Geometry and Invariance in Stochastic Dynamics: Verona, Italy, March 25-29, 2019

Preface
Random Transformations and Invariance in Stochastic Dynamics
Contents
Some Recent Developments on Lie Symmetry Analysis of Stochastic Differential Equations
	1 Introduction
	2 Symmetries of SDEs Driven by Brownian Motion
		2.1 Strong Symmetries
		2.2 Weak Symmetries
		2.3 Extended Symmetries
	3 Symmetries of SDEs Driven by Discrete Time processes
		3.1 Gauge Symmetries
		3.2 Weak Symmetries of Discrete Time SDEs
	4 Applications
		4.1 Reduction and Reconstruction of Symmetric SDEs
		4.2 Symmetries of SDEs and Symmetries of Corresponding Kolmogorov equations
		4.3 Weak Symmetries of Numerical Schemes for SDEs
	5 Other Results on the Lie Symmetry Analysis of Stochastic Systems
		5.1 Random Symmetries
		5.2 Variational SDEs and Noether Theorem
		5.3 Finite Dimensional Reduction of SPDEs
	References
Markov Processes with Jumps on Manifolds and Lie Groups
	1 Introduction
	2 Markov and Feller Processes
	3 Stochastic Differential Equations on Manifolds
	4 The Positive Maximum Principle and Courrège Theory
		4.1 The Courrège Theorem on Euclidean Space and Manifolds
		4.2 The Courrège Theorem on Lie Groups
	5 Invariant Markov Processes
	6 Inhomogeneous Lévy Processes
	7 Decomposition of Invariant Markov Processes
	References
Asymptotic Expansion for a Black–Scholes Model with Small Noise Stochastic Jump-Diffusion Interest Rate
	1 Introduction
	2 The Asymptotic Expansion
		2.1 The General Setting
		2.2 The Asymptotic Character of the Expansion of the Solution Xεt of the SDE in Powers of ε
	3 The Black–Scholes Model with Stochastic Interest Rate
	4 Conclusions
	References
Stochastic Geodesics
	1 Introduction
	2 Geodesics on Riemannian Manifolds
	3 The Frame Bundle and the Laplacians
	4 Stochastic Analysis on Manifolds
	5 Stochastic Geodesics
	6 Stochastic Geodesics on Lie Groups
	7 Relation with Stochastic Forward-Backward Differential Equations
	References
A Note on Supersymmetry and Stochastic Differential Equations
	1 Introduction
	2 Super-Geometry and Gaussian Super-Fields
		2.1 Some Notions of Super-Geometry
		2.2 Construction of the Super-Field
		2.3 Relation with SDEs
	3 Supersymmetry and the Supersymmetric Field
		3.1 The Supersymmetry
		3.2 Localization of Supersymmetric Averages
	References
Quasi-shuffle Algebras in Non-commutative Stochastic Calculus
	1 Introduction
	2 Karandikar's Axioms and Quasi-shuffle Algebras
		2.1 Itô Calculus for Semimartingales
		2.2 Singular Quasi-shuffle Algebras and Stratonovich Calculus
		2.3 Shuffle Algebra and Continuous Semimartingales
	3 Chronological Calculus for Stochastic Integration
		3.1 Chronological Calculus and Pre-Lie Algebra
		3.2 Chronological Itô Calculus
	References
Higher Order Derivatives of Heat Semigroups on Spheres and Riemannian Symmetric Spaces
	1 Introduction
	2 Brownian Motion on Spheres as Symmetric Spaces
		2.1 The Sphere as a Symmetric Space
		2.2 Derivatives of the Heat Semigroup
		2.3 Higher Derivatives of Ptf
	3 Decomposition and Conditioning of TξtTξt
		3.1 Decomposition of the Flow
		3.2 Expectations of Representations of Random Matrices: An Elementary Lemma
		3.3 Calculation for Sn
	4 Main Result for Sn
		4.1 Two Alternative Approaches
	5 Extensions
		5.1 Higher Order Derivatives
		5.2 More General Diffusion Semi-groups
		5.3 Questions
	References
Rough Homogenisation with Fractional Dynamics
	1 Introduction
		1.1 Main Results
	2 Homogenization via Rough Continuity
		2.1 CLT for Stationary Processes
		2.2 Fractional Ornstein Uhlenbeck as Fast Dynamics
		2.3 The Rough Path Topology
		2.4 Homogenization via Rough Continuity
		2.5 Examples Satisfying Assumption 2.7
	3 Recent Progress on Slow/Fast Markovian Dynamics
		3.1 The Basic Averaging Principal
		3.2 Quantitative Locally Uniform LLN
		3.3 Averaging with Hörmander's Conditions
		3.4 Diffusion Homogenisation Theory
	4 Appendix
	References
Stochastic Geometric Mechanics with Diffeomorphisms
	1 Noether's Theorem in Geometric Mechanics
		1.1 Euler-Poincaré Reduction
		1.2 Sobolev Class Diffeomorphisms
		1.3 Stochastic Advection by Lie Transport (SALT)
		1.4 Semidirect Product Group Adjoint and Coadjoint Actions
	2 Deterministic Geometric Fluid Dynamics
	3 Stochastic Geometric Fluid Dynamics
	References
McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations
	1 Introduction and Motivations
		1.1 General Considerations
		1.2 Some Motivating Examples
		1.3 Structure of the Paper
	2 McKean Representations of Non-linear Fokker-Planck Equations
		2.1 Probabilistic Representation of Linear Fokker-Planck Equations
		2.2 McKean Probabilistic Representation of Non-linear Fokker-Planck Equation
	3 McKean Feynman-Kac Representations for Non-conservative and Non-linear PDEs
	4 McKean Representation of a Fokker-Planck Equation with Terminal Condition
	5 Probabilistic Representation with Jumps for Non-conservative PDEs
	6 McKean SDEs in Random Environment
		6.1 The (S)PDE and the Basic Idea
		6.2 Well-Posedness of the SPDE
		6.3 McKean Equation in Random Environment
	7 McKean Representation of Stochastic Control Problems
		7.1 Stochastic Control Problems and Non-linear Partial Differential Equations
		7.2 McKean Type Representation in a Toy Control Problem Example
	References
Bernstein Processes, Isovectors and Mechanics
	1 Introduction
	2 Summary
	3 A Summary of Schrödinger's Problem and Its Solution
	4 The Method of Isovectors
		4.1 Examples
	5 Bernstein Processes
	6 Parametrization of a One-Factor Affine Model in Stochastic Finance
	7 Generalized Brownian Bridges
	8 Conclusion
	References
On the Positivity of Local Mild Solutions to Stochastic Evolution Equations
	1 Introduction
	2 Assumptions and Main Result
	3 Auxiliary Results
	4 Proof of Theorem 2.2
		4.1 Approximation of the solution
		4.2 Positivity
	5 Positivity of Forward Rates
	References
Invariance of Poisson Point Processes by Moment Identities with Statistical Applications
	1 Introduction
	2 Moments of Poisson Point Processes
	3 Slivnyak-Mecke Identity
	4 Nonlinear Slivnyak-Mecke Identities
	5 Random Sets in Stochastic Geometry
	6 Multiparameter Integrals in Random Graphs
	7 Random-Connection Model
	8 Moments of Poisson Shot Noise Processes
	9 Computation of Joint Moments
	10 Gram-Charlier Expansions
	References