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

