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.

## Backward Kolmogorov's equation. n this section we are repeatedly using the formulas ( Chain rule ) and ( Ito_formula ) without further reference. The filtration is generated by . The reference is [Kohn] .

A one-dimensional process is given by SDE for some smooth functions and and standard Brownian motion .

Let be an integrable function .

Proposition

(Backward Kolmogorov equation) A function given by is a solution of the problem Proof

We calculate  Note that  We apply the operation to the equation (*) and obtain for any . We conclude Indeed, if this is not true around some then we use freedom of to set at and obtain a contradiction for some sufficiently close to .

Proof

Alternatively, consider the following local argument  Thus   Hence, After applying the Ito formula we conclude  The local argument is better because it does not need assumption of smoothness of at the final stage of argument.

Proposition

(Backward Kolmogorov for running payoff) The function is a solution of the problem (Backward Kolmogorov with running payoff)

Proof

We calculate    Similarly to the previous proof,  Proposition

(Backward Kolmogorov for discounted payoff) The function is a solution of the problem Proof

We calculate     Hence,   Proposition

(Backward Kolmogorov with localization) Let be a set in the value space of . We define the exit time : Let be an integrable function.

The function is a solution of the problem Proof

The proof was already given for the situation when is away from the boundary. On the boundary the statement is obvious.

Remark

If is the arrival time to the level then is given by the equation A. Multidimensional backward Kolmogorov's equation.
 B. Representation of solution for elliptic PDE using stochastic process.
 Notation. Index. Contents.