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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.

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 $X_{t}$ . The reference is [Kohn] . A one-dimensional process $X_{t}$ is given by the SDE MATH for smooth functions MATH and MATH and standard Brownian motion $dW_{t}$ .

Proposition

(Backward Kolomogorov equation) The function MATH is a solution of the problem MATH

Proof

We calculate MATH MATH Note that MATH MATH We apply the operation MATH to the equation (*) and obtain MATH for any $t,T,x$ . We conclude MATH Indeed, if this is not so around some MATH then we use freedom of $t,T,x$ to set $\left( t,x\right) $ at MATH and obtain a contradiction for some $T$ sufficiently close to $t_{0}$ .

Proof

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

Proposition

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

MATH (Backward Kolmogorov with running payoff)

Proof

We calculate MATH MATH MATH MATH Similarly to the previous proof MATH MATH

Proposition

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

Proof

We calculate MATH MATH MATH MATH MATH Hence, MATH MATH MATH

Proposition

(Backward Kolmogorov with localization) Let $D$ be a set in the value space of $X_{t}$ . We define the exit time $\tau$ : MATH

Let MATH be an integrable function. The function MATH is a solution of the problem MATH

Proof

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

Remark

If $\tau$ is the arrival time to the level $x=L$ then MATH is given by the equation MATH




A. Multidimensional backward Kolmogorov's equation.
B. Representation of solution for elliptic PDE using stochastic process.

Notation. Index. Contents.


















Copyright 2007