arXiv subjects

Cristina Di Girolami

Publications and source records attributed to Cristina Di Girolami.

10 recordsLinked to original sources

arXiv2026

Dam Management in the Era of Climate Change

Climate change has a dramatic impact, particularly by concentrating rainfall into a few short periods, interspersed with long dry spells. In this context, the role of dams is crucial. We consider the optimal control of a dam, where the water level must neither exceed a designated safety threshold nor fall below a minimum level to ensure functionality and sustainability for the downstream river. To model dry spells and intense rainfall events, commonly referred to as water bombs, we introduce a Hawkes process, a well-known example of a self-exciting process characterized by time-correlated intensity, which endogenously reproduces the clustering of events. The problem is formulated as an optimal switching problem with constraints. We establish existence results and propose numerical methods for approximating the solution. Finally, we illustrate the main achievements of this approach through numerical examples focusing in particular on the sensitivity of the self-exciting parameter describing the importance of both water bombs and dry-spells. For the parameter configurations considered in this paper, the optimal water level inside the dam decreases as the self-exciting parameter increases. This numerical finding suggests that, under the considered calibration, the management response is driven more strongly by overtopping risk than by drought risk. In conclusion, dams will increasingly lose their role as water reserves and take on a greater role in flood protection.

math.OC↗

arXiv2025

The Design of Optimal Re-Insurance Contracts when Losses are Clustered

This paper investigates the form of optimal reinsurance contracts in the case of clusters of losses. The underlying insured risk is represented by a marked Hawkes process, where the intensity of the jumps depends not only on the occurrence of previous jumps but also on the size of the jumps, which represents the financial magnitude of the loss. The reinsurance contracts are applied to each loss at the time of occurrence, but their structure is assumed to be constant. We derive closed-form formulas within the meanvariance framework. Additionally, we demonstrate that the optimal contract is not the classical excess-loss (deductible) form. The optimal contract is piecewise linear with three ranges: first, no reinsurance below a certain threshold; second, reinsurance with a slope greater than 1; and finally, full reinsurance. When the marked process converges to a Poisson process, we recover the optimality of the deductible form.

math.OC↗

arXiv2025

High risk aversion Merton's problem without transversality conditions

This paper revisits the classical Merton portfolio choice problem over infinite horizon for high risk aversion, addressing technical challenges related to establishing the existence and identification of optimal strategies. Traditional methods rely on perturbation arguments and/or impose restrictive conditions, such as large discount rates and/or bounded strategies, to ensure well-posedness. Our approach leverages the problem's homogeneity to directly solve the associated Hamilton-Jacobi-Bellman equation and verify the optimality of candidate strategies without requiring transversality conditions.

math.OC↗

arXiv2020

About classical solutions of the path-dependent heat equation

This paper investigates two existence theorems for the path-dependent heat equation, which is the Kolmogorov equation related to the window Brownian motion, considered as a C([--T, 0])-valued process. We concentrate on two general existence results of its classical solutions related to different classes of final conditions: the first one is given by a cylindrical non necessarily smooth r.v., the second one is a smooth generic functional.

math.PR↗

arXiv2014

Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations

The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the study of strict solutions of Kolmogorov path dependent equations associated with "windows" of diffusion processes. One makes the link between the Banach space approach and the so called functional stochastic calculus. When no strict solutions are available one describes the notion of strong-viscosity solution which alternative (in infinite dimension) to the classical notion of viscosity solution.

math.PR↗

arXiv2013

The covariation for Banach space valued processes and applications

This article focuses on a new concept of quadratic variation for processes taking values in a Banach space $B$ and a corresponding covariation. This is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace $χ$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of $\bar ν_0$-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.

math.PR↗

arXiv2013

Generalized covariation for Banach space valued processes, Itô formula and applications

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $χ$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_πB_{2})^{\ast}$), we define the so-called $χ$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $χ$-covariation is called $χ$-quadratic variation. The notion of $χ$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $χ$ is the whole space $(B_{1}\hat{\otimes}_πB_{1})^{\ast}$ then the $χ$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $χ$-covariation of various processes for several examples of $χ$ with a particular attention to the case $B_{1}=B_{2}=C([-τ,0])$ for some $τ>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-τ,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-τ,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.

math.PR↗

arXiv2011

Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes

This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of $χ$-covariation which is a generalized notion of covariation for processes with values in two Banach spaces $B_{1}$ and $B_{2}$. $χ$ refers to a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$. We investigate some $C^{1}$ type transformations for various classes of stochastic processes admitting a $χ$-quadratic variation and related properties. If $\X^1$ and $\X^2$ admit a $χ$-covariation, $F^i: B_i \rightarrow \R$, $i = 1, 2$ are of class $C^1$ with some supplementary assumptions then the covariation of the real processes $F^1(\X^1)$ and $F^2(\X^2)$ exist. A detailed analysis will be devoted to the so-called window processes. Let $X$ be a real continuous process; the $C([-τ,0])$-valued process $X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-τ,0]$, is called {\it window} process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for $B=\R$. If $\X=X$ is a real valued Dirichlet process and $F:B \rightarrow \R$ of class $C^{1}(B)$ then $F(\X)$ is still a Dirichlet process. If $\X=X$ is a weak Dirichlet process with finite quadratic variation, and $F: C^{0,1}([0,T]\times B)$ is of class $C^{0,1}$, then $[ F(t, \X_t) ] $ is a weak Dirichlet process. We specify corresponding results when $B=C([-τ,0])$ and $\X=X(\cdot)$. This will consitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As applications, we give a new technique for representing path-dependent random variables.

math.PR↗

arXiv2011

On stochastic calculus related to financial assets without semimartingales

This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class $\mathcal{A}$ of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus related to $\mathcal{A}$-martingales with some examples is developed. Some applications to no-arbitrage, viability, hedging and the maximization of the utility of an insider are expanded. We finally revisit some no arbitrage conditions of Bender-Sottinen-Valkeila type.

math.PR↗

arXiv2010

Clark-Ocone type formula for non-semimartingales with finite quadratic variation

We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $τ>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-τ,0]$. If $X$ is a real finite quadratic variation process, an appropriated Itô formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.

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