Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths

Probabilistically distorted risk-sensitive infinite-horizon dynamic programming

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

In this paper, we study the asymptotic behavior of dynamic coherent risk measures in general settings regardless of specific representations of the risk measures. In particular, we develop three different types laws of large numbers (LLN) for the average values of portfolios. These LLNs capture the limiting behavior of time-consistent dynamic coherent risk measures under appropriate conditions. Our results apply to general probability spaces with a sequence of financial returns characterized by a set of probability measures. We show that the limit of these averages will generally be multivalued within an identified set. We give examples to illustrate the potential applicability of our results and derive asymptotic results on estimation for the risk of returns of financial assets using a time-consistent dynamic coherent risk measure induced by a class of g-expectations.

In this paper, we take a new perspective to describe the model uncertainty, and thus propose two new classes of risk measures under model uncertainty. To be precise, we use an auxiliary random variable to describe model uncertainty. By proposing new sets of axioms under model uncertainty, we axiomatically introduce and characterize conditional coherent and convex risk measures under a random environment, respectively. As examples, we also discuss the connections of the introduced conditional coherent risk measures under random environments with two existing risk measures. This paper mainly gives some theoretical results, and it is expected to make meaningful complement to the study of coherent and convex risk measures under model uncertainty.

A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and g-expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic g-expectations in the framework of the continuous-time asset pricing.

In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

Coherent risk measures have been introduced by Artzner et al. in financial markets where the set of states Ω is finite. This has been extended by Delbaen to cases where Ω is infinite. The financial markets are constantly evolving and the classical definition of risk measures is static. Hence, the need to study its complexity and evolution. Complexity theory focuses on the interdisciplinary study of its systems. In its evolution, the study of complex adaptive systems and the emergent phenomena associated with them are of fundamental importance. In this paper, we consider the coherent risk measure and their interpretation as complex adaptive system. KEYWORD: Coherent risk measure; complex adaptive system.

Inverse portfolio problem with coherent risk measures

Time consistent dynamic risk processes

Despite the fact that the value-at-risk (VaR) measure has been adopted as a standard risk measure in the financial industry, it has a number of deficiencies recognized by financial professionals. It is not a coherent risk measure. This is because it does not satisfy the subadditivity property requirement of a coherent risk measure. That is, there are cases in which the portfolio VaR is larger than the sum of the VaRs of the portfolio constituents, supporting the view that VaR cannot be used as a true risk measure. Unlike VaR, the average value-at-risk measure (AVaR)—also referred to as conditional value-at-risk and expected shortfall—is a coherent risk measure and has other advantages that result in its greater acceptance in risk modeling. Keywords: average value-at-risk; coherent risk measures; average value-at-risk; conditional value-at-risk; expected shortfall; The Student's; distribution

A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.

Different approaches to defining dynamic market risk measures are available in the literature. Most are focused or derived from probability theory, economic behavior or dynamic programming. Here, we propose an approach to define and implement dynamic market risk measures based on recursion and state economy representation. The proposed approach is to be implementable and to inherit properties from static market risk measures.

A note on the Swiss Solvency Test risk measure

Summary Probabilistic methods for reserves estimation, including uncertainty quantification and probabilistic aggregation, have gained widespread acceptance in the oil and gas industry, since the first comprehensive guidelines were issued by the Society of Petroleum Engineers (SPE) (Swinkels 2001). The probabilistic methods now used in the oil industry, as proposed in these guidelines, are similar to those also used in portfolio theory and risk management by the finance industry. A significant amount can be learned from the extensive experience with probabilistic methods and quantification of risk with measures like value-at-risk (VAR), in financial risk management. Especially, the guidelines issued by the Basel II Accord (Bank for International Settlements 2006) and the discussions since the 2008 financial crisis contain important lessons. In this paper, we examine a fundamental question: "Is the P90 reserves value an appropriate measure for quantifying the reserves’ downside?" For the P90 reserves value to be considered a good measure of the reserves’ downside, it needs to possess a number of basic characteristics involving P90 reserves for each field and the probabilistically aggregated P90 reserves for the portfolio of fields. Analogous to the definition of a coherent risk measure used in the finance industry, we define these characteristics for P90 reserves. The P90 reserves are as good a risk measure as VAR used in the financial industry. However, like VAR, it is not a coherent risk measure. A possible uncertainty scenario, in which one of these necessary characteristics does not hold, is given. An alternative measure of risk for quantifying the reserves’ downside, defined as the average reserves over the confidence interval higher than P90, is presented. This is a coherent risk measure. In this paper, we highlight the appropriateness and limitations of using the P90 reserves estimate as a measure of the reserves’ downside. Understanding of the limitations posed by using the P90 reserves value is vital in management of reserves risk.

The production life of oil reservoirs starts under significant uncertainty regarding the actual economical return of the recovery process due to the lack of oil field data. Consequently, investors and operators make management decisions based on a limited and uncertain description of the reservoir. In this work, we propose a new formulation for robust optimization of reservoir well controls. It is inspired by the multiple shooting (MS) method which permits a broad range of parallelization opportunities and output constraint handling. This formulation exploits coherent risk measures, a concept traditionally used in finance, to bound the risk on constraint violation. We propose a reduced sequential quadratic programming (rSQP) algorithm to solve the underlying optimization problem. This algorithm exploits the structure of the coherent risk measures, thus a large set of constraints are solved within sub-problems. Moreover, a variable elimination procedure allows solving the optimization problem in a reduced space and an iterative active-set method helps to handle a large set of inequality constraints. Finally, we demonstrate the application of constraints to bound the risk of water production peaks rather than worst-case satisfaction.