Cover

Series Title

Title Page

Imprints

Contents

Preface

List of contributors

1 Self-organisation and emergence

Mario Nicodemi, Yu-Xi Chau, Christopher Oates, Anas Rana and Leigh Robinson

1.1 Random walks

1.1.1 Introduction

1.1.2 One-dimensional discrete random walk

1.1.3 Applications to finance

1.1.4 Random walks on mathbb Z superscript d lattices

1.1.5 Conformation of polymers

1.2 Markov processes

1.2.1 Definitions

1.2.2 Markov property

1.2.3 Continuous-time random walks

1.2.4 Applications of Markov processes

1.3 Lattice models

1.3.1 Lattice gas model

1.3.2 Ising model of ferromagnetism

1.3.3 Percolation

1.3.4 Universality

1.4 Self-organised criticality

1.4.1 Overview

1.4.2 The OFC model

1.4.3 Empirical data and the OFC model

1.4.4 Does SOC exist?

1.5 Networks

1.5.1 Definitions

1.5.2 Random graphs

1.5.3 Degree distribution

1.5.4 Clustering coefficient

1.5.5 Network vulnerability

1.5.6 The small-world effect

References

2 Complexity and chaos in dynamical systems

Yulia Timofeeva

2.1 Dynamical models

2.2 First-order systems

2.2.1 Fixed points and stability

2.2.2 Linear stability analysis

2.2.3 Local existence and uniqueness (in mathbb R)

2.2.4 Potentials

2.2.5 Numerical simulation of dynamical systems

2.2.6 Bifurcations

2.2.7 Flows on the circle

2.3 Linear systems in mathbb R superscript 2

2.3.1 Harmonic oscillator

2.3.2 Classification of fixed points

2.4 Linear systems in mathbb R superscript n

2.4.1 Normal forms

2.5 Nonlinear systems in mathbb R superscript 2 (in mathbb R superscript n)

2.5.1 Stability

2.5.2 Linearisation

2.6 Nonlinear oscillations

2.6.1 Limit cycles in mathbb R superscript 2

2.6.2 Poincaré--Bendixson theorem

2.6.3 Relaxation oscillators

2.7 Perturbation methods

2.7.1 Regular perturbation theory and its failures

2.7.2 Method of multiple scales (two-timing)

2.8 Coupled oscillators

2.9 Poincaré maps

2.9.1 Linear stability of limit cycle

2.10 Introduction to chaos

2.10.1 Lorenz equations

2.10.2 Properties of the Lorenz equations

2.10.3 Chaos on a strange attractor

2.10.4 Exponential divergence

2.10.5 Defining chaos

2.10.6 Defining attractor and strange attractor

2.10.7 Lorenz map

2.10.8 Tests for chaos

2.11 One-dimensional maps

2.11.1 Fixed points and linear stability

2.11.2 Logistic map

2.11.3 Routes to chaos

2.11.4 State space reconstruction

2.12 Fractals and fractal dimensions

2.13 More on bifurcations

2.13.1 Local bifurcations

2.13.2 Global bifurcations of cycles

Illustrative bibliography

3 Interacting stochastic particle systems

Stefan Grosskinsky

3.1 Basic theory

3.1.1 Markov processes

3.1.2 Continuous-time Markov chains and graphical representations

3.1.3 Three basic IPS

3.1.4 Semigroups and generators

3.1.5 Stationary measures and reversibility

3.1.6 Simplified theory for Markov chains

3.2 The asymmetric simple exclusion process

3.2.1 Stationary measures and conserved quantities

3.2.2 Symmetries and conservation laws

3.2.3 Currents and conservation laws

3.2.4 Hydrodynamics and the dynamic phase transition

3.3 Zero-range processes

3.3.1 From ASEP to ZRPs

3.3.2 Stationary measures

3.3.3 Equivalence of ensembles and relative entropy

3.3.4 Phase separation and condensation

3.4 The contact process

3.4.1 Mean-field rate equations

3.4.2 Stochastic monotonicity and coupling

3.4.3 Invariant measures and critical values

3.4.4 Results for Lambda = mathbb Z superscript d

3.4.5 Duality

References

4 Statistical mechanics of complex systems

Ellák Somfai

4.1 Introduction to information theory

4.2 The maximum entropy framework superscript 1

4.3 Applications of the maximum entropy framework

4.4 Fluctuations and thermodynamics

4.5 Phase transitions

4.6 Surface growth superscript 5

4.7 Collective biological motion: flocking

References

5 Numerical simulation of continuous systems

Colm Connaughton

5.1 Introduction

5.1.1 Partial differential equations in complexityscience

5.1.2 What is this chapter about?

5.2 Ordinary differential equations (ODEs)

5.2.1 Ordinary differential equations

5.2.2 Dynamical systems

5.2.3 Approximation of derivatives by finite differences

5.2.4 Timestepping algorithms 1: Euler methods

5.2.5 Timestepping algorithms 2: predictor–correctormethods

5.2.6 Timestepping algorithms 3: Runge–Kuttamethods

5.2.7 Adaptive timestepping

5.2.8 Snakes in the grass

5.3 Partial differential equations (PDEs)

5.3.1 Introduction to PDEs and their mathematical classification

5.3.2 Non-dimensionalisation

5.3.3 First-order PDEs: method of characteristics

5.3.4 Similarity solutions and travelling waves

5.3.5 Travelling waves

5.3.6 Similarity solutions

5.4 The diffusion equation

5.4.1 Forward time centred space (FTCS) method

5.4.2 Source terms and boundary conditions in the FTCS method

5.4.3 Consistency and stability of the FTCS method

5.4.4 The Crank–Nicholson method

5.4.5 Stability of the Crank–Nicholson method

5.4.6 Similarity solutions as attractors: rescaling

5.5 Hyperbolic PDEs

5.5.1 The advection equation revisited

5.5.2 Lax method and CFL criterion

5.5.3 Conservation laws and Lax–Wendroff method

5.5.4 The wave equation

5.5.5 Boundary conditions for the wave equation

References

6 Stochastic methods in economics and finance

Vassili N. Kolokoltsov

6.1 Utility theory

6.1.1 Order, utility and expected utility

6.1.2 Utility on monetary outcomes

6.1.3 Dominance

6.1.4 Utility on R superscript d subscript +

6.2 Variance – spread – risk

6.2.1 Variance and correlation

6.2.2 Volatility and correlation estimators

6.2.3 Waiting time paradox

6.2.4 Hedging via futures

6.2.5 Other measures of risk

6.3 Optimal betting (Kelly's system) and money management

6.4 Portfolio, CAPM and factor models

6.4.1 Portfolio optimization

6.4.2 Portfolio optimization with a risk-free asset

6.4.3 Capital Asset Pricing Model (CAPM)

6.4.4 Performance evaluation

6.4.5 CAPM as pricing formula

6.4.6 Multi-factor models (arbitrage pricing)

6.4.7 The Wilkie model

6.5 Derivative securities in discrete time: arbitrage pricing and hedging

6.5.1 Efficient market hypothesis

6.5.2 Binomial model: basic assumptions

6.5.3 Conditional expectation and martingales on Bernoulli spaces

6.5.4 Binomial model: definition and risk-neutral laws

6.5.5 Binomial model: replication and risk-neutral evaluation of contingent claims

6.5.6 Put-call parity

6.5.7 Put-call parity in capital structure (Merton's model for credit risk)

6.5.8 Exotic options

6.5.9 Options on a dividend-paying stock

6.5.10 Setting up (or fitting to the data) a binomial model

6.5.11 Deflator: connecting actual and risk-neutral probability

6.5.12 Trading with options

6.5.13 American options

6.6 Introducing the central limit theoremand fat tails

6.6.1 Generating functions

6.6.2 Asymptotic normality

6.6.3 Fat (or heavy) tails

6.7 Black–Scholes option pricing

6.7.1 Log-normal random variables

6.7.2 Equivalent normal laws

6.7.3 Continuous-time models for asset pricing

6.7.4 Black–Scholes theory

6.8 Credit risk and Gaussian copula

6.8.1 Probability of survival and hazard rate

6.8.2 Credit default swaps (CDS)

6.8.3 Transformation of probability laws

6.8.4 Gaussian copulas

6.8.5 Markov model for default probabilities and defaultable bonds

6.9 Revision exercises

6.10 Solutions to exercises (sketch)

6.11 Concluding remarks

References

7 Space-time phases

Robert S. MacKay

7.1 Stochastic dynamics--probabilistic cellular automata (PCA)

7.1.1 Stavskaya's PCA

7.1.2 The majority voter PCA

7.1.3 General properties for phases of PCA

7.2 Deterministic case--coupled map lattices (CML)

7.2.1 Examples with non-unique space-time phase

7.2.2 Statistical phases for uniformly hyperbolic attractors of finite-dimensional deterministic dynamical systems

7.2.3 Uniformly hyperbolic dynamics on networks

7.2.4 Natural measures on uniformly hyperbolic attractors for coupled map lattices

7.2.5 Conclusion

References

8 Selfish routing

Robert S. MacKay

8.1 Introduction

8.2 Basic theory of Nash flows

8.3 Bounding the price of anarchy

8.3.1 Affine cost functions

8.3.2 General cost functions

8.4 Avoiding Braess' paradox

8.5 Possible research directions

References

Index