Paper: arXiv 2412.17526

Authors: Eduardo Abi Jaber, Christian Bayer, Simon Breneis

Abstract

We show that the state spaces of multifactor Markovian processes, coming from approximations of nonnegative Volterra processes, are given by explicit linear transformation of the nonnegative orthant. We demonstrate the usefulness of this result for applications, including simulation schemes and PDE methods for nonnegative Volterra processes.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical characterization of the state space for multifactor Markovian approximations of Volterra processes using stochastic differential equations and linear transformations. While mathematically dense and novel in resolving open questions regarding simulation validity and PDE domain meshing, it is primarily a theoretical contribution with limited empirical validation or real-world backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Identify state spaces of multifactor Volterra approximations] --> B[Input: Nonnegative Volterra Processes & Kernel Functions]
    B --> C[Method: Construct Markovian Multifactor Approximations]
    C --> D[Computation: Apply explicit linear transformations to nonnegative orthant]
    D --> E[Outcome: Explicit characterization of valid state spaces]
    E --> F[Applications: Efficient Stochastic Simulation & PDE Methods]

Cited References (22)

  1. Affine V olterra processes (2019)
  2. Optimal approximations of power laws with exponentials: application to volatility models with long memory (2007)
  3. Generalized F eller processes and M arkovian lifts of stochastic V olterra processes: the affine case (2020)
  4. The characteristic function of rough H eston models (2019)
  5. Linear-Quadratic control for a class of stochastic V olterra equations: solvability and approximation (2021)
  6. American options in the V olterra H eston model (2022)
  7. Lifting the H eston model (2019)
  8. A time-stepping deep gradient flow method for option pricing in (rough) diffusion models (2025)
and 14 more
  1. Fractional B rownian motion and the M arkov property. (1998)
  2. Invariance of stochastic control systems with deterministic arguments (2004)
  3. Stochastic viability of convex sets (2007)
  4. Markovian approximations of stochastic V olterra equations with the fractional kernel (2023)
  5. Multifactor approximation of rough volatility models (2019)
  6. Approximation of Stochastic Volterra Equations with kernels of completely monotone type (2024)
  7. Strong convergence rates for M arkovian representations of fractional B rownian motion (2019) — link
  8. Markovian structure of the V olterra H eston model (2019)
  9. Weak M arkovian approximations of rough H eston — → 2309.07023
  10. Efficient option pricing in the rough H eston model using weak simulation schemes (2024)
  11. Second-order weak approximations of CKLS and CEV processes by discrete random variables (2021)
  12. Stochastic invariance of closed sets with non- L ipschitz coefficients (2019)
  13. DOLFINx : the next generation FEniCS problem solving environment (2023) — link
  14. Numerical integration of the extended variable generalized L angevin equation with a positive P rony representable memory kernel (2013)

1 of 22 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.