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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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.
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 $X_{t}$ given by the SDE

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

Theorem

(Forward Kolmogorov equation for diffusion (Ito) process). If the process $X_{t}$ 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 MATH .





Notation. Index. Contents.


















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