From Probability to Finance: Lecture Notes of BICMR Summer School on Financial Mathematics

Preface
Contents
Continuous-State Branching Processes with Immigration
	1 Laplace Transforms of Measures
	2 Construction of CB-Processes
	3 Some Basic Properties
	4 Positive Integral Functionals
	5 Construction of CBI-Processes
	6 Structures of Sample Paths
	7 Martingale Problem Formulations
	8 Stochastic Equations for CBI-Processes
	9 Local and Global Maximal Jumps
	10 A Coupling of CBI-Processes
	References
Enlargement of Filtration in Discrete Time
	1 Some Well-Known Results and Definitions
		1.1 Basic Definitions
		1.2 mathbbH-Martingales
		1.3 Doob\'s Decomposition and Applications
		1.4 Projections
		1.5 Multiplicative Decomposition
		1.6 Stochastic Exponential Process
		1.7 Stopping Times and Local Martingales
		1.8 Change of Probability
		1.9 Some Facts on Finance
		1.10 Enlargement of Filtration
	2 Initial Enlargement
		2.1 Bridge
		2.2 Viability
		2.3 Initial Enlargement with a mathbbZ-Valued Random Variable ξ
		2.4 Example: Supremum of a Random Walk
	3 Introduction to Progressive Enlargement
		3.1 General Results
		3.2 Immersion in Progressive Enlargement
	4 Progressive Enlargement Before τ
		4.1 Semimartingale Decomposition
		4.2 Viability
	5 Progressive Enlargement After τ
		5.1 General Case
		5.2 Honest Times
		5.3 Viability
	6 Pseudo-Stopping Times
	7 An Optional Representation for Martingales in a Progressive Enlargement
	8 Other Enlargements
		8.1 Enlargement with a Pair (ζ,τ)
		8.2 Enlargement with a Process
	9 Credit Risk
	References
Clustering Effects via Hawkes Processes
	1 Introduction
	2 Point Processes: An Overview
		2.1 What is a Point Processes?
		2.2 Compensator and Intensity
		2.3 The Poisson Process
		2.4 Change of Probability
	3 Hawkes Processes
		3.1 A Time Change Approach to Hawkes Process
		3.2 A Change of Probability Approach of the Hawkes Process
		3.3 Maximum Likelihood Estimation
		3.4 Limit Behaviour
	4 Moments
		4.1 Moments of the Intensity
		4.2 Recurrence Formulae for Moments
	5 Generalization of Hawkes Processes
		5.1 Marked Hawkes Processes
		5.2 Nonlinear Hawkes Processes
	6 Financial Applications of Hawkes Processes
		6.1 Calibration of Hawkes Processes
		6.2 Interest Rate Model with Hawkes Jumps
		6.3 Hawkes Processes for Credit Modelling
		6.4 Clustering for Trading Orders and Optimal Execution
	7 A Branching Processes Point of View
		7.1 Branching Property and Integral Representation
		7.2 Hawkes Processes and CBI Processes
		7.3 Extensions
	References
Bernstein Copulas and Composite Bernstein Copulas
	1 Introduction
	2 Bernstein Copulas
		2.1 Definition of Bernstein Copulas
		2.2 The BC Generated by a Copula Function
		2.3 Convergence of Bernstein Copulas
		2.4 The Density Function of the BC
		2.5 Dependence Measures
		2.6 Baker\'s Distribution—One Family of Distributions  with BCs
		2.7 Random Numbers Generation
	3 Composite Bernstein Copula
		3.1 Definition of Composite Bernstein Copula
		3.2 Properties of the Composite Bernstein Copula
		3.3 Reproduction Property
		3.4 Bivariate Tail Dependence
		3.5 Probabilistic Expressions for the CBC
		3.6 The Simulation Method for CBC
		3.7 Some Families of the CBC
	4 Simulation Study and Stock Data Analysis
		4.1 A Simulation Study
		4.2 Stock Data Analysis
	5 Conclusions
	References
Wealth Transfers, Indifference Pricing, and XVA Compression Schemes
	1 Introduction
		1.1 Abbreviations
	2 XVA Framework
		2.1 Agents
		2.2 Cash Flows
		2.3 Valuation Operator
		2.4 Contra-Assets and Contra-Liabilities
		2.5 Cost of Capital
		2.6 Funds Transfer Price
		2.7 A General Result
	3 Wealth Transfers Triggered by Market Incompleteness
		3.1 The Limiting Case of Complete Markets
		3.2 DVA Wealth Transfer Triggered by Shareholders Not Being Able to Redeem Bank Debt
		3.3 CVACL Wealth Transfer Triggered by Shareholders Bankruptcy Costs
		3.4 FVA Wealth Transfer Triggered by the Impossibility to REPO Derivatives
		3.5 MVA Wealth Transfer Triggered by Different Funding Policies for Initial Margins
		3.6 KVA Wealth Transfer Triggered by The Cost of Capital Which Is Required by the Impossibility of Hedging out Counterparty Default Losses
	4 XVA Formulas and Wealth Transfers in a Static Setup
		4.1 Cash Flows
		4.2 Static XVA Formulas
	5 Derivative Management: From Hedging to XVA Compression
		5.1 Capital/Collateral Optimisation of Inter-Dealer Trades
		5.2 Optimal Liquidation of the CCP Portfolio of a Defaulted Clearing Member
		5.3 XVA Compression Cycles
	References