Abstract
In risk theory, risks are often modeled by risk measures which allow quantifying the risks and estimating their possible outcomes. Risk measures rely on measure theory, where the risks are assumed to be random variables with some distribution function. In this work, we derive a novel topological-based representation of risks. Using this representation, we show the differences between diversifiable and non-diversifiable. We show that topological risks should be modeled using two quantities, the risk measure that quantifies the predicted amount of risk, and a distance metric which quantifies the uncertainty of the risk.
Highlights
The mathematical formulation of risks is based purely on probability
We present a special type of set-valued risk measures that classifies these two types using concepts from general topology
We have investigated the use of topology for studying risks
Summary
The mathematical formulation of risks is based purely on probability. Let Y be a random variable on the probability space (Ω, F , P), where Ω represents the space of all possible outcomes, F is the σ-algebra, and P is the probability measure. For any TR random set X, there exists a TR set-valued measure $ that quantifies the amount of loss expected from the risk, and it is defined by $ : X ∈ R =⇒$ ( X ) ∈ R ⊆ Rd that is a d-dimensional risk measure.
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