Stochastic Processes, Statistical Methods, and Engineering Mathematics: SPAS 2019, Västerås, Sweden, September 30–October 2

Preface
Contents
Contributors
Part I Stochastic Processes and Analysis
1 An Improved Asymptotics of Implied Volatility in the Gatheral Model
	1.1 Introduction
	1.2 Previous Results
	1.3 The Expansion of Order 3
	1.4 A Sketch of a Proof
		1.4.1 Preliminaries
		1.4.2 Order 3
	1.5 Conclusions and Future Work
	References
2 Ruin Probability for Merged Risk Processes with Correlated Arrivals
	2.1 Introduction
	2.2 The Compound Poisson Model with Phase Type Claims
		2.2.1 Phase Type Distribution
		2.2.2 Ruin Probability for Phase Type Claims
	2.3 Correlated Poisson Arrivals with Exponential Claims
		2.3.1 Transition to a Single Poisson Compound Process
		2.3.2 Numerical Example
		2.3.3 An Auxiliary Result
	2.4 Correlated Poisson Arrivals with Phase Type Claims
		2.4.1 The Phase Type Distribution of Claims of the Merged Process
		2.4.2 Example
		2.4.3 Ruin Probability in Case of Phase Type Claims
	2.5 Conclusion
	References
3 Method Development for Emergent Properties in Stage-Structured Population Models with Stochastic Resource Growth
	3.1 Introduction
	3.2 The Deterministic Stage Model
	3.3 Stochastic Stage Model
	3.4 The Recovery Potential
	3.5 Probability of Extinction for Stochastic Case
	3.6 Resilience for Stage Model
	3.7 Simulation Results
		3.7.1 Stage-Structured Biomass Dynamics and Yield
		3.7.2 Impact on Size Structure and Biomass
		3.7.3 The Stock Recovery Potential
		3.7.4 The Probability of Extinction
		3.7.5 Resilience
	3.8 Conclusions and Discussion
	3.9 Appendix 3.1. Stage-Structured Biomass Model with Logistic Resource Dynamics
	3.10 Appendix 3.2. Proof of Uniqueness of the Solution w*J(R)
	References
4 Representations of Polynomial Covariance Type Commutation Relations by Linear Integral Operators on Lp Over Measure Spaces
	4.1 Introduction
	4.2 Preliminaries and Notations
	4.3 Representations by Linear Integral Operators
	References
5 Computable Bounds of Exponential Moments of Simultaneous Hitting Time for Two Time-Inhomogeneous Atomic Markov Chains
	5.1 Introduction
	5.2 Notation
	5.3 Geometric Drift Condition
		5.3.1 Drift Condition for Inhomogeneous Markov Chains
		5.3.2 Constructing a Sequence that Dominates Return Time
	5.4 Main Result
	5.5 Auxiliary Lemmas
	References
6 Valuation and Optimal Strategies for American Options Under a Markovian Regime-Switching Model
	6.1 Introduction
	6.2 A Markovian Regime-Switching Model
	6.3 Pricing of American Options
	6.4 Some Properties for Optimal Strategy
		6.4.1 Preliminaries
		6.4.2 Lemmata
		6.4.3 Properties
		6.4.4 Optimal Strategy
	6.5 Numerical Examples
		6.5.1 Model Implementation
		6.5.2 Numerical Results
	6.6 Conclusion and Future Research
	References
7 Inequalities for Moments of Branching Processes in a Varying Environment
	7.1 Introduction
	7.2 Main Results
	7.3 Auxiliary Results
	7.4 Proof of the Main Results
	References
8 A Law of the Iterated Logarithm for the Empirical Process Based Upon Twice Censored Data
	8.1 Introduction
	8.2 Product-Limit Estimator
	8.3 Results
	8.4 Simulation
	8.5 Proofs of the Lemmas
	References
9 Investigating Some Attributes of Periodicity in DNA Sequences via Semi-Markov Modelling
	9.1 Introduction
	9.2 The Basic Framework
		9.2.1 The Homogeneous Case
		9.2.2 The Case of Partial Non Homogeneity
	9.3 Quasiperiodicity
	9.4 Illustrations of Real and Synthetic Data
		9.4.1 DNA Sequences of Synthetic Data
		9.4.2 DNA Sequences of Real Data
	9.5 Conclusion
	References
10 Limit Theorems of Baxter Type for Generalized Random Gaussian Processes with Independent Values
	10.1 Introduction
	10.2 The Covariance Functional of a Generalized Random Process with Independent Values
	10.3 The Families of Test Functions
	10.4 Convergence of Baxter Sums
	10.5 Conditions of Singularity of Measures
	References
11 On Explicit Formulas of Steady-State Probabilities for the [ M/M/c/c+m ]-Type Retrial Queue
	11.1 Introduction
	11.2 Mathematical Model and Ergodicity Condition
	11.3 Steady-State Distribution
	11.4 Numerical Results
	11.5 Conclusion and Future Research
	References
12 Testing Cubature Formulae on Wiener Space Versus Explicit Pricing Formulae
	12.1 Introduction and Background
	12.2 Implementation of Cubature Formulae on Wiener Space
		12.2.1 SDE and Stochastic Integral (Itô)
		12.2.2 SDE and Stochastic Integral (Stratonovich)
		12.2.3 Itô Integral Versus Stratonovich Integral
	12.3 Construction of Cubature Formulae on Wiener Space
		12.3.1 The Trajectories of SDE (12.6)
	12.4 Cubature Formula of Degree 5
		12.4.1 Black–Scholes Versus Cubature Pricing Formula (Degree 5)
		12.4.2 Construction of a Trinomial Model Based on the Cubature Formula
	12.5 Cubature Formula of Degree 7
		12.5.1 Black–Scholes Versus Cubature Formula (Degree 7)
	12.6 Conclusion and Future Works
	References
13 Gaussian Processes with Volterra Kernels
	13.1 Introduction
	13.2 Gaussian Volterra Processes and Their Smoothness Properties
	13.3 Gaussian Volterra Processes with Sonine Kernels
		13.3.1 Fractional Brownian Motion and Sonine Kernels
		13.3.2 A General Approach to Volterra Processes with Sonine Kernels
	13.4 Examples of Sonine Kernels
	13.5 Appendix
		13.5.1 Inequalities for Norms of Convolutions and Products
		13.5.2 Continuity of Trajectories and Hölder Condition
		13.5.3 Application of Fractional Calculus
		13.5.4 Existence of the Solution to Volterra Integral Equation Where the Integral Operator Is an Operator of Convolution with Integrable Singularity at 0
	References
14 Stochastic Differential Equations Driven by Additive Volterra–Lévy and Volterra–Gaussian Noises
	14.1 Introduction
	14.2 Brief Description of Volterra–Lévy Processes
	14.3 Moment Upper Bounds and Hölder Properties of Volterra–Lévy Processes
		14.3.1 General Upper Bounds for the Incremental Moments
		14.3.2 Incremental Moments and Hölder Continuity Under Power Restrictions on the Kernel g
		14.3.3 Application of the Upper Bounds for the Incremental Moments to Volterra–Lévy Processes of Three Types
		14.3.4 Examples of Volterra–Lévy Processes with Power Restrictions on the Kernel
		14.3.5 Sonine Pairs and Two Kinds of Volterra–Gaussian Processes
	14.4 Equations with Locally Lipschitz Drift of Linear Growth
	14.5 Equations with Volterra–Gaussian Processes
		14.5.1 Girsanov Theorem. Definition of Weak and Strong Solutions
		14.5.2 Weak Existence and Weak Uniqueness
		14.5.3 Pathwise Uniqueness of Weak Solution. Existence and Uniqueness of Strong Solution
	References
15 Fixed Point Results of Generalized Cyclic Contractive Mappings in Multiplicative Metric Spaces
	15.1 Introduction
		15.1.1 Multiplicative Metric Spaces
		15.1.2 Fixed Points of Maps in Multiplicative Metric Space
	15.2 Cyclic Contraction Mappings
		15.2.1 Fixed Point Results of Cyclic Contraction Mappings
		15.2.2 Well-Posedness Results for Cyclic Contractive Maps
		15.2.3 Limit Shadowing Property for Cyclic Contractive Maps
		15.2.4 Periodic Points of Cyclic Contractive Maps
	References
16 Fixed Points of T-Hardy Rogers Type Mappings and Coupled Fixed Point Results in Multiplicative Metric Spaces
	16.1 Introduction
	16.2 Fixed Points of T-Hardy Rogers Type Contractive Maps
		16.2.1 Well-Posedness Results for T-Hardy Rogers Type Contractions
		16.2.2 Limit Shadowing Property for T-Hardy Rogers Type Contractions
		16.2.3 Periodic Point Property for T-Hardy Rogers Type Contractive Maps
	16.3 Coupled Fixed Points in Multiplicative Metric Spaces
		16.3.1 Coupled Fixed Points
		16.3.2 Coupled Fixed Point Results
		16.3.3 Well-Posedness Result for Coupled Maps
		16.3.4 Application
	References
17 Some Periodic Point and Fixed Point Results in Multiplicative Metric Spaces
	17.1 Introduction
	17.2 Periodic Point Results
	17.3 Cyclic Contractions
	17.4 Applications
	References
18 Bochner Integrability of the Random Fixed Point of a Generalized Random Operator and Almost Sure Stability of Some Faster Random Iterative Processes
	18.1 Introduction and Preliminaries
	18.2 Bochner Integrability of the Fixed Point of a Generalized Random Operator
	18.3 Almost Sure T-Stability Results
	18.4 Application to Random Nonlinear Integral Equation of the Hammerstein Type
	References
19 An Approach to the Absence of Price Bubbles Through State-Price Deflators
	19.1 Introduction
	19.2 Securities Market Model and State-Price Deflators
	19.3 About the Existence of Price Bubbles
	19.4 Vector Spaces Associated to Marketed Strategies
	19.5 Conclusion
	References
20 Form Factors for Stars Generalized Grey Brownian Motion
	20.1 Introduction
	20.2 Generalized Grey Brownian Motion in Arbitrary Dimensions
		20.2.1 Construction of the Mittag-Leffler Measure
		20.2.2 Generalized Grey Brownian Motion
	20.3 Form Factors for Different Classes of Star Generalized Grey Brownian Motion
	20.4 Form Factors for Star Fractional Brownian Motion
	20.5 Conclusion
	References
21 Flows of Rare Events for Regularly Perturbed Semi-Markov Processes
	21.1 Introduction
	21.2 First-Rare-Event Times for Perturbed Semi-Markov Processes
		21.2.1 First-Rare-Event Times
		21.2.2 Asymptotically Uniformly Ergodic Markov Chains
		21.2.3 Necessary and Sufficient Conditions of Weak Convergence for First-Rare-Event Times
	21.3 Counting Processes Generated by Flows of Rare Events
		21.3.1 Counting Processes for Rare-Events
		21.3.2 Necessary and Sufficient Conditions of Convergence for Counting Processes Generated by Flows of Rare Events
	21.4 Markov Renewal Processes Generated by Flows of Rare Events
		21.4.1 Return Times and Rare Events
		21.4.2 Necessary and Sufficient Conditions of Convergence For Markov Renewal Processes Generated by Flows of Rare Events
	21.5 Vector Counting Processes Generated by Flows of Rare Events
		21.5.1 Vector Counting Process for Rare Events
		21.5.2 Necessary and Sufficient Conditions of Convergence for Vector Counting Process Generated by Flows of Rare Events
	References
Part II Statistical Methods
22 An Econometric Analysis of Drawdown Based Measures
	22.1 Introduction
	22.2 Risk Measures
	22.3 Mathematical Models
		22.3.1 ARMA Model
		22.3.2 GARCH Model
		22.3.3 EGARCH Model
	22.4 Application
	22.5 Conclusions
	References
23 Forecasting and Optimizing Patient Enrolment in Clinical Trials Under Various Restrictions
	23.1 Introduction
	23.2 Enrolment Modelling
		23.2.1 Modelling Unrestricted Enrolment
		23.2.2 Modelling Enrolment on Country Level
		23.2.3 Modelling Global Enrolment
	23.3 Modelling Enrolment with Restrictions
		23.3.1 Modelling Enrolment with Restrictions in One Centre
		23.3.2 Modelling Enrolment with Restrictions on Country Level
		23.3.3 Forecasting Global Enrolment Under Country Restrictions
		23.3.4 Using Historic Data for Better Prediction of the Enrolment Rates for the New Trials
	23.4 Optimal Enrolment Design
		23.4.1 Unrestricted Enrolment
		23.4.2 Restricted Enrolment
	23.5 Conclusions
	23.6 Appendix
		23.6.1 Approximation of the Convolution of PG Variables
		23.6.2 Calculation of the Mean of the Restricted Process
		23.6.3 Calculation of the 2nd Moment of the Restricted Process
	References
24 Algorithms for Recalculating Alpha and Eigenvector Centrality Measures Using Graph Partitioning Techniques
	24.1 Introduction
		24.1.1 Notation and Abbreviations
		24.1.2 Graph Concepts
	24.2 The Alpha Centrality Algorithm
		24.2.1 Stages for Algorithm Formulation
		24.2.2 Computing Other Centrality Measures by Using the α-Centrality Measure Algorithm
		24.2.3 Example
	24.3 The Eigenvector Centrality for Large Directed Graphs
		24.3.1 The Eigenvector Centrality Algorithm
		24.3.2 Reformulation of the Power Method
		24.3.3 Computing Eigenvector Centrality of a Graph Componentwise
	24.4 Discussion and Conclusion
	References
25 On Statistical Properties of the Estimator of Impulse Response Function
	25.1 Introduction
	25.2 The Estimator of an Impulse Response Function and Its Properties
	25.3 Trigonometric Basis
	25.4 Square Gaussian Random Variables and Processes
	25.5 On the Rate of Convergence of the Estimator of Impulse Response Function
	25.6 Testing Hypotheses on the Impulse Response Function
	25.7 Simulation Study
	References
26 Connections Between the Extreme Points for Vandermonde Determinants and Minimizing Risk Measure in Financial Mathematics
	26.1 Introduction
	26.2 Money Market Account
	26.3 Derivatives and Arbitrage Pricing
	26.4 Pricing Derivatives
		26.4.1 Discount Bonds and Coupon-Bearing Bonds
		26.4.2 Yield to Maturity
		26.4.3 Spot Rate
		26.4.4 Forward Yields and Forward Rates
		26.4.5 Forward and Future Contracts
	26.5 Options
	26.6 Optimization Model in Finance
		26.6.1 Extreme Points of the Vandermonde Determinant on Various Surfaces Defined as Efficient Frontiers
	26.7 Vandermonde Matrix, Determinant and Portfolio Construction
		26.7.1 Optimum Value of Generalized Variance V[] with Extreme Points of Vandermonde Determinant
	26.8 Conclusion
	References
27 Extreme Points of the Vandermonde Determinant and Wishart Ensemble on Symmetric Cones
	27.1 Introduction
		27.1.1 Gaussian and Chi-Square Distributions
		27.1.2 Laplace Transform and The Wishart Density
		27.1.3 The Gindikin Set and Wishart Joint Eigenvalue Distribution
	27.2 A Quick Jump into Wishart Distribution on Symmetric Cones
	27.3 Extreme Points of the Degenerate Wishart Distribution and Vandermonde Determinant
	27.4 Conclusion
	References
28 Option Pricing and Stochastic Optimization
	28.1 Introduction
	28.2 Problem of Investor
		28.2.1 Problem Statement
		28.2.2 Stochastic Optimization and Probability Functionals
	28.3 Applying Investor Problem for Option Pricing
		28.3.1 Investor Optimal Price
		28.3.2 Examples
		28.3.3 Numerical Results
	28.4 Summary
	References
Part III Engineering Mathematics
29 Stochastic Solutions of Stefan Problems with General Time-Dependent Boundary Conditions
	29.1 Introduction
		29.1.1 Random Walk and the Heat Equation
		29.1.2 A Random Walk Model with Boundary Conditions
	29.2 The Stefan Problem
		29.2.1 The Stefan Condition
		29.2.2 Modelling the Moving Boundary
		29.2.3 Stefan Problem with an Incoming Heat Flux
	29.3 Numerical Results for Stefan Problems
		29.3.1 Stefan Problem with Constant Boundary Condition f(t)=T0
		29.3.2 Stefan Problem with a Special Boundary Condition f(t)=et-1
		29.3.3 Stefan Problem with a Special Heat Flux Boundary Condition h(t)=-q0/sqrtt
		29.3.4 Stefan Problem with Oscillating Boundary Condition
		29.3.5 Stefan Problem with Boundary Condition According to Daytime Temperature Variations
	29.4 Discussion
	29.5 Conclusions
	References
30 Numerical Upscaling via the Wave Equation with Perfectly Matched Layers
	30.1 Introduction
	30.2 The Wave Approach to Approximate a0
	30.3 Perfectly Matched Layer for the Second Order Wave Equation
		30.3.1 The New Local Problem Based on the Wave Equation Combined with PML
	30.4 Numerical Discretization
	30.5 Computational Results
	30.6 Concluding Remarks
	References
31 Homotopy Analysis Method (HAM) for Differential Equations Pertaining to the Mixed Convection Boundary-Layer Flow over a Vertical Surface Embedded in a Porous Medium
	31.1 Introduction
	31.2 Governing Equations
	31.3 Homotopy Analysis Solution
	31.4 Results and Discussion
	31.5 Concluding Remarks
	References
32 Magnetic Force Calculation Between Truncated Cone Shaped Permanent Magnet and Soft Magnetic Cylinder Using Hybrid Boundary Element Method
	32.1 Introduction
	32.2 Problem Definition
		32.2.1 Force Calulation Betwen Circular Loops Loaded with Magnetization Charges
		32.2.2 Force Calulation Betwen Permanent Magnet and Soft Magnetic Cylinder
	32.3 Numerical Results
	32.4 Conclusion
	References
33 A Mathematical Model for Harvesting in a Stage-Structured Cannibalistic System
	33.1 Introduction
	33.2 Model Description
		33.2.1 Biological Assumptions
		33.2.2 The Schematic Diagram of the Model
		33.2.3 The Model
		33.2.4 Stability Analysis
	33.3 Numerical Simulation
		33.3.1 Comparing Harvesting Scenarios
	33.4 Discussion and Conclusion
	References
34 On the Approximation of Physiologically Structured Population Model with a Three Stage-Structured Population Model in a Grazing System
	34.1 Introduction
		34.1.1 Consumer-Resource Systems
		34.1.2 Unstructured and Structured Population Models
		34.1.3 Description of the Grazing System
	34.2 Physiologically Structured Population Models
		34.2.1 Size-Structured Population Models
		34.2.2 Formulation of the Size-Structured Population Model
		34.2.3 Vital Rates
		34.2.4 Modification of the Physiologically Structured Population Model
	34.3 Stage-Structured Population Model
	34.4 Discussion and Conclusion
	References
35 Magnetohydrodynamic Casson Nanofluid Flow Over a Nonlinear Stretching Sheet with Velocity Slip and Convective Boundary Conditions
	35.1 Introduction
	35.2 Mathematical Formulation
	35.3 Results and Discussion
	35.4 Velocity Profiles
	35.5 Temperature Profiles
	35.6 Concentration Profiles
	35.7 Conclusion
	References
36 Mathematical and Computational Analysis of MHD Viscoelastic Fluid Flow and Heat Transfer Over Stretching Surface Embedded in a Saturated Porous Medium
	36.1 Introduction
	36.2 Mathematical Formulation
	36.3 Boundary Conditions
		36.3.1 Prescribed Surface Temperature (PST)
		36.3.2 Prescribed Wall Heat Flux (PHF)
	36.4 Dimensionless Quantities
	36.5 Reduced Non-linear Ordinary Differential Equations
	36.6 Reduced Boundary Conditions
		36.6.1 Prescribed Surface Temperature (PST)
		36.6.2 Prescribed Heat Flux (PHF)
	36.7 Results and Discussion
	36.8 Conclusion
	References
37 Numerical Solution of Boundary Layer Flow Problem of a Maxwell Fluid Past a Porous Stretching Surface
	37.1 Introduction
	37.2 Mathematical Formulation
	37.3 Physical Quantities
	37.4 Numerical Solution of the Problem
	37.5 Results and Discussion
	37.6 Conclusions
	References
38 Effect of Electromagnetic Field on Mixed Convection of Two Immiscible Conducting Fluids in a Vertical Channel
	38.1 Introduction
	38.2 Mathematical Formulation
	38.3 Analytical Solutions
		38.3.1 Special Cases
	38.4 Results and Discussion
	38.5 Conclusion
	38.6 Nomenclature
	References
39 Stochastic Smart Grid Meter for Industry 4.0—From an Idea to the Practical Prototype
	39.1 Introduction
	39.2 Multibit SFADC
	39.3 Base Conditions and Limitations
	39.4 Mathematical Model of the SFADC Measurement Uncertainty
	39.5 Multibit SMI
	39.6 Stochastic Digital Electrical Energy Meter
	39.7 Hardware Prototype of 4-Bit SDEEM
	39.8 Measurement Results
	39.9 Conclusion
	References
40 Mathematical Basis of the Stochastic Digital Measurement Method
	40.1 Introduction
	40.2 Measurement of the Mean Value of the Signal
	40.3 Measurement of the Mean Value of the Product of Two Signals Using Two-Bit SDMM
		40.3.1 Measurement in Time Domain
		40.3.2 Measurement in Fourier Domain
	40.4 Stochastic Digital DFT Processor (SDDFT Processor)
	40.5 Application of SDMM
	40.6 Discussion
	40.7 Conclusion
	References
Subject Index
Index
Author Index
Author Index