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Weak Convergence of Stochastic Integrals on Skorokhod Space in Skorokhod's J1 and M1 Topologies

Andreas Sojmark, Fabrice Wunderlich

Abstract

We provide criteria for Itô integration to behave continuously with respect to Skorokhod's J1 and M1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M1 setting and unify existing theories in the J1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M1 tightness in fact implies J1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M1 and J1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.

Weak Convergence of Stochastic Integrals on Skorokhod Space in Skorokhod's J1 and M1 Topologies

Abstract

We provide criteria for Itô integration to behave continuously with respect to Skorokhod's J1 and M1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M1 setting and unify existing theories in the J1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M1 tightness in fact implies J1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M1 and J1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.
Paper Structure (36 sections, 29 theorems, 157 equations, 1 figure)

This paper contains 36 sections, 29 theorems, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convergence in the $M_1$ topology
  3. Overview of the paper
  4. Convergence of stochastic integrals
  5. Conditions on the integrands and integrators
  6. Weak convergence of stochastic integrals on Skorokhod space
  7. Convergence in probability and relaxed conditions on integrands
  8. Convergence of simple integrals
  9. Martingale integrators
  10. Exploding integrals for uniformly vanishing martingales
  11. $M_1$ versus $J_1$ tightness of local martingales
  12. Preserving local martingality in the limit
  13. An alternative in place of the AVCI condition
  14. Weak continuity of Itô integrals revisited
  15. A particular class of integrators and integrands
  16. ...and 21 more sections

Key Result

Proposition 2.5

Let $H^n,X^n$ be adapted càdlàg processes defined on filtered probability spaces $(\Omega^n, \mathcal{F}^n, \mathbb{F}^n, \mathbb{P}^n)$. Suppose $(|X^n|^*_t)_{n\ge 1}$ is tight, for all $t>0$, and $(H^n,X^n) \Rightarrow_{f.d.d.} (H,X)$ on a dense subset $D\subseteq [0,\infty)$, for some $H,X \in {\

Figures (1)

  • Figure 1: Graphical representation of the functions $\Psi^n$ and $\Lambda^n$ with a realisation of $\tau$.

Theorems & Definitions (70)

  • Remark 1.1: Convergence together
  • Definition 2.1: Consecutive increment function
  • Definition 2.2: Asymptotically vanishing consecutive increments
  • Definition 2.3: Good decompositions
  • Example 2.4
  • Proposition 2.5: Preservation of the semimartingale property
  • Theorem 2.6: Weak 'continuity' of Itô integrals
  • Remark 2.7
  • Proposition 2.8: Sufficient conditions for AVCI
  • Remark 2.9: Weak and strong $M_1$ topology
  • ...and 60 more