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stochastic differential equations, jesper carlsson [pdf]

Stochastic Differential Equations by Jesper Carlsson, Kyoung-Sook Moon, Anders Szepessy, Ra´ul Tempone and Georgios Zouraris

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Stochastic Differential Equations by Jesper Carlsson , Kyoung-Sook Moon , Anders Szepessy , Ra´ul Tempone and Georgios Zouraris .

Stochastic Differential Equations by Jesper Carlsson , Kyoung-Sook Moon , Anders Szepessy , Ra´ul Tempone and Georgios Zouraris cover the following topics.

Mathematical Models and their Analysis, Stochastic Integrals, Stochastic Differential Equations, The Feynman-K?ac Formula and the Black-Scholes Equation
  • 1. Introduction to Mathematical Models and their Analysis
    1.1 Noisy Evolution of Stock Values
    1.2 Molecular Dynamics
    1.3 Optimal Control of Investments
    1.4 Calibration of the Volatility
    1.5 The Coarse-graining and Discretization Analysis

  • 2. Stochastic Integrals
    2.1 Probability Background
    2.2 Brownian Motion
    2.3 Approximation and Definition of Stochastic Integrals

  • 3. Stochastic Differential Equations
    3.1 Approximation and Definition of SDE
    3.2 Itˆo’s Formula
    3.3 Stratonovich Integrals
    3.4 Systems of SDE

  • 4. The Feynman-K?ac Formula and the Black-Scholes Equation
    4.1 The Feynman-K?ac Formula
    4.2 Black-Scholes Equation

  • 5. The Monte-Carlo Method
    5.1 Statistical Error
    5.2 Time Discretization Error

  • 6. Finite Difference Methods
    6.1 American Options
    6.2 Lax Equivalence Theorem

  • 7. The Finite Element Method and Lax-Milgram’s Theorem
    7.1 The Finite Element Method
    7.2 Error Estimates and Adaptivity
    7.2.1 An A Priori Error Estimate
    7.2.2 An A Posteriori Error Estimate
    7.2.3 An Adaptive Algorithm
    7.3 Lax-Milgram’s Theorem

  • 8. Markov Chains, Duality and Dynamic Programming
    8.1 Introduction
    8.2 Markov Chains
    8.3 Expected Values
    8.4 Duality and Qualitative Properties
    8.5 Dynamic Programming
    8.6 Examples and Exercises

  • 9. Optimal Control and Inverse Problems
    9.1 The Determinstic Optimal Control Setting
    9.1.1 Examples of Optimal Control
    9.1.2 Approximation of Optimal Control
    9.1.3 Motivation of the Lagrange formulation
    9.1.4 Dynamic Programming and the HJB Equation
    9.1.5 Characteristics and the Pontryagin Principle
    9.1.6 Generalized Viscosity Solutions of HJB Equations
    9.1.7 Maximum Norm Stability of Viscosity Solutions
    9.2 Numerical Approximation of ODE Constrained Minimization
    9.2.1 Optimization Examples
    9.2.2 Solution of the Discrete Problem
    9.2.3 Convergence of Euler Pontryagin Approximations
    9.2.4 How to obtain the Controls
    9.2.5 Inverse Problems and Tikhonov Regularization
    9.2.6 Smoothed Hamiltonian as a Tikhonov Regularization
    9.2.7 General Approximations
    9.3 Optimal Control of Stochastic Differential Equations
    9.3.1 An Optimal Portfolio
    9.3.2 Dynamic Programming and HJB Equations
    9.3.3 Relation of Hamilton-Jacobi Equations and Conservation Laws
    9.3.4 Numerical Approximations of Conservation Laws and HamiltonJacobi Equations

  • 10. Rare Events and Reactions in SDE
    10.1 Invariant Measures and Ergodicity
    10.2 Reaction Rates
    10.3 Reaction Paths

  • 11. Molecular Dynamics
    11.1 Molecular dynamics at constant temperature: Zwanzig’s model and derivation of Langevin dynamics
    11.2 The Gibbs distribution derived from dynamic stability
    11.3 Smoluchowski dynamics derived from Langevin dynamics
    11.4 Macroscopic conservation laws for compressible fluids motivated from molecular dynamics
    11.4.1 A general potential

  • 12. Appendices
    12.1 Tomography Exercise
    12.2 Molecular Dynamics

  • 13. Recommended Reading

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