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.

## Connection between Hamiltonian and Lagrangian.

ssume that the function solves the Euler-Lagrange equation. We introduce the function Assume further that the equation

 (generalized impulse equation)
has a unique smooth solution
 (generalized impulse solution)
It follows that consequently We define a Hamiltonian associated with the Lagrangian as follows:

Claim

Under assumption that the solves the EL equation ( Euler Lagrange equation ) and the and are defined as above, the and solve the Hamilton equations: and

Proof

We have the relationships satisfied at and the definition We calculate the derivatives , and at accordingly:

 Notation. Index. Contents.