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More about usStochastic 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.

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

2.1 Probability Background

2.2 Brownian Motion

2.3 Approximation and Definition of Stochastic Integrals

3.1 Approximation and Definition of SDE

3.2 Itˆo’s Formula

3.3 Stratonovich Integrals

3.4 Systems of SDE

4.1 The Feynman-K?ac Formula

4.2 Black-Scholes Equation

5.1 Statistical Error

5.2 Time Discretization Error

6.1 American Options

6.2 Lax Equivalence 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

8.1 Introduction

8.2 Markov Chains

8.3 Expected Values

8.4 Duality and Qualitative Properties

8.5 Dynamic Programming

8.6 Examples and Exercises

9.1 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

10.1 Invariant Measures and Ergodicity

10.2 Reaction Rates

10.3 Reaction Paths

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

12.1 Tomography Exercise

12.2 Molecular Dynamics

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