Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 $x\left( s\right) $ solves the Euler-Lagrange equation. We introduce the function MATH Assume further that the equation

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

Claim

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

Proof

We have the relationships MATH satisfied at $x\left( s\right) $ and the definition MATH We calculate the derivatives MATH , MATH and $\frac{d}{ds}H$ at $x\left( s\right) $ accordingly: MATH MATH





Notation. Index. Contents.


















Copyright 2007