Optional projection under equivalent local martingale measures

Finitely Additive Equivalent Martingale Measures

Let L be a linear space of real bounded random variables on the probability space (Ω,A, P0). There is a finitely additive probability P on A, such that P ∼ P0 and EP (X) = 0 for all X ∈ L, if and only if c EQ(X) ≤ ess sup(−X), X ∈ L, for some constant c > 0 and (countably additive) probability Q on A such that Q ∼ P0. A necessary condition for such a P to exist is L − L+∞ ∩ L + ∞ = {0}, where t...

Robust stochastic control and equivalent martingale measures

We study a class of robust, or worst case scenario, optimal control problems for jump diffusions. The scenario is represented by a probability measure equivalent to the initial probability law. We show that if there exists a control that annihilates the noise coefficients in the state equation and a scenario which is an equivalent martingale measure for a specific process which is related to th...

Equivalent martingale measures and ramifica - tions

Equivalent martingale measures and optimal market completions

Optimal fictitious completions of an incomplete financial market are shown to be associated with exponential martingales (not just local martingales) and, therefore, to "an optimal equivalent martingale measure'. Results of independent interest, in the theory of continuous-time martingales, are derived as well. iirshurgb, PA 15213-3890

In discrete time a local martingale is a martingale under an equivalent probability measure

1 Result and Discussion We consider a discrete-time infinite horizon model with an adapted d-dimensional process S = (S t) given on a stochastic basis (Ω, F, F = (F t) t=0,1,... , P). The notations used: M(P), M loc (P) and P are the sets of d-dimensional martingales, local martingales and predictable (i.e. (F t−1)-adapted) processes; H · S t = j≤t H j ∆S j. To our knowledge, this result was ne...