Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
A. Forward and backward propagators.
B. Feller process and semi-group resolvent.
C. Forward and backward generators.
D. Forward and backward generators for Feller process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Forward and backward propagators.


(Propagator and transitional probability) Let $A$ be a subset of $\QTR{cal}{R}^{n}$ , $X_{t}$ is a $\QTR{cal}{R}^{n}$ -valued stochastic process with the property ( Markov property ) and $f$ is a function MATH .

We define the "transition function" $\pi$ and the two-parameter families of operators MATH .

1. The function $\pi$ acts MATH as follows

MATH (Transition function)

2. We define the family of forward propagators MATH acting on probability measures MATH as follows

MATH (Forward propagator)
for any set MATH (Measure $m$ carried forward by the process $X_{t}$ ).

3. We define the family of backward propagators MATH acting on Borel measurable functions MATH as follows

MATH (Backward propagator)

4. The transition function $\pi$ is called "homogeneous" if MATH depends only on $t-s,x$ and $B$ . In such case we use the notations MATH


(Basic properties of propagator 1)

1. Let $m\left( B\right) =$ Prob MATH then MATH Prob MATH , $s<t$ .

2. Let MATH then MATH , $s<t$ .


(1). According to the formula ( Total_probability_rule ) with MATH and $C_{\alpha}$ covering all MATH MATH The dot marks the position where the operator acts.

(2) According to the formula ( Chain_rule ) MATH


Let $s<u<t$ , then


1. To carry the measure from $s$ to $t$ is to carry it from $\,s$ to $u$ and then to carry it from $u$ to $t$ :

MATH (Kolmogorov-Chapman equation)

2. MATH . (Chain rule (compare with the formula ( Chain_rule )).


1. By definition of $P^{\ast}$ , MATH By the formula ( Total_probability_rule ) we have MATH hence, MATH We change the order of integration (see the proposition ( Fubini theorem )) MATH and use the definition of $P^{\ast}$ : MATH MATH

2. The technique of the proposition ( Basic properties of propagator 1 )-2 applies with little changes.


(Basic properties of propagator 2)

1. MATH , $s<t.$

2. MATH , $s<u<t$ .

3. MATH , $s<t$ .

4. MATH , $s<u<t$ .






Note that when any of the operators $P$ acts on the transitional probability MATH the following rules apply. The backward operator $P_{t_{0}t_{1}}$ acts on the backward space argument $x$ and extends the backward time argument $t_{1}$ backward in time. The forward *-operator MATH acts on the forward space set $B$ and extends the forward parameter $t_{2}$ forward in time. The precise manner of the extension is natural in every case.

The next observation is on the symmetry of $P$ and $P^{\ast}$ .


We introduce the notation MATH for the natural scalar product of function and measure: MATH


The propagators $P_{st}$ and $P_{st}^{\ast}$ are adjoint operators with respect to the scalar product MATH : MATH for any integrable $f$ and probability measure $m$ .


By the definitions, MATH

Notation. Index. Contents.

Copyright 2007