Enlargement of Filtration in Discrete Time

Abstract

We present some results on enlargement of filtration in discrete time. Many results known in continuous time extend immediately in a discrete time setting. Here, we provide direct proofs which are much more simpler. We study also arbitrages conditions in a financial setting and we present some specific cases, as immersion and pseudo-stopping times for which we obtain new results.

The insider problem in the trinomial model: a discrete-time jump process approach

“…They spawned two main models that have been extensively discussed since in the literature: the initial enlargement model under Jacod's hypothesis which assumes the equivalence between the conditional laws of G with respect to F and the law of G (see Jacod [41]), and the progressive enlargement model (where G is N 0 -valued and G is the smallest filtration satisfying usual conditions and making G a stopping time) with honest times (see Barlow [10], Jeulin and Yor [43]). Besides, all related results extend immediately in a discrete time setting as highlighted by Blanchet-Scalliet, Jeanblanc and Romero in [15], most of them simply stemming from Doob's decomposition. For a comprehensive review of the deep results on the enrichment of filtrations the reader can refer to the lecture notes of Mansuy and Yor [48] and with a view to financial purposes in the book of Aksamit and Jeanblanc [5].…”

“…As stated in Lemma 1.4 of Blanchet et al [15], for two F-adapted processes U and K, U, K P 0 = 0 and ∆ U, K where we have got the second line by conditioning with respect to F t−1 and by defining the family {a j t,k, , (k, ) ∈ E 2 } by a j t,k, = E P j [∆Z j (t,k) ∆R j (t, ) ], i.e. a j t,1,1 = λ j p j t (1−λp j t ), a j t,1,−1 = 0, a j t,−1,1 = (λ j ) 2 p j t (1−p j t ) and a j t,−1,−1 = λ j (1 − p j t ) (1 − λ j ) 1 − λ j p j t .…”

“…The difficulties inherent in the preservation of semimartingales are directly lifted as a consequence of Doob's decomposition. Indeed it is clear (see Blanchet-Scalliet, Jeanblanc and Romero [15]) that any integrable process is a special semimartingale in any filtration with respect to which it is adapted. Furthermore, the hypotheses of section 2.2 in Blanchet-Scalliet et al [15] are fulfilled in our context; thus, for some j ∈ J and a given (P j , F)-martingale X j , the process (X G,j t ) t∈T defined by X G,j 0 = X j 0 , and for t ∈ T * ,…”

The robust pricing–hedging duality for American options in discrete time financial markets

“…Clearly each random time satisfying the immersion property is a pseudo‐stopping time. Thus, keeping the equality true, the pseudo‐stopping time property in the definition of above can be replaced by a stronger condition characterizing the immersion property, See section 3.1.2 of Blanchet‐Scalliet, Jeanblanc, and Romero () for the discrete time context of progressive enlargement of filtration and Aksamit and Li () for connections between pseudo‐stopping times, the immersion property, and projections.…”

“…See section 3.1.2 of Blanchet-Scalliet, Jeanblanc, and Romero (2016) for the discrete time context of progressive enlargement of filtration and Aksamit and Li (2016) for connections between pseudostopping times, the immersion property, and projections.…”