Sur des problemes de regularisation, de recollement et d'interpolation en theorie des processus

Dellacherie, Claude; Lenglart, Érik

doi:10.1007/bfb0092793

Cited by 21 publications

(31 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Preliminary results on the value family. Let us first introduce the definition of an admissible family of random variables indexed by stopping times in T 0,T (or T 0,T -system in the vocabulary of Dellacherie and Lenglart [6]).…”

Section: 1mentioning

confidence: 99%

“…Prop. A.6] which still holds in the case of a general filtration) givesU (τ ) ≥ lim n→+∞ E f τ,τn (U (τ n )) = E f τ,τ ( lim n→+∞ U (τ n )) = lim n→+∞ U (τ n ) a.s.By Lemma 5 of Dellacherie and Lenglart[6] 10 , the family (U (S)) is thus right-uppersemicontinuous (along stopping times). The value family V = (V (S), S ∈ T 0,T ) defined in(5.1)is an E f -supermartingale family.…”

mentioning

confidence: 90%

See 1 more Smart Citation

Optimal stopping with f-expectations: The irregular case

Grigorova

Imkeller

Ouknine

et al. 2020

Stochastic Processes and their Applications

View full text Add to dashboard Cite

We consider the optimal stopping problem with nonlinear f -expectation (induced by a BSDE) without making any regularity assumptions on the payoff process ξ and in the case of a general filtration. We show that the value family can be aggregated by an optional process Y . We characterize the process Y as the E f -Snell envelope of ξ. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with ξ as the obstacle. To do this, we first establish some useful properties of irregular RBSDEs, in particular an existence and uniqueness result and a comparison theorem.where T S,T denotes the set of stopping times valued a.s. in [S, T ] and E f S,τ (·) denotes the conditional f -expectation/evaluation at time S when the terminal time is τ .The above non-linear problem has been introduced in [14] in the case of a Brownian filtration and a continuous financial position/pay-off process ξ and applied to the (nonlinear) pricing of American options. It has then attracted considerable interest, in particular, 2 Note that in the case of a not necessarily non-negative pay-off process ξ this result holds up to a translation by the martingale XS := E[ess sup τ ∈T 0,T ξ − τ |FS] (cf. e.g. Appendix A in [30]). More precisely, the property holds forṽ := v + X andξ = ξ + X.

show abstract

Section: 1mentioning

confidence: 99%

mentioning

confidence: 90%

Optimal stopping with f-expectations: The irregular case

Grigorova

Imkeller

Ouknine

et al. 2020

Stochastic Processes and their Applications

View full text Add to dashboard Cite

show abstract

“…Let us recall the definition of an admissible family of random variables indexed by stopping times in T 0 (or T 0 -system in the vocabulary of Dellacherie and Lenglart [8]). Moreover, for each S ∈ T , there exists a sequence of controls (ν n ) n∈N with ν n ∈ V S for all n, such that the sequence (E ν n S,T (η)) n∈N is nondecreasing and satisfies:…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

European Options in a Nonlinear Incomplete Market Model with Default

Grigorova¹,

Sulem²

2020

SIAM J. Finan. Math.

View full text Add to dashboard Cite

This paper studies the superhedging prices and the associated superhedging strategies for European options in a non-linear incomplete market model with default. We present the seller's and the buyer's point of view. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow non-linear dynamics with a non-linear driver f. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum, over a suitable set of equivalent probability measures Q ∈ Q, of the fevaluation/expectation under Q of the payoff. We also provide a characterization of the seller's (superhedging) price process as the minimal supersolution of a constrained BSDE with default and a characterization in terms of the minimal weak supersolution of a BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. Our results rely on first establishing a non-linear optional and a non-linear predictable decomposition for processes which are E f-strong supermartingales under Q, for all Q ∈ Q.

show abstract

“…From their Eq. (23) we also infer that ξ can actually be chosen via the following threshold principle (2) or, equivalently, via…”

Section: A Stochastic Representation Problemmentioning

confidence: 99%

“…By Proposition 2 in Dellacherie and Lenglart [2], to show that ξ is indeed pathwise lower semi-continuity from the right, it suffices to prove lim n ξ S n ≥ ξ S , for any sequence of stopping times S n ↓ S such that ζ lim n ξ S n exists almost surely. Now, for such a sequence of stopping times, we indeed have that, for ε > 0,…”

Section: A Stochastic Representation Problemmentioning

confidence: 99%

On Gittins’ index theorem in continuous time

Bank

Küchler

2007

Stochastic Processes and their Applications

View full text Add to dashboard Cite

We give a new and comparably short proof of Gittins' index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.

show abstract

Sur des problemes de regularisation, de recollement et d'interpolation en theorie des processus

Cited by 21 publications

References 9 publications

Optimal stopping with f-expectations: The irregular case

Optimal stopping with f-expectations: The irregular case

European Options in a Nonlinear Incomplete Market Model with Default

On Gittins’ index theorem in continuous time

Contact Info

Product

Resources

About