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 for the heat equation.

he fundamental solution for the heat operator takes the form Here the is the step function (see the formula ( step function )). Using the fact one can verify directly that

Problem

Consider the problem where the functions and are smooth.

Solution

We extend with 0 for t<0 and introduce Here the is assumed to be smooth and solve the above problem for t>0. The function is the step function (see the formula ( step function )). The is defined for all t and x and might be discontinuous across . Therefore The came from the -differentiation of the step function on the boundary. Consequently, if we set then such function solves The convolution is taken with respect to both variables and :

Problem

Consider the problem where the functions are smooth.

Solution

The solves the above problem. It is defined on . We assume it to be smooth until the solution is revealed and we verify such conjecture. We extend it to the entire plane according to the following rule Hence, the jumps across and but the -derivative does not jump across . We have

Hence,

Problem

Consider the problem where the functions are smooth.

Solution

The is defined on and solves the above problem. We extend it to the entire plane according to the rule Hence, the jumps across and does not jump across . The -derivative jumps across . We have Hence,

 Notation. Index. Contents.