3 research outputs found
Functional central limit theorems for rough volatility
We extend Donsker's approximation of Brownian motion to fractional Brownian
motion with Hurst exponent and to Volterra-like processes. Some
of the most relevant consequences of our `rough Donsker (rDonsker) Theorem' are
convergence results for discrete approximations of a large class of rough
models. This justifies the validity of simple and easy-to-implement Monte-Carlo
methods, for which we provide detailed numerical recipes. We test these against
the current benchmark Hybrid scheme \cite{BLP15} and find remarkable agreement
(for a large range of values of~). This rDonsker Theorem further provides a
weak convergence proof for the Hybrid scheme itself, and allows to construct
binomial trees for rough volatility models, the first available scheme (in the
rough volatility context) for early exercise options such as American or
Bermudan.Comment: 30 pages, 11 figure
At the mercy of the common noise: blow-ups in a conditional McKean–Vlasov Problem
We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to ‘blow-up’ phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean–Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean–Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise
Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times
We consider a family of McKean-Vlasov equations arising as the large particle
limit of a system of interacting particles on the positive half-line with
common noise and feedback. Such systems are motivated by structural models for
systemic risk with contagion. This contagious interaction is such that when a
particle hits zero, the impact is to move all the others toward the origin
through a kernel which smooths the impact over time. We study a rescaling of
the impact kernel under which it converges to the Dirac delta function so that
the interaction happens instantaneously and the limiting singular
McKean--Vlasov equation can exhibit jumps. Our approach provides a novel method
to construct solutions to such singular problems that allows for more general
drift and diffusion coefficients and we establish weak convergence to relaxed
solutions in this setting. With more restrictions on the coefficients we can
establish an almost sure version showing convergence to strong solutions. Under
some regularity conditions on the contagion, we also show a rate of convergence
up to the time the regularity of the contagion breaks down. Lastly, we perform
some numerical experiments to investigate the sharpness of our bounds for the
rate of convergence.Comment: 43 pages, 4 figure