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I. Basic math.
1. Conditional probability.
2. Normal distribution.
3. Brownian motion.
4. Poisson process.
5. Ito integral.
6. Ito calculus.
7. Change of measure.
8. Girsanov's theorem.
9. Forward Kolmogorov's equation.
10. Backward Kolmogorov's equation.
A. Multidimensional backward Kolmogorov's equation.
B. Representation of solution for elliptic PDE using stochastic process.
11. Optimal control, Bellman equation, Dynamic programming.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Representation of solution for elliptic PDE using stochastic process.


roposition

Let $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ with a $C^{1}$ -boundary $\partial U$ . Let MATH and $f|_{\partial U}=0$ , MATH . The solution of the boundary problem MATH is given by MATH where $W_{t}$ is standard Brownian motion in $\QTR{cal}{R}^{n}$ and MATH is the first time when the process $x+W_{t}$ exits $U$ .

Proof

We use the technique of the section ( Backward equation ). Let MATH

where MATH . Thus MATH , MATH . We calculate as in the section ( Backward equation ) and under assumption that $x$ is away from the boundary MATH : MATH MATH It remains to note that MATH Indeed, MATH for any $h$ .





Notation. Index. Contents.


















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