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Unformatted text preview: Stochastic Differential Equations: Models and Numerics 1 Jesper Carlsson Kyoung-Sook Moon Anders Szepessy Raul Tempone Georgios Zouraris February 2, 2010 1 This is a draft. Comments and improvements are welcome. Contents 1 Introduction to Mathematical Models and their Analysis 4 1.1 Noisy Evolution of Stock Values . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Optimal Control of Investments . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Calibration of the Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 The Coarse-graining and Discretization Analysis . . . . . . . . . . . . . 8 2 Stochastic Integrals 11 2.1 Probability Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Approximation and Definition of Stochastic Integrals . . . . . . . . . . . 13 3 Stochastic Differential Equations 23 3.1 Approximation and Definition of SDE . . . . . . . . . . . . . . . . . . . 23 3.2 It os Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Stratonovich Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Systems of SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 The Feynman-K ac Formula and the Black-Scholes Equation 38 4.1 The Feynman-K ac Formula . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 The Monte-Carlo Method 44 5.1 Statistical Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Time Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6 Finite Difference Methods 54 6.1 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.2 Lax Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7 The Finite Element Method and Lax-Milgrams Theorem 62 7.1 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Error Estimates and Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . 67 7.2.1 An A Priori Error Estimate . . . . . . . . . . . . . . . . . . . . . . 67 1 7.2.2 An A Posteriori Error Estimate . . . . . . . . . . . . . . . . . . . 69 7.2.3 An Adaptive Algorithm . . . . . . . . . . . . . . . . . . . . . . . 70 7.3 Lax-Milgrams Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8 Markov Chains, Duality and Dynamic Programming 76 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.3 Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.4 Duality and Qualitative Properties . . . . . . . . . . . . . . . . . . . . . . 81 8.5 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . .8....
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