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.
 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.

## Forward Kolmogorov's equation.

e are considering a one-dimensional process given by the SDE

 (SDE X)
where is increment of standard Brownian motion and are some smooth real valued functions. We introduce transitional distribution , according to the relationship
 (SDE X p)
We are interested in evolution of as time progresses. Let be any smooth function of both variables rapidly decaying in . We have We apply the formula ( Ito formula ). Since , the term disappears under the expectation sign. We utilize the definition ( SDE_X_p ): We perform integration by parts: By definition of , hence We substitute this equality into the relationship In addition, the integration by parts in -variable does not produce boundary terms due to fast decay of at -infinities. Therefore, We conclude
 (Forward Kolmogorov)
for all and all .

Theorem

(Forward Kolmogorov equation for diffusion (Ito) process). If the process is given by the SDE ( SDE for X ) then the function ( Distribution of X ) evolves according to the PDE ( Forward Kolmogorov ) with the initial condition .

 Notation. Index. Contents.