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Optional splitting formula in a progressively enlarged filtration

Shiqi Song (2014)

ESAIM: Probability and Statistics

Let 𝔽 F be a filtration andτbe a random time. Let 𝔾 G be the progressive enlargement of 𝔽 F withτ. We study the following formula, called the optional splitting formula: For any 𝔾 G-optional processY, there exists an 𝔽 F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being ℬ [ 0 , ∞ ] ⊗ 𝒪 ( 𝔽 ) ℬ[0,∞]⊗x1d4aa;(F) measurable, such that Y = Y ' 1 [ 0 , τ ) + Y ' ' ( τ ) 1 [ τ , ∞ ) . Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,τk). We are interested in this formula because of its fundamental role in many...

Representation of Itô integrals by Lebesgue/Bochner integrals

Qi Lü, Jiongmin Yong, Xu Zhang (2012)

Journal of the European Mathematical Society

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding Itô integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later...

Small deviations of iterated processes in the space of trajectories

Andrei Frolov (2013)

Open Mathematics

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.

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