“Aut Caesar, aut nihil.” ― Cesare Borgia

Fokker-Planck Equations (2024)

Reference: Linear and Nonlinear Fokker-Planck Equations: Analysis & Probabilistic Counterparts, Marco Reheimer

Ⅰ. Linear FPEs: Existence and Uniqueness

Ⅱ. Superposition Principle

Ⅲ. Connection with Markov Process

Ⅳ. Nonlinear FPEs: Existence & Uniqueness and its Superposition Principle

Ⅴ. Flow Selection for NLFPEs

Ⅵ. Connections with Nonlinear Markov Process

Continuous Time Mathematical Finance (2024)

Reference: Introduction on Financial Math II, Peter Bank

Ⅰ. Preliminary & Settings

Ⅱ. Hedging and Pricing: Basic THMs and Applications on Kunita Watanabe Decompositions

Ⅲ. Financial Modeling: Brownian Financial Models, Volatility Models and Term Structure Models

Ⅳ. Financial Optimization: Metron’s Problem & HJB Equations and Convex Dual Method

Mean Field Games (2024)

References: [1].Mean field games and interacting particle systems, Daniel Lacker [2].Mean field games with common noise, Rene Carmona, Francois Delarue, Daniel Lacker

Ⅰ. Prelininary: Intro., Convergence and Metric, Stochastic Control

Ⅱ. Mckean Vlasov Equations: Limit THM, PDE System, Loss of Markovian, Common Noise and Long Time Behavior

Ⅲ. Static MFG: Limit THM and Multiple Type of Agents

Ⅳ. Stochatic Differential Game: Two Player Games and n-Player Games

Ⅴ. Stochastic MFG: Limit THM, Semilinear MFG Example, Multiple Typer of Agents

Ⅵ. The Master Equations: Calculus on Measure Space, Ito Formula, Verfication THM

Ⅶ. MFG with Common Noise

Rough Path Analysis and Signatures (2024)

References: [1].Rough Path Theory, Andrew L. Allan [2]. A Course on Rough Paths, Peter K. Friz, Martin Hairer [3].An Introduction to the Theory of Rough Paths, Xi Geng [4]. Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Peter K. Friz, Nicolas B. Victoir

Malliavin Calculus (2023)

References: [1].An Introduction to Malliavin Calculus , Markus Kunze (primary) [2].Introduction to Malliavin Calculus, M. Hairer

Ⅰ. Wiener Chaos and its Decomposition

Ⅱ. Malliavin Derivative: Properties, Divergence Operator and Ornstein-Uhlenbeck Semigroup

Ⅲ. Stochastic Calculus: Mutiple Wiener Integrals, Calculus for White Noise and Ito Integrals

Ⅳ. Smoothness of Probability Laws: Absolutely Continuous, Smoothness, SDEs and Hormander’s THM

Stochastic Partial Differential Equations (2023)

References:[1] An Introduction to Stochastic PDEs, Martin Hairer [2] Stochastic Partial Differential Equation, Étienne Pardoux

Ⅰ.Gaussian Measure Theory: Cameron-Martin Space, Image of Gaussian Measures and Cylindrical Wiener Processes

Ⅱ. Linear SPDEs: Ito Calculus in Hilbert spce, Regularity of Solutions and its Long Time Behaviors

Ⅲ. Semilinear SPDEs: Existence and Uniqueness, Sobolev Embedding and Some Examples

Ⅳ. SPDEs Driven by Space-Time White Noises and Super Brownian Motions

Diffusive Processes and its Applications (2022)

Reference: Stochastic Differential Equations, Øksendal B.

Ⅰ.Ito Integral and Ito Process

Ⅱ.Example:Filtering Problem

Ⅲ.Diffusion Process: Properties, Generators, Feynman-Kac Formula, Mart. Problem, Random Time Change, Girsanov THM of Diffusions

Ⅳ.Boundary Value Problem: Poisson-Dirichlet Problem

Ⅴ.Optimal Stopping

Ⅵ.Application in Stochastic Control: HJB Equations and Terminal Conditions

Ⅶ.Application in Finance: Attainable and Complete, Option Pricing

Complement: Stationary Distribution of Diffusions

Mathematical Finance (2023)

References: [1].Mathematical Finance, N. H. Bingham (primary) [2].Mathematical Finance, Steve Lalley [3].Lectures on Malliavin calculus and its applications to finance, Eulalia Nualart

Ⅰ. Economic Background

Ⅱ. Finance in Discrete Time: Arbitrage, Completeness, Discrete Time Black-Scholes Model and Option

Ⅲ. Finance in Continuous Time: GBMs, Black-Scoles Model and BS PDE, Barrier Option and CM Formula

Ⅳ. Insurance Math: Ruin Problem

Ⅴ.Foregin Exchange

Stochastic Calculus (2022)

References:[1]Brownian Motion, Martingales, and Stochastic Calculus, Le Gall, Jean-François(primary) [2] Continuous martingales and Brownian motion, Daniel Revuz, Marc Yor [3]Brownian Motion and Stochastic Calculus, Ioannis Karatzas, Steven E. Shreve

Ⅰ. Continuous Martingales

Ⅱ. Continuous Semimartingales

Ⅲ. Stochastic Integration: Construction, Ito Formula, Representation of Marts, and Girsanov Thm

Ⅳ. Local Time and Generalized Ito Formula

Ⅴ. General Theory of Markov Process

Ⅵ. Levy Processes

Ⅶ. SDEs: Existence & Uniqueness, Yamada Theory and Local Solutions & Blowup

Complement on Motivation of Gaussian white noise and Origin of BM

Complement on Continuous Local Mart

Complement on Stochastic Integration