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Optional strong semimartingale inequalities for the strong Snell envelopes

  • Mouna Haddadi EMAIL logo and Youssef Ouknine
Published/Copyright: August 3, 2024
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Random Operators and Stochastic Equations
From the journal Random Operators and Stochastic Equations Volume 32 Issue 3

Abstract

On finite time interval [ 0 , T ] , let X be an optional semimartingale of class ( D ) , and Z its strong Snell envelope, which is the smallest optional strong supermartingale bounding X above (except on evanescent set). In this article, we provide the several characterizations of strong Snell envelopes to establish the main result which is the following inequality:

∥ Z 1 - Z 2 ∥ ℍ p ≤ C p ⁢ ∥ X 1 - X 2 ∥ ℍ p ,

where X 1 and X 2 are two optional semimartingales of class ( D ) belonging to the space ℍ p ( p > 1 ), Z 1 and Z 2 are their Snell envelopes, C p is an absolute constant.

Keywords: Optional supermartingale; Snell envelope; optional section theorem; Mertens decomposition; Doob’s optional sampling theorem
MSC 2020: 60G07

Communicated by Vyacheslav L. Girko

Acknowledgements

The authors would like to thank the Editor-in-chief as well as the two anonymous referees for their comments and remarks.

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Received: 2023-07-15
Accepted: 2024-04-04
Published Online: 2024-08-03
Published in Print: 2024-09-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/rose-2024-2014
Keywords for this article
Optional supermartingale; Snell envelope; optional section theorem; Mertens decomposition; Doob’s optional sampling theorem

Articles in the same Issue

  1. Frontmatter
  2. Sandwiched filtrations
  3. Stochastic processes and mean square calculus on fractal curves
  4. Random attractors for three-dimensional stochastic globally modified Navier–Stokes equations driven by additive noise on unbounded domains
  5. Existence results for boundary value problems of Hadamard fractional differential equations on unbounded domain
  6. Stochastic maximum principle for partially observed optimal control problem of McKean–Vlasov FBSDEs with Teugels martingales
  7. Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type
  8. Optional strong semimartingale inequalities for the strong Snell envelopes
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