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.
 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 be a bounded subset of with a -boundary . Let and , . The solution of the boundary problem is given by where is standard Brownian motion in and is the first time when the process exits .

Proof

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

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

 Notation. Index. Contents.