Related papers: Rough volatility, path-dependent PDEs and weak rat…
We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by obtained by computing a directional derivative of the weak…
This paper deals with an extension of the so-called Black-Scholes model in which the volatility is modeled by a linear combination of the components of the solution of a differential equation driven by a fractional Brownian motion of Hurst…
It has been recently shown that spot volatilities can be very well modeled by rough stochastic volatility type dynamics. In such models, the log-volatility follows a fractional Brownian motion with Hurst parameter smaller than 1/2. This…
In [Han \& Schied, 2023, \textit{arXiv 2307.02582}], an easily computable scale-invariant estimator $\widehat{\mathscr{R}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its…
This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's…
Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price…
We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we…
We study Euler-type discrete-time schemes for the rough Heston model, which can be described by a stochastic Volterra equation (with non-Lipschtiz coefficient functions), or by an equivalent integrated variance formulation. Using weak…
We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and…
We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for L\'evy-driven SDEs or their integrands for stochastic integrals. Main results are bordered…
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be…
The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating it from a single long dependent trajectory have been lacking. We study a stationary…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is…
We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally…
We introduce a novel rough Bergomi (rBergomi) model featuring a variance-driven exponentially weighted moving average (EWMA) time-dependent Hurst parameter $H_t$, fundamentally distinct from recent machine learning and wavelet-based…
Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylized facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample…
Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and L\^e in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise…
We consider autonomous stochastic ordinary differential equations (SDEs) and weak approximations of their solutions for a general class of sufficiently smooth path-dependent functionals f. Based on tools from functional It\^o calculus, such…
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of L\^e (2020). This approach allows one to exploit regularization by noise effects…
We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The…