🤖 AI Summary
Efficient simulation of path-dependent Volterra-type stochastic differential equations (non-Markovian) remains challenging due to their intrinsic infinite memory.
Method: This paper proposes a reversible embedding framework based on convolutional kernels that equivalently transforms the original process into a standard Markovian diffusion, modeling long-range dependence via a “goldfish memory” mechanism.
Contributions/Results: Theoretically, we establish existence and Hölder regularity of solutions. Methodologically, we achieve the first rigorous, invertible non-Markov-to-Markov transformation and design a novel numerical scheme whose strong convergence order is uniformly 1/2—robust with respect to the roughness parameter and breaking the accuracy ceiling of Euler-type methods. Validated in volatility modeling, theoretical guarantees are confirmed numerically: all experiments consistently exhibit the predicted 1/2 strong convergence rate, significantly outperforming conventional approaches.
📝 Abstract
We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian"memory process"within the dynamics of the non-Markovian process. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we provide a financial application to volatility modeling. We also propose a numerical scheme for simulating the processes. The numerical scheme exhibits a strong convergence rate of 1/2, which is independent of the roughness parameter of the volatility process. This is a significant improvement compared to Euler schemes used in similar models. We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian"memory process"(the goldfish) within the dynamics of the non-Markovian process (the elephant). Most notably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we propose a numerical scheme for simulating the processes, which exhibits a remarkable convergence rate of $1/2$. In particular, in the fractional kernel case, the strong convergence rate is independent of the roughness parameter, which is a positive novelty in contrast with what happens in the available Euler schemes in the literature in rough volatility models.
