Kristof Wiedermann (TU Wien)
Three perspectives on the failure of the Markov property for stochastic Volterra integral equations
Abstract: Memory-driven stochastic dynamics naturally arise in many applications ranging from population dynamics to rough volatility models and electricity spot prices. In stochastic Volterra integral equations (SVIEs), path dependence is formally encoded by convolutions with the Volterra kernel, which typically precludes the Markov property. However, since a formal proof has so far only been obtained for the Gaussian case, we present three independent approaches to rigorously verify that general SVIE solutions are non-Markovian. In our first method, we show that non-Markovianity can be established via a reduction to the well-studied Gaussian case through a small-time CLT for SVIEs. Secondly, we prove via moment methods designed specifically for affine drifts that the time-homogeneous Markov property only holds for the exponential Volterra kernel. Our final method utilizes non-degenerate perturbations of the SVIE which are realized via suitable projections of an abstract Hilbert space-valued Markovian lift. The latter provides a powerful tool for embedding memory into an augmented state space and our study of its effective dimension gives new insights into how the kernel governs the degree of non-Markovianity of the SVIE. Moreover, we establish the absolute continuity of such non-degenerate perturbations, allowing us to derive the failure of even the time-inhomogeneous Markov property for general d-dimensional SVIEs.
Our results cover a broad class of SVIEs of practical relevance with diagonal locally square integrable singular or regular kernels, non-constant initial curves and Hölder continuous drift and diffusion coefficients. This reflects the intrinsic infinite-dimensionality of memory effects in SVIEs and highlights the need for analytical and probabilistic tools beyond the classical Markovian framework.
Based on joint work with Martin Friesen and Stefan Gerhold.

Luca Pelizzari (University of Vienna)
Rough PDEs for local stochastic volatility models
Abstract: In this talk, we introduce a pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non- Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.
This is based on joint work with P. Bank, C. Bayer and P. Friz.

Ulrich Horst (HU Berlin)
Mean Field Portfolio Games with Epstein-Zin Preferences
Abstract: We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary stochastic maximum principle tailored to Epstein-Zin utility and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for the equilibrium investment and consumption policies.
The talk is based on joint work with Guanxing Fu.