Quantitative Analysis.
Trading Platform.
Python for Excel.
Author.

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I.Basic math.
II.Pricing and Hedging.
III.Explicit techniques.
IV.Data Analysis.
V.Implementation tools.
VI.Basic Math II.
VII.Implementation tools II.
1.Calculational Linear Algebra.
2.Wavelet Analysis.
3.Finite element method.
4.Construction of approximation spaces.
5.Time discretization.
6.Variational inequalities.
7.Lattice approach to derivative pricing.
A.Basic definitions of the lattice approach.
B.Markov Generator for smooth process. Dyson decomposition. Kolmogorov equations.
C.Path-Integral representation of Markov generator.
D.Markov generator for piecewise smooth process.
E.Fast exponentiation.
Bibliography.
Forum Notation Index Contents

Path-Integral representation of Markov generator.

efinition

Symbolic path MATH is a sequence of sites in MATH such that MATH for all $j$. Let $\Gamma_{m}$ be the set of all symbolic paths in MATH.

We intend to transform the proposition ( Dyson decomposition theorem) into a sum over symbolic paths. For this reason we study a technique for removing terms of the form MATH from the integrals in the proposition ( Dyson decomposition theorem).

Consider the integral of the formMATH Let's change the order of integration of the k-th and (k+1)-th variables.


Figure
Change of order of integration.

MATH(order of integration one)
So we pull the function $f_{k}$ out of the integration chain. Let's pull the function $f_{k+1}$ out by changing the order of integration of the $s_{k+1}$ and $s_{k+2}$ variables. We have
MATH(order of integration two)
 We can do it again for the k+2 variable:MATH

Our next task is to examine the chainsMATH in the situation when all the functions $f_{k}$ are equal. Hence, we are interested in studyingMATH

Let us change order of integration in the integral MATH:MATH and rename variables of integration:MATH We now add the last result and the original definition of $J_{2}$:MATH Hence,MATH We proceed to calculate $J_{3}$:MATH Consequenty,

MATH(chained integral n factorial)

Proposition

(Path-Integral representation). The Markov propagator admits the following representationMATH where $t_{q+1}=T.$

Proof

We start from the proposition ( Dyson decomposition theorem):MATH We apply the results ( order of integration one),( order of integration two) to pull out every term of the form MATH. Then we apply ( chained integral n factorial). The proposition follows by grouping of terms.





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