversion.
result Dual moments provide an equivalent index of absolute risk aversion in non-EU models.
Study optimal dividends in dual risk model with stochastic interest rate.
problem Optimal dividend strategy in dual risk model with stochastic interest rate.
method Geometric Brownian motion or exponential Lévy process for discounting factor.
result Closed form solutions can be obtained for optimal dividends.
Dual risk models are popular for modeling a venture capital or high tech company, for which the running cost is deterministic and the profits arrive stochastically over time. Most of the existing literature on dual risk models concentrated on the optimal dividend strategies. In this paper, we propose to study the optim…
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
In a dual risk model, the premiums are considered as the costs and the claims are regarded as the profits. The surplus can be interpreted as the wealth of a venture capital, whose profits depend on research and development. In most of the existing literature of dual risk models, the profits follow the compound Poisson …
Study a dual risk model with innovation delays, focusing on ruin probability and time.
problem Analyzing risk models with innovation delays affecting ruin probability and time.
method Delayed dual risk model with innovation delays, focusing on ruin probability and time.
result Closed-form formulas for ruin probability and time in some special cases.
Develops dual story for risk apportionment, interpreting preferences between lottery pairs.
problem Understanding preferences between different lottery pairs.
method Specifying model-free preferences towards nested classes of lottery pairs, developing dual story.
result Intuitive interpretation and full characterization of dual counterparts of prudence and temperance.
Dual representations for systemic risk measures using acceptance sets.
problem Measuring systemic risk in financial systems.
method Developed dual representations for systemic risk measures based on acceptance sets.
result Simple and self-contained proof of dual representations for utility-based risk measures.
Study dual representations for quasiconvex systemic risk measures.
problem Finding dual representations for quasiconvex systemic risk measures.
method Abstract infinite-dimensional setting, explicit formula for penalty function, nonstandard minimax inequality.
result Explicit formula for the penalty function of quasiconvex compositions.
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk m…
New insights into risk aversion for complex decision models.
problem Understanding risk aversion in non-monotone decision models.
method Characterization of probabilistic risk aversion for generalized rank-dependent functions.
result Probabilistic risk aversion is determined by the distortion function, which is convex or scaled quantile-spread mixtures.
New financial model revises risk measure under NA condition.
problem Revising classical financial mathematics with coherent risk measure on L0. method Developed a new version of the fundamental theorem of asset pricing and provided dual representations.
result Set of risk-hedging prices is closed under NA condition.
The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for a dual risk model. It is well known that the value function of this optimal control problem do…
Develops risk measures on Lipschitz spaces for financial positions.
problem Lack of standard cash-additive methods in Lipschitz spaces.
method Proposes Lipschitz-free space, uses additivity along benchmark-deviation instruments.
result Derives dual representations for convex and coherent risk measures.
This paper concerns the dual risk model, dual to the risk model for insurance applications, where premiums are surplus-dependent. In such a model premiums are regarded as costs, while claims refer to profits. We calculate the mean of the cumulative discounted dividends paid until ruin, if the barrier strategy is applie…
Generalizes risk sharing models to a continuum of agents.
problem Risk sharing among a large number of heterogeneous agents.
method Modeling agents as points in a measure space, using risk measures on a probability space, and deriving dual representations.
result Explicit formulas for specific risk measures (entropic and expected shortfall) and applications to Pareto efficiency.
This paper analyzes privacy-preserving methods for sparse model optimization.
problem Privacy-preserving sparse model optimization with non-differentiable norms.
method Differential privacy techniques applied to Frank-Wolfe and objective perturbation algorithms.
result Excess risk bounds for Frank-Wolfe and objective perturbation algorithms are derived.
Dual optimization connects ERM-fDR to normalization function.
problem Empirical risk minimization with f-divergence regularization.
method Dual formulation, Legendre-Fenchel transform, implicit function theorem, nonlinear ODE.
result Computational method to calculate normalization function efficiently.
We present a general framework for measuring the liquidity risk. The theoretical framework defines a class of risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a given security or a portfolio…
In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we anal…
Paper develops risk statistics for portfolios considering regulator-based risk.
problem Traditional risk statistics fail to describe regulator-based risk.
method Develop dual representation for regulator-based risk statistics.
result Derived dual representation for regulator-based risk statistics.
Paper explores how risk-averse individuals' willingness to pay for insurance varies with risk probability.
problem Understanding how risk-averse individuals' willingness to pay for insurance varies with risk probability.
method Analyzes willingness to pay (WTP) for partial risk reduction within the dual theory of decision.
result In dual theory, reducing the probability of risk and providing insurance can be complementary if the surplus increases with risk reduction.
Investigates set-valued risk measures for processes and vectors, proving equivalence and providing new dual representations.
problem Investigates set-valued risk measures for processes and vectors.
method Utilizes equivalence of risk measures for processes and vectors and their penalty function formulations.
result Provides new dual representation for risk measures for processes in the set-valued framework.
A method for risk valuation using backward stochastic differential equations.
problem Risk evaluation in financial markets.
method Dual representation and stochastic control problem conversion, followed by dynamic programming.
result Piecewise-constant dual control provides a good approximation for risk valuation.
A fundamental problem in risk management is the robust aggregation of different sources of risk in a situation where little or no data are available to infer information about their dependencies. A popular approach to solving this problem is to formulate an optimization problem under which one maximizes a risk measure …
Proposes a computational framework for real-time risk assessment and prioritization.
problem Real-time risk assessment and prioritization for uncertain outcomes.
method Develops a computational framework based on satisficing measure for real-time risk assessment and prioritization. Applies sample average approximation and primal-dual stochastic approximation algorithms.
result Demonstrates the effectiveness of the proposed framework in real-time risk assessment and prioritization.
The paper develops algorithms to minimize risk and regret in uncertain decisions.
problem Minimizing risk and regret in multistage decisions under uncertainty.
method Established dual representations and used Lagrangian duality theory to develop progressive hedging algorithms.
result Modified progressive hedging algorithm can handle new linkage constraints.
Study systemic risk measures adjusted to financial markets.
problem Systemic risk in financial systems with market adjustments.
method Dual representation for convex robust systemic risk measures adjusted to the financial market.
result Relation to no-arbitrage conditions.
Extends portfolio optimization with two quasiconvex risk measures.
problem Optimizing portfolios with dual risk measures for multiple stakeholders.
method Dual problem formulation, bisection algorithm, duality results.
result Approximately optimal solutions can be achieved with prescribed optimality gap.
The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can…
Researchers develop multi-utility representations for incomplete preferences linked to risk measures.
problem Handling incomplete preferences induced by set-valued risk measures.
method Established dual representations of set-valued risk measures to create parsimonious and well-behaved multi-utility representations.
result Unified dual representations of set-valued risk measures, linking them to scalar risk measures.
A new approach to risk-sensitive reinforcement learning tackles computational challenges.
problem Computational challenges in estimating risk-sensitive policies for MDPs with finite state and action spaces.
method Proposes a new risk measure called 'caution' and uses a stochastic primal-dual method with KL divergence.
result Demonstrates improved reliability in reward accumulation without additional computational costs.
Optimal hedging framework with variational preferences under convex risk measures.
problem Optimal hedging with variational preferences under convex risk measures.
method Theoretical hedging optimization framework with dual representation of risk measures and utilities.
result Derivation of optimality and indifference pricing conditions.
New algorithms exploit data's strong convexity for fast linear convergence without explicit regularization.
problem Empirical risk minimization with convex loss functions.
method Primal-dual first-order algorithms that exploit data's strong convexity.
result Adaptive primal-dual algorithms achieve linear convergence without explicit regularization.
The paper defines and analyzes scalar risk measures in markets with transaction costs.
problem Defining and analyzing scalar risk measures in markets with transaction costs.
method Dual representation of scalar risk measures, time consistency, backward recursion.
result A weaker notion of time consistency for scalar risk measures in markets with frictions is defined and proven equivalent to a backward recursion.
Study quasiconvex risk measures in volatile financial markets.
problem Financial risk measurement in markets with variable volatility.
method Defined quasiconvex risk measures on Lp(⋅) space with p(⋅) as a random variable. result Deduced dual representation for the defined quasiconvex risk measures.
This paper shows how to calculate risk measures for sums of two counter-monotonic risks.
problem Calculating risk measures for sums of two counter-monotonic risks.
method Using a fixed distortion function and expressing the risk measure of a sum as the sum of two related measures of the marginals.
result The risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals.
New framework measures systemic risk with variable market volatility.
problem Classical risk measures fail to capture market volatility complexity.
method Proposes a new framework on Lp(⋅) space with random exponents. result Derives dual representations of systemic risk quantification.
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
Paper uses deep learning for systemic risk measures.
problem Computing optimal capital allocations for systemic risk.
method Deep learning algorithms to solve primal and dual problems.
result Deep learning provides fair risk allocations.
New concept of attitude towards probability introduced in risk sharing problems.
problem Risk sharing problems and attitudes towards probability.
method Generalized definition of probability premium, local approximation, rank-dependent utility model, dual theory.
result Attitude towards probability can be first-order or second-order, depending on the model.
Set-valued risk measures on Ldp with 0≤p≤∞ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Sc…
New fair risk allocation method for banks.
problem Fair allocation of systemic risk measures.
method Dual representation and utility maximization problem.
result Optimal risk allocation is fair for individual banks.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded \cd processes, we show that this framework provides a systematic approach…
The strong Fatou property is crucial for risk measures' dual representations.
problem Ensuring nice dual representations of risk measures.
method Exploring Fatou-type properties and inf-convolutions of law-invariant or surplus-invariant risk measures.
result Every quasiconvex law-invariant functional on a rearrangement invariant space with the strong Fatou property is σ(X, L∞)-lower semicontinuous.
Develops risk measures for markets with constraints and costs.
problem Risk measures in markets with portfolio constraints and transaction costs.
method Embeds portfolio constraints and transaction costs into securities market; provides comprehensive analysis of risk measures properties.
result Establishes dual representations for convex and quasiconvex risk measures.
Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate ris…
Dual representation and properties of expectile-based expected shortfall studied.
problem Studying the expectile-based expected shortfall as a risk measure.
method Provided dual representation in terms of Bochner integral, showed boundedness properties, and computed for selected distributions.
result Explicit dual representation and boundedness properties of expectile-based expected shortfall.