Probability measure: Difference between revisions
add to the definition of risk neutral measure that: the expected value in the definition has to be taken with respect to that same risk neutral measure, i.e. calculated using the correpsonding risk neutral probability density function
(add Template:Probability fundamentals using AWB) 
(add to the definition of risk neutral measure that: the expected value in the definition has to be taken with respect to that same risk neutral measure, i.e. calculated using the correpsonding risk neutral probability density function) 

==Example applications==
''Market measures'' which assign probabilities to [[financial market]] spaces based on actual market movements are examples of probability measures which are of interest in [[mathematical finance]], e.g. in the pricing of [[financial derivative]]s.<ref>''Quantitative methods in derivatives pricing'' by Domingo Tavella 2002 ISBN 0471394475 [http://books.google.com/books?id=dHIMulKy8dYC&pg=PA11 page 11]</ref> For instance, a [[riskneutral measure]] is a probability measure which assumes that the current value of assets is the [[expected value]] of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and [[discounted]] at the [[riskfree rate]]. If there is a unique probability measure that must be used to price assets in a market, then the market is called a [[complete market]].<ref>''Irreversible decisions under uncertainty'' by Svetlana I. Boyarchenko, Serge Levendorskiĭ 2007 ISBN 3540737456 [http://books.google.com/books?id=lpsrP5mQG_QC&pg=PA11 page 11]</ref>
Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in [[statistical mechanics]] is a measure space, such measures are not always probability measures.<ref name=stern/> In general, in statistical physics, if we consider sentences of the form "the probability of a system S assuming state A is p" the geometry of the system does not always lead to the definition of a probability measure [[congruence relationunder congruence]], although it may do so in the case of systems with just one degree of freedom.<ref name=gut/>
