Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 A. Space of distributions. Weak derivative.
 B. Fundamental solution.
 C. Fundamental solution for the heat equation.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Fundamental solution.

et be a linear differential operator: where and are functions .

Definition

A fundamental solution for the partial derivatives operator is the distribution with the following property

Note that the equation

has the solution where the * is the convolution operation

Indeed, since differentiation commutes with convolution,

Example

(Fundamental solution for ODE). Consider the ODE operator where the and are functions . We conjecture a fundamental solution of the form where the is the step function (see the formula ( step function )) and the is some unknown smooth function. We substitute into and calculate the derivatives The functions are supported at 0. Hence, if we set then We conclude that the is defined by the following Cauchy problem

 Notation. Index. Contents.