martingale theory In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given ...

, Émery topology is a

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

on the space of

semimartingale In probability theory, a real-valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...

s. The topology is used in

financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the Finance#Quantitative_finance, financial field. In general, there exist two separate ...

. The class of

stochastic integral Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...

s with general predictable integrands coincides with the closure of the set of all simple integrals. The topology was introduced in 1979 by the French mathematician Michel Émery.

Definition

Let

(\Omega,\mathcal,\,P)

be a

filtered Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...

probability space, where the filtration satisfies the usual conditions and

T\in (0,\infty)

. Let

\mathcal(P)

be the space of real semimartingales and

\mathcal(1)

the space of simple predictable processes

H

with

, H, =1

. We define :

\, X\, _:=\sup\limits_\mathbb\left (H\cdot X)_t, \right)\right

Then

(\mathcal(P),d)

with the metric

d(X,Y):=\, X-Y\, _

is a

complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...

and the induced topology is called Émery topology.{{cite journal, first1=M. , last1=De Donno, first2=M., last2=Pratelli, title=A theory of stochastic integration for bond markets, journal=Annals of Applied Probability, volume=15, number=4, pages=2773 - 2791, date=2005, doi=10.1214/105051605000000548, arxiv=math/0602532

References

Martingale theory