Modern risk systems
Books by Arcady Novosyolov

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Novosyolov A.A. Mathematical modeling of financial risks: measuring theory. Novosibirsk: Nauka, 2001. (in Russian)

Abstract

The book is devoted to mathematical risk theory - an emerging branch of probability theory with multiple applications to economics, finance and other areas of human activities, related to decision-making under uncertainty. Much attention is paid to measuring risk, that is, quantitative description of preferences over sets of probability distributions. An axiomatic approach for nonlinear preferences is being proposed, which is an extension of that by J. von Neumann and O. Morgenstern. Portfolio analysis and risk processes are also considered.

The book is intended for a broad range of readers interested in application of probability theory to social sciences, including finance, insurance and individual decision-making in general, and for graduate students studying mathematics and finance.

Contents

  1. Preface
  2. Preliminaries
    1. Relations
    2. Monotone functionals
    3. Reverse of an ordered set
    4. Probability distributions and generalized measures
    5. Distributions on ordered sets
    6. Orders on sets of distributions
  3. Risk theory
    1. Decision-making problem
    2. Risk measures
    3. Monotonicity of risk measures
      1. Expectation and variance
      2. Expected utility measure
      3. Distorted probability measure
    4. Convexity of risk measures
      1. Expectation and variance
      2. Expected utility measure
      3. Distorted probability measure
    5. Computing risk measures
      1. Discretizing distributions
      2. Expected utility measure
      3. Distorted probability measure
  4. Portfolio analysis
    1. Problem statement
    2. Second order portfolios
      1. Simplest portfolio
      2. Mixed functional
      3. Markowitz problem
      4. Attitude to risk
    3. Expected utility method
      1. Problem statement
      2. Normal distribution and expected utility
  5. Building risk measures
    1. Subjective probability
      1. Likelihood relation
      2. Existence of the probability distribution
    2. Preference relation
      1. Assumptions
      2. Existence of a risk measure
    3. Boundedness of sets of distributions
      1. Linear preference relation
      2. Nonlinear preference relation
  6. Risk processes
    1. Classical risk process
      1. Definition
      2. Ruin of a process
    2. Aggregate risk process
      1. Definition
      2. Properties of aggregate process
      3. Equation for survival probability
      4. Simplest risk process
    3. Solution of survival equation
      1. Absorbing process
      2. Discrete case
      3. General case
    4. Mutual approximation of processes
      1. Aggregation operator
      2. Approximation for paths of processes
  7. Conclusion

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