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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
A. Change of measure-based verification of Girsanov's theorem statement.
B. Direct proof of 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.

Direct proof of Girsanov's theorem.


e follow the setup of the two previous sections ( Girsanov setup ) and ( Girsanov change of measure ) but give the direct derivation of the expression for the changed probability measure. We consider a "small time interval" version of the formula ( Change of measure ):

MATH The last formula is the equation that we need to satisfy by selecting a process $h$ . The $h_{t-\Delta t}$ , $B_{t-\Delta t}$ , MATH are deterministic functions from point of view of MATH . The MATH and MATH are random variables from the point of view of MATH . These variables represent a small change over the time interval $\Delta t$ . The distribution of MATH is known, MATH for a column of iid standard normal variables $\xi$ in the "original measure". The distribution of MATH is what we are trying to find from the above equation. We seek $h_{t}$ as a variable adapted to the filtration generated by $B_{t}$ . Hence, we express the last "original measure" expectation as follows MATH where $n$ is dimensionality of $B_{t}$ . In the last integral the $x$ is the integration parameter over all possible values of MATH and MATH is a function of $x$ because $h_{t}$ is $B_{t}$ -adapted.

The last integral is supposed to be equal to MATH for any smooth and sharply decaying function $\phi$ . We seek $h_{t}$ such that the $B_{t}$ would be a standard Brownian motion in the new (changed) measure: MATH MATH We need the former and the latter integrals to be equal. This gives us a recipe for construction of the $h_{t}$ .

We introduce the convenience notation MATH and MATH and state that we are seeking a MATH satisfying MATH for any smooth $\phi:R\mapsto R$ . To make conclusions we need to have the same expression as an argument of $\phi$ . Hence, we make a change of variable $y=x+\theta\Delta t$ in the right integral. It becomes MATH Therefore, $\kappa$ has to satisfy MATH for any smooth $\phi$ . Hence, $\kappa$ must satisfy MATH or MATH Changing to the original variable $x$ we obtain MATH or, in the original notation, MATH Using the above SDE and the boundary condition MATH we conclude

MATH (Girsanov kernel)

We summarize with the following statement.

Theorem

(Girsanov's theorem) Let MATH where the $W_{t}$ is a column of $\QTR{cal}{F}_{t}$ -adapted iid standard Brownian motions with respect to some MATH -given probability measure and $\theta_{s}$ is an $\QTR{cal}{F}_{t}-$ adapted integrable process. Then $B_{t}$ is a standard Brownian motion with resect to the "changed" probability given by the expectation MATH , see ( Definition_of_change_of_measure ). The $h_{t}$ is given by the formula ( Girsanov kernel ).





Notation. Index. Contents.


















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