Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 $\QTR{cal}{E}$ for the heat operator MATH takes the form MATH Here the MATH is the step function (see the formula ( step function )). Using the fact MATH one can verify directly that MATH

Problem

Consider the problem MATH where the functions $f$ and $u_{0}$ are smooth.

Solution

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

Problem

Consider the problem MATH where the functions $\psi,u_{0}$ are smooth.

Solution

The $u$ solves the above problem. It is defined on MATH . We assume it to be smooth until the solution is revealed and we verify such conjecture. We extend it to the entire plane $\left( t,x\right) $ according to the following rule MATH Hence, the $v$ jumps across $t=0$ and $x=0$ but the $x$ -derivative does not jump across $x=0$ . We have MATH MATH

Hence, MATH MATH MATH

Problem

Consider the problem MATH where the functions $\psi,u_{0}$ are smooth.

Solution

The $u$ is defined on MATH and solves the above problem. We extend it to the entire plane $\left( t,x\right) $ according to the rule MATH Hence, the $v$ jumps across $t=0$ and does not jump across $x=0$ . The $x$ -derivative jumps across $x=0$ . We have MATH MATH Hence, MATH MATH MATH





Notation. Index. Contents.


















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