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.

efinition

(Propagator and transitional probability) Let be a subset of , is a -valued stochastic process with the property ( Markov property ) and is a function .

We define the "transition function" and the two-parameter families of operators .

1. The function acts as follows

 (Transition function)

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

 (Forward propagator)
for any set (Measure carried forward by the process ).

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

 (Backward propagator)

4. The transition function is called "homogeneous" if depends only on and . In such case we use the notations

Proposition

(Basic properties of propagator 1)

1. Let Prob then Prob , .

2. Let then , .

Proof

(1). According to the formula ( Total_probability_rule ) with and covering all The dot marks the position where the operator acts.

(2) According to the formula ( Chain_rule )

Proposition

Let , then

Claim

1. To carry the measure from to is to carry it from to and then to carry it from to :

 (Kolmogorov-Chapman equation)

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

Proof

1. By definition of , By the formula ( Total_probability_rule ) we have hence, We change the order of integration (see the proposition ( Fubini theorem )) and use the definition of :

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

Proposition

(Basic properties of propagator 2)

1. ,

2. , .

3. , .

4. , .

Proof

1.

2.

3.

4.

Note that when any of the operators acts on the transitional probability the following rules apply. The backward operator acts on the backward space argument and extends the backward time argument backward in time. The forward *-operator acts on the forward space set and extends the forward parameter forward in time. The precise manner of the extension is natural in every case.

The next observation is on the symmetry of and .

Notation

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

Proposition

The propagators and are adjoint operators with respect to the schalar product : for any integrable and probability measure .

Proof

By the definitions,

 Notation. Index. Contents.