On the predictable representation property of martingales associated with Lévy processes

Tella, Paolo Di; Engelbert, H. J.

doi:10.1080/17442508.2014.932051

Cited by 5 publications

(8 citation statements)

References 23 publications

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“…One can observe that, under the hypotheses of Theorem 4.13, the family of processes given by X, Y and s≤t ∆X i s ∆Y j s , i, j ≥ 1 is invariant under covariation. In particular when X and Y are martingales this family is a compensated-stable covariation family of martingales in the space of square-integrable martingales (see [15], [14]). Moreover Theorem 4.13 provides the minimal number of martingales needed for the predictable representation of this family.…”

Section: Discussionmentioning

confidence: 99%

“…for ξ A and ξ B in L 2 (M, P, F) and L 2 (N, P, H) respectively. The equalities in (14) imply that P (A ∩ B) differs from P (A)P (B) by the expression…”

Section: The Martingale Casementioning

confidence: 99%

“…which generates G T . To this end, note that equalities in (14) hold under Q so that Q(A ∩ B) differs from P (A)P (B) by the expression…”

Section: The Martingale Casementioning

confidence: 99%

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Enlargement of filtration and predictable representation property for semi-martingales

Calzolari

Torti

2015

Stochastics

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We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a semi-martingale X does not transfer from the reference filtration F to a larger filtration G if the information starts growing up to a positive time. Then we study the case G = F ∨ H when there exists a second special semi-martingale Y enjoying the predictable representation property with respect to H. We establish conditions under which the triplet (X, Y, [X, Y ]) enjoys the predictable representation property with respect to G.

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Section: Discussionmentioning

confidence: 99%

“…for ξ A and ξ B in L 2 (M, P, F) and L 2 (N, P, H) respectively. The equalities in (14) imply that P (A ∩ B) differs from P (A)P (B) by the expression…”

Section: The Martingale Casementioning

confidence: 99%

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Enlargement of filtration and predictable representation property for semi-martingales

Calzolari

Torti

2015

Stochastics

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“…In this sense, the general condition (2.2) links well with results on PRP and BSDEs for Lévy processes in [NS00,NS01] who study a specific Teugels martingale basis consisting of compensated power jump processes for Lévy processes which satisfy exponential moment conditions. For a systematic analysis of related PRP results, comprising general Lévy processes, see [DTE15b,DTE15a].…”

Section: )mentioning

confidence: 99%

On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples

Büttner

Kentia

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and timeinhomogeneous compensator. The BSDE generator function can be non-convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori L ∞ -bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions.

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“…We shall require that the family X is compensated-covariation stable , i.e., that for every X ,Y ∈ X the process [X ,Y ] − X ,Y again belongs to X , [X ,Y ] denoting the covariation process of X and Y . This property of compensated-covariation stability of families of martingales has been introduced in Di Tella [5] and Di Tella & Engelbert [6], where the predictable representation property is studied.…”

Section: Introductionmentioning

confidence: 99%

The chaotic representation property of compensated-covariation stable families of martingales

Tella

Engelbert

2016

Ann. Probab.

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In the present paper, we study the chaotic representation property for certain families X of square integrable martingales on a finite time interval [0, T ]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family X of square integrable martingales having deterministic mutual predictable covariation X,Y for all X,Y ∈ X . The main result of the present paper is stated in Theorem 5.8 below: If X is a compensated-covariation stable family of square integrable martingales such that X,Y is deterministic for all X,Y ∈ X and, furthermore, the system of monomials generated by X is total in L 2 (Ω, F X T , P), then X possesses the chaotic representation property with respect to the σ -field F X T . We shall apply this result to the case o f Lévy processes. Relative to the filtration F L generated by a Lévy process L, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Lévy processes, several examples of concrete families X of martingales including Teugels martingales. and then we defineWe call J the space of iterated integrals generated by X .(iv) By J T we denote the linear subspace of L 2 (P) of terminal variables of iterated integrals from J . The linear spaces J (α 1 ,...,α n ) n,T and J n,T of random variables in L 2 (P) are introduced analogously from the spaces of processes J (α 1 ,...,α n ) n and J n , respectively, n ≥ 0. Now we state the definition of the chaotic representation property on the space L 2 (P).Definition 3.6. We say that X = {X (α) , α ∈ Λ} possesses the chaotic representation property (CRP) on the Hilbert space L 2 (P) = L 2 (Ω, F , P) if the linear space J T (cf. Definition 3.5 (iii)) is equal to L 2 (P).We stress that, because the spaces (L 2 (P), · 2 ) and (H 2 , · H 2 ) can be identified, we can equivalently claim that X possesses the CRP if J = H 2 .Proposition 3.4 (ii) yields that J n,T (n ≥ 1) (resp., J n (n ≥ 1)) are pairwise orthogonal closed subspaces of L 2 (P) (resp., H 2 ). Furthermore, it can easily be checked that J 0,T (resp., J 0 ) is orthogonal to J n,T (resp., J n ) for all n ≥ 1. This immediately leads to the following equivalent description of the CRP.Proposition 3.7. (i) It holds J T = ∞ n=0 J n,T (resp., J = ∞ n=0 J n ). (ii) The family X possesses the CRP if and only if L 2 (P) = ∞ n=0 J n,T (resp., H 2 = ∞ n=0 J n ).

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On the predictable representation property of martingales associated with Lévy processes

Cited by 5 publications

References 23 publications

Enlargement of filtration and predictable representation property for semi-martingales

Enlargement of filtration and predictable representation property for semi-martingales

On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples

The chaotic representation property of compensated-covariation stable families of martingales

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