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Optional splitting formula in a progressively enlarged filtration
Let be a filtration and be a random time. Let
be the progressive enlargement of with . We
study the validity of the following formula, called optional splitting formula
: For any -optional process , there exist a
-optional process and a function defined on
being
measurable, such that
Y=Y'\ind_{[0,\tau)}+Y"(\tau)\ind_{[\tau,\infty)} We are interested in this
formula, because it has been taken for granted in number of recent works in
credit risk modeling, whilst such a formula can not be true in general.
Sufficient conditions will be given for the validity of the above formula as
well as of its extension in the case of multiple random times
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