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.

Hamilton-Jacobi Equations.

he reference for most of material here is [Evans] .

The main result of this chapter is negative: even though there is a Lagrangian for the heat equation nevertheless the Hamiltonian technique does not apply. However, the method of characteristics is separately useful.

We are considering a PDE of the form

 (First order PDE)
where the is an unknown function , the variables are from , the refers to the vector of derivatives and the function is smooth. We will use the following notation for the function 's arguments:

 A. Characteristics.
 B. Hamilton equations.
 C. Lagrangian.
 D. Connection between Hamiltonian and Lagrangian.
 E. Lagrangian for heat equation.
 Notation. Index. Contents.