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Functional CLTs for subordinated Lévy models in physics, finance, and econometrics

Andreas Søjmark, Fabrice Wunderlich

Abstract

We present a simple unifying treatment of a broad class of applications from statistical mechanics, econometrics, mathematical finance, and insurance mathematics, where (possibly subordinated) Lévy noise arises as a scaling limit of some form of continuous-time random walk (CTRW). For each application, it is natural to rely on weak convergence results for stochastic integrals on Skorokhod space in Skorokhod's J1 or M1 topologies. As compared to earlier and entirely separate works, we are able to give a more streamlined account while also allowing for greater generality and providing important new insights. For each application, we first elucidate how the fundamental conclusions for J1 convergent CTRWs emerge as special cases of the same general principles, and we then illustrate how the specific settings give rise to different results for strictly M1 convergent CTRWs.

Functional CLTs for subordinated Lévy models in physics, finance, and econometrics

Abstract

We present a simple unifying treatment of a broad class of applications from statistical mechanics, econometrics, mathematical finance, and insurance mathematics, where (possibly subordinated) Lévy noise arises as a scaling limit of some form of continuous-time random walk (CTRW). For each application, it is natural to rely on weak convergence results for stochastic integrals on Skorokhod space in Skorokhod's J1 or M1 topologies. As compared to earlier and entirely separate works, we are able to give a more streamlined account while also allowing for greater generality and providing important new insights. For each application, we first elucidate how the fundamental conclusions for J1 convergent CTRWs emerge as special cases of the same general principles, and we then illustrate how the specific settings give rise to different results for strictly M1 convergent CTRWs.
Paper Structure (8 sections, 8 theorems, 30 equations)

This paper contains 8 sections, 8 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Functional CLTs for moving averages and CTRWs
  3. Stable Lévy noise in statistical mechanics
  4. Cointegrated processes in econometrics
  5. Stability of the financial gain in mathematical finance
  6. On J1 stability of the financial gains
  7. On M1 stability of the financial gains
  8. Reserves with risky investments in insurance mathematics

Key Result

Proposition 2.1

Let $X^n$ be an uncorrelated CTRW (i.e., as defined in defi:CTRW with $c_j=0$ for all $j\ge 1$) with $\alpha \in (0,2]$, $\beta \in (0,1)$ and scaling limit $X=((Z^-)_{D^{-1}})^+$ from eq:coupled_CTRW_J1_conv which reduces to $X=Z_{D^{-1}}$ if the CTRWs are uncoupled. Let also $f_n,f$ be such that $

Theorems & Definitions (16)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • ...and 6 more