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Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions

Yingli Wang, Chunhao Cai, Ping He, QingHua Wang

TL;DR

This work studies the scaling limit of heavy-tailed, nearly unstable cumulative INAR($\infty$) processes, showing that with a tail $\eta_n \sim \frac{K}{n^{1+\alpha}}$ and $\alpha\in(\tfrac12,1)$, the properly renormalized discrete process converges to a rough fractional diffusion, realized as an integrated fractional CIR process. The authors develop discrete renewal-theoretic tools and a martingale central limit framework to prove tightness and identify the limit: a continuous martingale $Z$ with $Z_t=B_{Y_t}$ and a Hölder-continuous $Y$ solving a stochastic Volterra equation driven by the Mittag-Leffler kernel $f^{\alpha,\lambda}$. They further provide an efficient INAR($\infty$)–based simulation scheme for the limiting fractional CIR process, including a mechanism to encode a nonzero initial value via a time-dependent baseline intensity. Numerical validation against rough Heston benchmarks demonstrates the practical accuracy of the approach, with a Goodness-of-Fit test validating the univariate INAR($\infty$) approximation in European-option pricing contexts. The work offers a discrete microstructural foundation for rough volatility, enabling fast, exact-like simulation and scalable inference for long-memory, heavy-tailed self-exciting dynamics.

Abstract

In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR($\infty$) processes. These processes exhibit a power-law tail of the form $n^{-(1+α)}$ for $α\in (\frac{1}{2}, 1)$, and the $\ell^1$ norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present an efficient method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at https://github.com/gagawjbytw/INAR-rough-Heston.

Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions

TL;DR

This work studies the scaling limit of heavy-tailed, nearly unstable cumulative INAR() processes, showing that with a tail and , the properly renormalized discrete process converges to a rough fractional diffusion, realized as an integrated fractional CIR process. The authors develop discrete renewal-theoretic tools and a martingale central limit framework to prove tightness and identify the limit: a continuous martingale with and a Hölder-continuous solving a stochastic Volterra equation driven by the Mittag-Leffler kernel . They further provide an efficient INAR()–based simulation scheme for the limiting fractional CIR process, including a mechanism to encode a nonzero initial value via a time-dependent baseline intensity. Numerical validation against rough Heston benchmarks demonstrates the practical accuracy of the approach, with a Goodness-of-Fit test validating the univariate INAR() approximation in European-option pricing contexts. The work offers a discrete microstructural foundation for rough volatility, enabling fast, exact-like simulation and scalable inference for long-memory, heavy-tailed self-exciting dynamics.

Abstract

In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR() processes. These processes exhibit a power-law tail of the form for , and the norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present an efficient method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at https://github.com/gagawjbytw/INAR-rough-Heston.
Paper Structure (21 sections, 13 theorems, 130 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 21 sections, 13 theorems, 130 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Assumptions and Main Results
  3. Heuristic Motivation for the Scaling Limit
  4. A Microstructural Foundation for the Initial Value
  5. Numerical Validation and Application
  6. Simulation via INAR($\infty$) Approximation
  7. A Goodness-of-Fit Test and application in rough Heston models
  8. Proof of Lemmas
  9. Proof of Lemma \ref{['lemma:laplaceofrho']}
  10. Proof of Lemma \ref{['lemma:sumandnotsum']}
  11. Proof of Lemma \ref{['lemma:discreteKaramataTauberian']}
  12. Proof of Lemma \ref{['lemma:tightnessofyandlambda']}
  13. Proof of Lemma \ref{['lemma:YLambdaconvergeinprobability']}
  14. Proof of Lemma \ref{['lemma:discreterenewalequation']}
  15. Proof of Lemma \ref{['lemma:anotherexpressionoflambda']}
  16. ...and 6 more sections

Key Result

Lemma 1

Define a sequence of functions $(\rho^T)_{T\ge0}$ on $[0, \infty)$ by Each $\rho^T$ is a step function and a valid probability density function, since it is non-negative and integrates to one. As $T \to \infty$, the sequence of probability measures corresponding to these densities converges weakly to a probability measure on $[0, \infty)$ with the density $\lambda x^{

Figures (1)

  • Figure 1: A sample path of the fractional CIR process $(V_t = \dot{Y}_t)_{0 \leq t \leq 1}$, simulated via the INAR($\infty$) approximation. The simulation parameters are $\alpha = 0.62$, $\lambda_{\text{param}} = 1.0$, $\nu^* = 200.0$, and a target initial value $v_0 = 0.7$. The number of discrete steps is $T_{\text{steps}}=5000$. The red dashed line indicates the target initial value, which the simulated path closely matches at $t=0$.

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Discrete-type extended Karamata-Tauberian Theorem
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Criterion for $C$-tightness, Lemma 3.5 in horst2023convergence
  • Lemma 6
  • ...and 16 more