This appendix provides a very basic introduction to the language of probabil ity theory and stochastic processes We assume the reader is familiar with the general measure and integration theory we start this chapter with the notions that are specific to Probability Theory

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1STOCHASTIC PROCESSES This appendix provides a very basic introduction to the language of probabil- ity theory and stochastic processes. We assume the reader is familiar with the general measure and integration theory, we start this chapter with the notions that are specific to Probability Theory. 1.1 Random variables, laws A probability space is a triple (?,F , P ) where ? is a set, F is a ?-field of subsets of ? and P is a probability measure on F . A subset A of ? is negligible if there exists B ? F such that A ? B and P (B) = 0. Note that this notion is relative to F and P but we shall not mention them. The probability space (?,F , P ) is called complete if F contains all the negligible sets. An incomplete probability space (?,F , P ) can easily be completed in the following way. Let N denote the set of all negligible sets. Let F be the ?- algebra generated by F and N . It is easy to see that it coincides with the set of subsets of ? which are of the form B?N for some B ? F and N ? N . One extends P to a probability measure P on F by putting P (B ? N) = P (B). The probability space (?,F , P ) is then complete.

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Random variable

1STOCHASTICPROCESSESThisappendixprovidesaverybasicintroductiontothelanguageofprobabil-itytheoryandstochasticprocesses.Weassumethereaderisfamiliarwiththegeneralmeasureandintegrationtheory,westartthischapterwiththenotionsthatarespeciﬁctoProbabilityTheory.1.1Randomvariables,lawsAprobabilityspaceisatriple(Ω,F,P)whereΩisaset,Fisaσ-ﬁeldofsubsetsofΩandPisaprobabilitymeasureonF.AsubsetAofΩisnegligibleifthereexistsB∈FsuchthatA⊂BandP(B)=0.NotethatthisnotionisrelativetoFandPbutweshallnotmentionthem.Theprobabilityspace(Ω,F,P)iscalledcompleteifFcontainsallthenegligiblesets.Anincompleteprobabilityspace(Ω,F,P)caneasilybecompletedinthefollowingway.LetNdenotethesetofallnegligiblesets.LetFbetheσ-algebrageneratedbyFandN.ItiseasytoseethatitcoincideswiththesetofsubsetsofΩwhichareoftheformB∪NforsomeB∈FandN∈N.OneextendsPtoaprobabilitymeasurePonFbyputtingP(B∪N)=P(B).Theprobabilityspace(Ω,F,P)isthencomplete.Itiscalledthecompletionof(Ω,F,P).Fromnowon,allourprobabilityspacesareimplicitelyassumedtobecomplete.ArandomvariableisameasurablemapX:Ω7→RwhereRisequippedwithitsusualBorelσ-ﬁeldB(R).Thelaw,ordistribution,ofarandomvariableXisthe(probability)measureµX=X◦PonR,theimageofPunderX,µX(A)=PX−1(A)thatis,�forallA∈B(R).Veryoften,theabovequantityisdenotedbyP(X∈A).TheexpectationofanintegrablerandomvariableXistheintegralZZE[X]=X(ω)dP(ω)=xdµX(x).RΩ

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