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Quantitative Analysis
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Numerical Analysis
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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.

Ito integral.


iven a filtration $\QTR{cal}{F}_{t}$ , Ito integral is initially defined for "simple" processes of the form MATH where the variables $a_{i}$ are MATH measurable for each $i$ and MATH See the section ( Filtration_definition_section ) for the notations MATH and $\Omega$ . In other words, the value of $a_{t}$ remains constant during MATH and it is known with certainty at $t_{i}$ . For such $a_{t}$ the stochastic (Ito) integral is defined as MATH where $W_{t}$ is a standard Brownian motion adapted to the filtration $\QTR{cal}{F}_{t}$ . For each $i$ the $a_{i}$ is MATH measurable while the MATH is MATH measurable and MATH -independent. Hence, $a_{i}$ and MATH are independent random variables for each $i$ . This observation is the key tool when doing anything with stochastic integral. In particular, we use such observation when proving the following properties.

Proposition

1.


MATH is an $\QTR{cal}{F}_{t}-$ adapted martingale.

2. Ito isometry. MATH .

Proof

By definition, MATH We use the formula ( Chain_rule ): MATH MATH MATH MATH

Proposition

(Chebyshev's inequality). MATH

Proof

See the formula ( Chebyshev inequality ) with MATH .

With help of the Ito isometry and Chebyshev's inequality we expand the notion of stochastic integral to the class of $\QTR{cal}{F}_{t}-$ adapted processes $a_{t}$ with the property MATH We approximate (in MATH sense) any such process with a Cauchy sequence of simple processes. Then the stochastic integral of the process is a limit in probability of the Cauchy sequence of the simple integrals.





Notation. Index. Contents.


















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