Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
1. Black-Scholes formula.
2. Change of variables for Kolmogorov equation.
3. Mean reverting equation.
4. Affine SDE.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
9. Hamilton-Jacobi Equations.
A. Characteristics.
B. Hamilton equations.
C. Lagrangian.
D. Connection between Hamiltonian and Lagrangian.
E. Lagrangian for heat equation.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.


et MATH MATH be a smooth function. We introduce the action functional MATH taking vector functions of $s$ and producing a number. We consider a problem of minimization of $A$ across a variety of smooth functions $v$ with fixed values of $s~$ at the ends of the interval $\left[ 0,t\right] $ .

If a function $v$ minimizes $A\left[ v\right] $ then we must have for any perturbation function $\delta v$ and number $\varepsilon$ : MATH We integrate the first term by parts and use the fact that the perturbation must be zero at the ends of the interval: MATH The above is true for any MATH . Hence

MATH (Euler Lagrange equation)
when evaluated at the minimizing function $v$ . The last equation is called the Euler-Lagrange equation.

Notation. Index. Contents.

Copyright 2007