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.

## Lagrangian.

et be a smooth function. We introduce the action functional taking vector functions of and producing a number. We consider a problem of minimization of across a variety of smooth functions with fixed values of at the ends of the interval .

If a function minimizes then we must have for any perturbation function and number : We integrate the first term by parts and use the fact that the perturbation must be zero at the ends of the interval: The above is true for any . Hence

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

 Notation. Index. Contents.