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Giulia Di Nunno

Publications and source records attributed to Giulia Di Nunno.

At least 37 records · Page 2Linked to original sources
arXiv2023

Maximum principles for stochastic time-changed Volterra games

We study a stochastic differential game between two players, controlling a forward stochastic Volterra integral equation (FSVIE). Each player has to optimize his own performance functional which includes a backward stochastic differential equation (BSDE). The dynamics considered are driven by time-changed Lévy noises, with absolutely continuous time-change process. We prove a sufficient maximum principle to characterize Nash equilibria and the related optimal strategies. For this we use techniques of control under partial information, and the non-anticipating stochastic derivative. The zero-sum game is presented as a particular case.

math.PR↗
arXiv2023

SPDE bridges with observation noise and their spatial approximation

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.

math.NA↗
arXiv2022

Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary $λ$-Hölder continuous process, $λ\in(0,1)$. We prove that, under some mild moment assumptions on the Hölder constant of the noise, the $L^r(Ω;L^\infty([0,T]))$-rate of convergence is equal to $λ$. To exemplify, we consider numerical schemes for the generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models. The results are illustrated by simulations.

math.PR↗
arXiv2021

Sandwiched SDEs with unbounded drift driven by Hölder noises

We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of an arbitrary order. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some additional assumptions on the noise, we prove that the solution has moments of all orders. We complete the study providing a numerical scheme for the solution. As an illustration of our results and motivation for applications, we suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes.

math.PR↗
arXiv2020

Copula Measures and Sklar's Theorem in Arbitrary Dimensions

Although copulas are used and defined for various infinite-dimensional objects (e.g. Gaussian processes and Markov processes), there is no prevalent notion of a copula that unifies these concepts. We propose a unified approach and define copulas as probability measures on general product spaces. For this we prove Sklar's Theorem in this infinite-dimensional setting. We show how to transfer this result to various function space settings and describe how to model and approximate dependent probability measures in these spaces in the realm of copulas.

math.PR↗
arXiv2020

Sensitivity analysis in the infinite dimensional Heston model

We consider the infinite dimensional Heston stochastic volatility model proposed in \arXiv:1706:03500. The price of a forward contract on a non-storable commodity is modelled by a generalized Ornstein-Uhlenbeck process in the Filipović space with this volatility. We prove different representation formulas for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning of the Greeks. For this we use infinite dimensional Malliavin calculus and a randomization technique.

math.PR↗
arXiv2020

Stochastic differential equations driven by additive Volterra-Lévy and Volterra-Gaussian noises

We study the existence and uniqueness of solutions to stochastic differential equations with Volterra processes driven by Lévy noise. For this purpose, we study in detail smoothness properties of these processes. Special attention is given to two kinds of Volterra-Gaussian processes that generalize the compact interval representation of fractional Brownian motion and to stochastic equations with such processes.

math.PR↗
arXiv2019

Fully-dynamic risk-indifference pricing and no-good-deal bounds

The seller's risk-indifference price evaluation is studied. We propose a dynamic risk-indifference pricing criteria derived from a fully-dynamic family of risk measures on the $L_p$-spaces for $p\in [1,\infty]$. The concept of fully-dynamic risk measures extends the one of dynamic risk measures by adding the actual possibility of changing the risk perspectives over time. The family is then characterised by a double time index. Our framework fits well the study of both short and long term investments. In this dynamic framework we analyse whether the risk-indifference pricing criterion actually provides a proper convex price system, for which time-consistency is guaranteed. It turns out that the analysis is quite delicate and necessitates an adequate setting. This entails the use of capacities and an extension of the whole price system to the Banach spaces derived by the capacity seminorms. Furthermore, we consider the relationship of the fully-dynamic risk-indifference price with no-good-deal bounds. Recall that no-good-deal pricing guarantees that not only arbitrage opportunities are excluded, but also all deals that are "too good to be true". We shall provide necessary and sufficient conditions on the fully-dynamic risk measures so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. The use of no-good-deal bounds also provides a method to select the risk measures and then construct a proper fully-dynamic risk-indifference price system in the $L_2$-spaces.

math.PR↗
arXiv2018

A continuous auction model with insiders and random time of information release

In a unified framework we study equilibrium in the presence of an insider having information on the signal of the firm value, which is naturally connected to the fundamental price of the firm related asset. The fundamental value itself is announced at a future random (stopping) time. We consider two cases. First when the release time of information is known to the insider and then when it is unknown also to her. Allowing for very general dynamics, we study the structure of the insider's optimal strategies in equilibrium and we discuss market efficiency. In particular, we show that in the case the insider knows the information release time, the market is fully efficient. In the case the insider does not know this random time, we see that there is an equilibrium with no full efficiency, but where the sensitivity of prices is decreasing in time according with the probability that the announcement time is greater than the current time. In other words, the prices become more and more stable as the announcement approaches.

q-fin.PR↗
arXiv2017

Sensitivity analysis in a market with memory

A general market model with memory is considered in terms of stochastic functional differential equations. We aim at representation formulae for the sensitivity analysis of the dependence of option prices on the memory. This implies a generalization of the concept of delta.

math.PR↗
arXiv2017

Stochastic functional differential equations and sensitivity to their initial path

We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of {\it randomisation of the initial condition}.

math.PR↗
arXiv2016

Fractional calculus and path-wise integration for Volterra processes driven by Lévy and martingale noise

We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.

math.PR↗
arXiv2016

A Maximum Principle for Mean-Field SDEs with time change

Time change is a powerful technique for generating noises and providing flexible models. In the framework of time changed Brownian and Poisson random measures we study the existence and uniqueness of a solution to a general mean-field stochastic differential equation. We consider a mean-field stochastic control problem for mean-field controlled dynamics and we present a necessary and a sufficient maximum principle. For this we study existence and uniqueness of solutions to mean-field backward stochastic differential equations in the context of time change. An example of a centralised control in an economy with specialised sectors is provided.

math.PR↗
arXiv2016

Representation of convex operators and their static and dynamic sandwich extensions

Monotone convex operators and time-consistent systems of operators appear naturally in stochastic optimization and mathematical finance in the context of pricing and risk measurement. We study the dual representation of a monotone convex \emph{operator} when its domain is defined on a subspace of $L_p$, with $p\in [1,\infty]$, and we prove a sandwich preserving extension theorem. These results are then applied to study systems of such operators defined only on subspaces. We propose various dynamic sandwich preserving extension results depending on the nature of time: finite discrete, countable discrete, and continuous. Of particular notice is the fact that the extensions obtained are time-consistent.

math.FA↗
arXiv2016

A Malliavin-Skorohod calculus in $L^0$ and $L^1$ for additive and Volterra-type processes

In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the $L^0$ and $L^1$ settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in $L^1$. Our theory is then applied to extend the stochastic integration with respect to volatility modulated Lévy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect, e.g. to Volterra processes driven by $α$-stable processes with $α< 2$. The presentation focuses on jump type processes.

math.PR↗
arXiv2015

Hedging under worst-case-scenario in a market driven by time-changed Lévy noises

In an incomplete market driven by time-changed Lévy noises we consider the problem of hedging a financial position coupled with the underlying risk of model uncertainty. Then we study hedging under worst-case-scenario. The proposed strategies are not necessarily self-financing and include the interplay of a cost process to achieve the perfect hedge at the end of the time horizon. The hedging problem is tackled in the framework of stochastic differential games and it is treated via backward stochastic differential equations. Two different information flows are considered and the solutions compared.

math.PR↗
arXiv2014

Approximations of Stochastic Partial Differential Equations

In this paper we show that solutions of stochastic partial differential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.

math.PR↗