State Space Approaches for Modelling and Control in Financial Engineering Systems theory and machine learning methods 1st Edition by Gerasimos Rigatos 3319528663 9783319528663

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ISBN-10 :  3319528663 

ISBN-13 :  9783319528663

Author:  Gerasimos G. Rigatos 

The book conclusively solves problems associated with the control and estimation of nonlinear and chaotic dynamics in financial systems when these are described in the form of nonlinear ordinary differential equations. It then addresses problems associated with the control and estimation of financial systems governed by partial differential equations (e.g. the Black–Scholes partial differential equation (PDE) and its variants). Lastly it an offers optimal solution to the problem of statistical validation of computational models and tools used to support financial engineers in decision making. The application of state-space models in financial engineering means that the heuristics and empirical methods currently in use in decision-making procedures for finance can be eliminated. It also allows methods of fault-free performance and optimality in the management of assets and capitals and methods assuring stability in the functioning of financial systems to be established. Covering the following key areas of financial engineering: (i) control and stabilization of financial systems dynamics, (ii) state estimation and forecasting, and (iii) statistical validation of decision-making tools, the book can be used for teaching undergraduate or postgraduate courses in financial engineering. It is also a useful resource for the engineering and computer science community

State Space Approaches for Modelling and Control in Financial Engineering Systems theory and machine learning methods 1st Table of contents:

1 Systems Theory and Stability Concepts
1.1 Outline
1.2 Characteristics of the Dynamics of Nonlinear Systems
1.3 Computation of Isoclines
1.4 Stability Features of Dynamical Systems
1.4.1 The Phase Diagram
1.4.2 Stability Analysis of Nonlinear Systems
1.4.3 Local Stability Properties of a Nonlinear Model
1.5 Phase Diagrams and Equilibria
1.5.1 Phase Diagrams for Linear Dynamical Systems
1.5.2 Multiple Equilibria for Nonlinear Dynamical Systems
1.5.3 Limit Cycles
1.6 Bifurcations
1.6.1 Bifurcations of Fixed Points
1.6.2 Saddle-Node Bifurcations of Fixed Points in a One-Dimensional System
1.6.3 Pitchfork Bifurcation of Fixed Points
1.6.4 The Hopf Bifurcation
1.7 Chaos in Dynamical Systems
1.7.1 Chaotic Dynamics
1.7.2 Examples of Chaotic Dynamical Systems
2 Main Approaches to Nonlinear Control
2.1 Outline
2.2 Overview of Main Approaches to Nonlinear Control
2.3 Control Based on Global Linearization Methods
2.3.1 Overview of Differential Flatness Theory
2.3.2 Differential Flatness for Finite Dimensional Systems
2.4 Control Based on Approximate Linearization Methods
2.4.1 Approximate Linearization Round Temporary Equilibria
2.4.2 The Nonlinear H-Infinity Control
2.4.3 Approximate Linearization with Local Fuzzy Models
2.5 Control Based on Lyapunov Stability Analysis
2.5.1 Transformation of Nonlinear Systems into a Canonical Form
2.5.2 Adaptive Control Law for Nonlinear Systems
2.5.3 Approximators of System Unknown Dynamics
2.5.4 Lyapunov Stability Analysis for Dynamical Systems
3 Main Approaches to Nonlinear Estimation
3.1 Outline
3.2 Linear State Observers
3.3 The Continuous-Time Kalman Filter for Linear Models
3.4 The Discrete-Time Kalman Filter for Linear Systems
3.5 The Extended Kalman Filter for Nonlinear Systems
3.6 Sigma-Point Kalman Filters
3.7 Particle Filters
3.7.1 The Particle Approximation of Probability Distributions
3.7.2 The Prediction Stage
3.7.3 The Correction Stage
3.7.4 The Resampling Stage
3.7.5 Approaches to the Implementation of Resampling
3.8 The Derivative-Free Nonlinear Kalman Filter
3.8.1 Conditions for solving the estimation problem in single-input nonlinear systems
3.8.2 State Estimation with the Derivative-Free Nonlinear Kalman Filter
3.8.3 Derivative-Free Kalman Filtering for multivariable Nonlinear Systems
3.9 Distributed Extended Kalman Filtering
3.9.1 Calculation of Local Extended Kalman Filter Estimations
3.9.2 Extended Information Filtering for State Estimates Fusion
3.10 Distributed Sigma-Point Kalman Filtering
3.10.1 Calculation of Local Unscented Kalman Filter Estimations
3.10.2 Unscented Information Filtering for State Estimates Fusion
3.11 Distributed Particle Filter
3.11.1 Distributed Particle Filtering for State Estimation Fusion
3.11.2 Fusion of the Local Probability Density Functions
3.12 The Derivative-Free Distributed Nonlinear Kalman Filter
3.12.1 Overview
3.12.2 Fusing Estimations from Local Distributed Filters
3.12.3 Calculation of the Aggregate State Estimation
3.12.4 Derivative-Free Extended Information Filtering
4 Linearizing Control and Estimation for Nonlinear Dynamics in Financial Systems
4.1 Outline
4.2 Dynamic Model of the Chaotic Finance System
4.2.1 State-Space Model of the Chaotic Financial System
4.2.2 Chaotic Dynamics of the Finance System
4.3 Overview of Differential Flatness Theory
4.3.1 Conditions for Applying the Differential Flatness Theory
4.3.2 Transformation of Nonlinear Systems into Canonical Forms
4.4 Flatness-Based Control of the Chaotic Finance Dynamics
4.4.1 Differential Flatness of the Chaotic Finance System
4.4.2 Design of a Stabilizing Feedback Controller
4.5 Adaptive Fuzzy Control of the Chaotic Finance System Using ƒ
4.5.1 Problem Statement
4.5.2 Transformation of Tracking into a Regulation Problem
4.5.3 Estimation of the State Vector
4.5.4 The Additional Control Term uc
4.5.5 Dynamics of the Observation Error
4.5.6 Approximation of Unknown Nonlinear Dynamics
4.6 Lyapunov Stability Analysis
4.6.1 Design of the Lyapunov Function
4.6.2 The Role of Riccati Equation Coefficients in Hinfty Control Robustness
4.7 Simulation Tests
5 Nonlinear Optimal Control and Filtering for Financial Systems
5.1 Outline
5.2 Chaotic Dynamics in a Macroeconomics Model
5.2.1 Dynamic Model of the Chaotic Finance System
5.2.2 State-Space Model of the Chaotic Financial System
5.2.3 Chaotic Dynamics of the Finance System
5.3 Design of an H-Infinity Nonlinear Feedback Controller
5.3.1 Approximate Linearization of the Chaotic Finance System
5.3.2 Equivalent Linearized Dynamics of the Chaotic Finance System
5.3.3 The Nonlinear H-Infinity Control
5.3.4 Computation of the Feedback Control Gains
5.3.5 The Role of Riccati Equation Coefficients in Hinfty Control Robustness
5.4 Lyapunov Stability Analysis
5.4.1 Stability Proof
5.4.2 Robust State Estimation with the Use of the Hinfty Kalman Filter
5.5 Simulation Tests
6 Kalman Filtering Approach for Detection of Option Mispricing in the Black–Scholes PDE
6.1 Outline
6.2 Option Pricing Modeling with the Use of the Black–Scholes PDE
6.2.1 Option Pricing Modeling with the Use of Stochastic Differential Equations
6.2.2 The Black–Scholes PDE
6.2.3 Solution of the Black–Scholes PDE
6.2.4 Sensitivities of the European Call Option
6.2.5 Nonlinearities in the Black–Scholes PDE
6.2.6 Derivative Pricing
6.3 Estimation of Nonlinear Diffusion Dynamics
6.3.1 Filtering in Distributed Parameter Systems
6.4 State Estimation for the Black–Scholes PDE
6.4.1 Modeling in Canonical Form of the Nonlinear Black–Scholes Equation
6.4.2 State Estimation with the Derivative-Free Nonlinear Kalman Filter
6.4.3 Consistency Checking of the Option Pricing Model
6.5 Simulation Tests
6.5.1 Estimation with the Use of an Accurate Black–Scholes Model
6.5.2 Detection of Mispricing in the Black–Scholes Model
7 Kalman Filtering Approach to the Detection of Option Mispricing in Elaborated PDE Finance Models
7.1 Outline
7.2 Option Pricing in the Energy Market
7.2.1 Energy Market and Swing Options
7.2.2 Energy Options Pricing Models
7.3 Validation of the Energy Options Pricing Model
7.3.1 State Estimation with the Derivative-Free Nonlinear Kalman Filter
7.3.2 Consistency Checking of the Option Pricing Model
7.4 Simulation Tests
7.4.1 Estimation with the Use of an Accurate Energy Pricing Model
7.4.2 Detection of Mispricing in the Energy Pricing Model
8 Corporations’ Default Probability Forecasting Using the Derivative-Free Nonlinear Kalman Filter
8.1 Outline
8.2 Company’s Credit Risk Models
8.2.1 The Merton-KMV Credit-Risk Model
8.2.2 Computation of a Company’s Distance to Default
8.3 Estimation of the Market Value of the Company Using ƒ
8.3.1 State-Space Description of the Black–Scholes Equation
8.4 Forecasting Default with the Derivative-Free Nonlinear Kalman Filter
8.4.1 State Estimation with the Derivative-Free Nonlinear Kalman Filter
8.4.2 The Derivative-Free Nonlinear Kalman Filter as Extrapolator
8.4.3 Forecasting of the Market Value Using the Derivative-Free Nonlinear Kalman Filter
8.4.4 Assessment of the Accuracy of Forecasting with the Use of Statistical Criteria
8.5 Simulation Tests
9 Validation of Financial Options Models Using Neural Networks with Invariance to Fourier Transform
9.1 Outline
9.2 Option Pricing in the Energy Market
9.3 Neural Networks Using Hermite Activation Functions
9.3.1 Generalized Fourier Series
9.3.2 The Gauss–Hermite Series Expansion
9.3.3 Neural Networks Using 2D Hermite Activation Functions
9.4 Signals Power Spectrum and the Fourier Transform
9.4.1 Parseval’s Theorem
9.4.2 Power Spectrum of the Signal Using the Gauss–Hermite Expansion
9.5 Simulation Tests
10 Statistical Validation of Financial Forecasting Tools with Generalized Likelihood Ratio Approache
10.1 Outline
10.2 Neuro-Fuzzy Modelling
10.2.1 Problem Statement
10.2.2 Determination of the Number and Type of Fuzzy Rules
10.2.3 Stages of Fuzzy Modelling
10.2.4 Fuzzy Model Validation for the Avoidance of Overtraining
10.3 Fuzzy Model Validation with the Local Statistical Approach
10.3.1 The Exact Model
10.3.2 The Change Detection Test
10.3.3 Isolation of Parametric Changes with the Sensitivity Test
10.3.4 Isolation of Parametric Changes with the Min-Max Test
10.3.5 Model Validation Reduces the Need for Model Retraining
10.4 Detectability of Changes in Fuzzy Models
10.5 Simulation Results
10.5.1 Fuzzy Rule Base in Input Space Partitioning
10.5.2 Fuzzy Modelling with the Input Dimension Partition
11 Distributed Validation of Option Price Forecasting Tools Using a Statistical Fault Diagnosis Appr
11.1 Overview
11.2 State Estimation for the Black–Scholes PDE
11.2.1 State-Space Description of the Black–Scholes PDE
11.2.2 State Estimation with Kalman Filtering
11.3 Distributed Forecasting Model
11.4 Consistency of the Kalman Filter
11.5 Equivalence Between Kalman Filters and Regressor Models
11.6 Change Detection of the Fuzzy Kalman Filter Using the Local Statistical Approach
11.6.1 The Global χ2 Test for Change Detection
11.6.2 Isolation of Inconsistent Kalman Filter Parameters with the Sensitivity Test
11.6.3 Isolation of Inconsistent Kalman Filter Parameters with the Min–Max Test
11.7 Simulation Tests
11.7.1 Distributed State Estimation of the Black–Scholes PDE
11.7.2 Simulation Results
12 Stabilization of Financial Systems Dynamics Through Feedback Control of the Black-Scholes PDE
12.1 Outline
12.2 Transformation of the Black-Scholes PDE into Nonlinear ODEs
12.2.1 Decomposition of the PDE Model into Equivalent ODEs
12.2.2 Modeling in State-Space Form of the Black-Scholes PDE
12.3 Differential Flatness of the Black-Scholes PDE Model
12.4 Computation of a Boundary Conditions-Based Feedback Control Law
12.5 Closed Loop Dynamics
12.6 Simulation Tests
13 Stabilization of the Multi-asset Black–Scholes PDE Using Differential Flatness Theory
13.1 Outline
13.2 Boundary Control of the Multi-asset Black–Scholes PDE
13.3 Flatness-Based Control of the Multi-asset Black–Scholes PDE
13.4 Stability Analysis of the Control Loop
13.5 Simulation Tests
14 Stabilization of Commodities Pricing PDE Using Differential Flatness Theory
14.1 Outline
14.2 Models for Commodities Pricing
14.2.1 Elaborated Schemes for Trading Electric Power
14.2.2 Commodities Pricing with the Single-Factor PDE Model
14.2.3 Commodities Pricing with the Two-Factor PDE Model
14.2.4 Commodities Pricing with the Three-Factor PDE Model
14.3 Boundary Control of the Multi-factor Commodities Price PDE
14.4 Flatness-Based Control of the Multi-factor Commodities Price PDE
14.5 Stability Analysis of the Control Loop of the Multi-factor Commodities Price PDE
14.6 Simulation Tests
15 Stabilization of Mortgage Price Dynamics Using Differential Flatness Theory
15.1 Outline
15.2 Options Theory-Based PDE Model of Mortgage Valuation
15.3 Computation of the Mortgage Price PDE
15.4 Boundary Control of the Multi-factor Mortgage Price PDE
15.5 Flatness-Based Control of the Multi-factor Mortgage Price PDE
15.6 Stability Analysis of the Control Loop of the Multi-factor Mortgage Price PDE
15.7 Simulation Tests

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