Giulia Di Nunno

• Banos, David; Cordoni, Francesco; Di Nunno, Giulia; Di Persio, Luca & Røse, Elin Engen (2019). Stochastic systems with memory and jumps. Journal of Differential Equations.  ISSN 0022-0396.  266(9), s 5772- 5820 . doi: http://dx.doi.org/10.1016/j.jde.2018.10.052 Full text in Research Archive.
• Corcuera, José Manuel; Di Nunno, Giulia & Fajardo, Jose (2019). Kyle equilibrium under random price pressure. Decisions in Economics and Finance.  ISSN 1593-8883.  42(1), s 77- 101 . doi: 10.1007/s10203-019-00231-4 Full text in Research Archive. Show summary

  We study the equilibrium in the model proposed by Kyle (Econometrica 53(6):1315–1335, 1985) and extended to the continuous-time setting by Back (Rev Financ Stud 5(3):387–409, 1992). The novelty of this paper is that we consider a framework where the price pressure can be random. We also allow for a random release time of the fundamental value of the asset. This framework includes all the particular Kyle models proposed in the literature. The results enlighten the equilibrium properties shared by all these models and guide the way of finding equilibria in this context.

• Di Nunno, Giulia; Fiacco, Andrea & Karlsen, Erik Hove (2019). On the approximation of Lévy driven Volterra processes and their integrals. Journal of Mathematical Analysis and Applications.  ISSN 0022-247X.  476(1), s 120- 148 . doi: 10.1016/j.jmaa.2019.02.051 Full text in Research Archive. Show summary

  Volterra processes appear in several applications ranging from turbulence to energy finance where they are used in the modelling of e.g.temperatures and wind and the related financial derivatives. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. In this work we suggest to construct an approximating sequence of Lévy driven Volterra processes, by perturbation of the kernel function. In this way,one can obtain an approximating sequence of semimartingales. Then we consider fractional integration with respect to Volterra processes as integrators and we study the corresponding approximations of the fractional integrals. We illustrate the approach presenting the specific study of the Gamma-Volterra processes. Examples and illustrations via simulation are given.

• Baños, David; Di Nunno, Giulia; Haferkorn, Hannes Hagen & Proske, Frank Norbert (2018). Stochastic functional differential equations and sensitivity to their initial path, In Elena Celledoni; Giulia Di Nunno; Kurusch Ebrahimi-Fard & Hans Munthe-Kaas (ed.),  Computation and Combinatorics in Dynamics, Stochastics and Control.  Springer.  ISBN 978-3-030-01592-3.  2.  s 37 - 70
• Corcuera, José Manuel & Di Nunno, Giulia (2018). Kyle-Back's model with a random horizon. International Journal of Theoretical and Applied Finance.  ISSN 0219-0249.  21(2) . doi: 10.1142/S0219024918500164 Full text in Research Archive.
• Bion-Nadal, Jocelyne & Di Nunno, Giulia (2017). Fully-dynamic risk-indifference prices and no-good-deal bounds. arXiv.org.  ISSN 2331-8422. Full text in Research Archive. Show summary

  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 ∈ [1, ∞]. 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.

• Bion-Nadal, Jocelyne & Di Nunno, Giulia (2017). Representation of convex operators and their static and dynamic sandwich extensions. Journal of Convex Analysis.  ISSN 0944-6532.  24(4), s 1375- 1405
• Di Nunno, Giulia & Haferkorn, Hannes Hagen (2017). A maximum principle for mean-field SDEs with time change. Applied Mathematics and Optimization.  ISSN 0095-4616.  76(1), s 137- 176 . doi: 10.1007/s00245-017-9426-0 Full text in Research Archive.
• Di Nunno, Giulia & Vives, Josep (2017). A Malliavin-Skorohod calculus in L^0 and L^1 for additive and Volterra-type processes. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  89(1), s 142- 170 . doi: 10.1080/17442508.2016.1140767 Full text in Research Archive. Show summary

  In this paper we develop a Malliavin–Skorohod type calculus for additive processes in the L1 and L1 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 L1. 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 to e.g. Volterra processes driven by alfa-stable processes with alfa < 2. The presentation focuses on jump type processes.

• Di Nunno, Giulia & Karlsen, Erik Hove (2016). Hedging under worst-case-scenario in a market driven by time-changed Lévy noises, In Mark Podolskij; Robert Stelzer; Steen Thorbjørnsen & Almut E. D. Veraart (ed.),  The Fascination of Probability, Statistics and their Applications. In honour of Ole E. Barndorff-Nielsen.  Springer Science+Business Media B.V..  ISBN 978-3-319-25824-9.  Chapter 22.  s 465 - 499 Show summary

  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.

• Di Nunno, Giulia; Mishura, Yuliya & Ralchenko, Kostiantyn (2016). Fractional calculus and pathwise integration for Volterra processes driven by Lévy and martingale noise. Fractional Calculus and Applied Analysis.  ISSN 1311-0454.  19(6), s 1356- 1392 . doi: 10.1515/fca-2016-0071 Show summary

  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.

• Di Nunno, Giulia & Zhang, Tusheng (2016). Approximations of stochastic partial differential equations. The Annals of Applied Probability.  ISSN 1050-5164.  26(3), s 1443- 1466 . doi: 10.1214/15-AAP1122
• Benth, Fred Espen; Di Nunno, Giulia; Khedher, Asma & Schmeck, Maren Diane (2015). Pricing of Spread Options on a Bivariate Jump Market and Stability to Model Risk. Applied Mathematical Finance.  ISSN 1350-486X.  22(1), s 28- 62 . doi: 10.1080/1350486X.2014.948708
• Di Nunno, Giulia & Karlsen, Erik Hove (2015). Hedging under worst-case-scenario in a market driven by time-changed Lévy noises. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.
• Di Nunno, Giulia; Khedher, Asma & Vanmaele, Michèle (2015). Robustness of Quadratic Hedging Strategies in Finance via Backward Stochastic Differential Equations with Jumps. Applied Mathematics and Optimization.  ISSN 0095-4616.  72(3), s 353- 389 . doi: 10.1007/s00245-014-9283-z
• Di Nunno, Giulia & Vives, Josep (2015). A Malliavin-Skorohod calculus in L^0 and L^1 for additive and Volterra-type processes. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.
• Bion-Nadal, Jocelyne & Di Nunno, Giulia (2014). Representation of convex operators and their static and dynamic sandwich extension. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  (4) Full text in Research Archive.
• Corcuera, Jose Manuel; Di Nunno, Giulia; Farkas, Gergely & Øksendal, Bernt (2014). A continuous auction model with insiders and random time of information release. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  (3) Full text in Research Archive.
• Di Nunno, Giulia & Sjursen, Steffen A. Søreide (2014). BSDEs driven by time-changed Lévy noises and optimal control. Stochastic Processes and their Applications.  ISSN 0304-4149.  124(4), s 1679- 1709 . doi: 10.1016/j.spa.2013.12.010
• Di Nunno, Giulia & Sjursen, Steffen A. Søreide (2014). Information and optimal investment in defaultable assets. International Journal of Theoretical and Applied Finance.  ISSN 0219-0249.  17(8) . doi: 10.1142/S0219024914500502
• Di Nunno, Giulia & Zhang, Tusheng (2014). Approximations of Stochastic Partial Differential Equations. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  (1) Full text in Research Archive.
• Benth, Fred Espen; Di Nunno, Giulia & Khedher, Asma (2013). A note on convergence of option prices and their Greeks for Lévy models. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  85(6), s 1015- 1039 . doi: 10.1080/17442508.2012.736994
• Di Nunno, Giulia & Bion-Nadal, Jocelyne (2013). Dynamic no-good-deal pricing measures and extension theorems for linear operators on L-infinity. Finance and Stochastics.  ISSN 0949-2984.  17(3), s 587- 613 . doi: 10.1007/s00780-012-0195-y
• Di Nunno, Giulia & Sjursen, Steffen A. Søreide (2013). On Chaos Representation and Orthogonal Polynomials for the Doubly Stochastic Poisson Process, In  Seminar on Stochastic Analysis, Random Fields and Applications VII.  Birkhäuser Verlag.  ISBN 978-3-0348-0545-2.  Del 1 - kap 2.  s 23 - 54
• Benth, Fred Espen; Di Nunno, Giulia & Khedher, Asma (2012). Computation of Greeks in multifactor models with applications to power and commodity markets. Journal of Energy Markets.  ISSN 1756-3607.  5(4), s 3- 31
• Benth, Fred Espen; Di Nunno, Giulia & Khedher, Asma (2011). Robustness of option prices and their deltas in markets modelled by jump-diffusions. Communications on Stochastic Analysis.  ISSN 0973-9599.  5(2), s 285- 307
• Di Nunno, Giulia & Bion-Nadal, Jocelyne (2011). Extension theorems for linear operators on L_\infty and application to price systems. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  4

View all works in Cristin

• Celledoni, Elena; Di Nunno, Giulia; Ebrahimi-Fard, Kurusch & Munthe-Kaas, Hans (ed.) (2018). Computation and Combinatorics in Dynamics, Stochastics and Control. Springer.  ISBN 978-3-030-01592-3.  737 s. Show summary

  The Abel Symposia volume at hand contains a collection of high-quality articles written by the world’s leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory. In recent years we have witnessed a remarkable convergence between individual mathematical disciplines that approach deterministic and stochastic dynamical systems from mathematical analysis, computational mathematics and control theoretical perspectives. Breakthrough developments in these fields now provide a common mathematical framework for attacking many different problems related to differential geometry, analysis and algorithms for stochastic and deterministic dynamics. In the Abel Symposium 2016, which took place from August 16-19 in Rosendal near Bergen, leading researchers in the fields of deterministic and stochastic differential equations, control theory, numerical analysis, algebra and random processes presented and discussed the current state of the art in these diverse fields. The current Abel Symposia volume may serve as a point of departure for exploring these related but diverse fields of research, as well as an indicator of important current and future developments in modern mathematics.

• Benth, Fred Espen & Di Nunno, Giulia (ed.) (2016). Stochastics of Environmental and Financial Economics. Springer Science+Business Media B.V..  ISBN 978-3-319-23424-3.  360 s.

View all works in Cristin

• Di Nunno, Giulia (2019, 17. juli). ICIAM 2019 Su Buchin - Prize Prof. Giulia di Nunno. [TV].  WebsEdgeEducation.
• Di Nunno, Giulia (2019). Martingale random fields in time change models, the role of information in optimal portfolio problems.
• Di Nunno, Giulia (2019). Mathematics, Society, Economy and Development.
• Di Nunno, Giulia (2019). Optimal strategies in a market with memory. Show summary

  We consider a market model driven by Volterra type dynamics driven by a time-changed Levy noise. These dynamics allow both for memory features and clustering effects in the trading times. In this framework, we study an optimal portfolio problem, which is then tackled via maximum principle. To produce such results we use different kind of information flows that take care of the time-change in adequate way and we rely on the non-anticipating stochastic derivative for random fields. Moreover, we study the solutions of Volterra type SDEs and Volterra type BSDEs driven by time-changed Levy noises. Talk based on joint work with Michele Giordano, UiO.

• Di Nunno, Giulia (2019). Time-change in modelling, stochastic calculus and control. Show summary

  Time change is a powerful technique for generating noises and providing flexible models. Its main idea stands in the representation of complicated stochastic structures by some known processes and a randomly perturbed time line. We shall provide an excursus through related stochastic calculus, chaos structure and information, integral representations and stochastic differentiation. These will then be applied to backward stochastic differential equations and optimal control problems. Motivation of our work is taken from stochastic finance.

• Di Nunno, Giulia & Natalini, Roberto (2019, 01. august). Interview by Roberto Natalini with Giulia Di Nunno, 2019 Su Buchin Prize Winner. [Internett].  DIANOIA.
• Di Nunno, Giulia & Salomon, Mónica G. (2019, 27. august). Hay matemáticos de primer nivel trabajando a destajo en África.  El Pais.
• Di Nunno, Giulia (2018). A continuous auction model with insiders and random time of information release. Show summary

  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. This is joint work with Jose Manuel Corcuera, Gergely Farkas, Bernt Øksendal

• Di Nunno, Giulia (2018). Fully dynamic risk-indifference pricing and no-good-deal bounds. Show summary

  In an incomplete market with no a priori assumption on the underlying price dynamics, we focus on the problem of derivative pricing from the seller's perspective. We consider risk indifference pricing as an alternative to the classical utility indifference, so that the actual evaluations are done via risk measures. In addition we propose a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. This is based on the concept of fully-dynamic risk measure which extends the one of dynamic risk measure by adding the actual possibility of changing the risk perspectives over time. This entails an analysis on the questions of time-consistency in the risk and then the price evaluations. The framework proposed fits well the study of both short and long term investments. In this framework we study whether the risk indifference criterion actually provides a proper convex price system. We shall see that some conditions have to be fulfilled. Then we consider the relationship of fully dynamic risk-indifference price with no-good-deal bounds. We 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 measure so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. In this way no-good-deal bounds provide a way to select the risk measures to obtain a proper fully-dynamic risk-indifference price system. The presentation is based on various joint works with Jocelyne Bion-Nadal.

• Di Nunno, Giulia (2018). Integration with respect to Levy driven Volterra processes. Show summary

  Volterra processes appear in several applications ranging from turbulence to energy finance and biological modelling. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. We present some recent results within the framework of fractional calculus and Malliavin Calculus. As illustration we consider specifically the so-called Gamma-Volterra processes. The presentation is based on joint works with: Andrea Fiacco, Erik H. Karlsen and Josep Vives.

• Di Nunno, Giulia (2018). Kyle-Backs equilibrium model with a random time of information release. Show summary

  We consider the continuous-time version of Kyle-Back’s model, originally from 1985 and 1992. In Back’s model there is asymmetric information in the market in the sense that there is an insider having information on the real value of the asset. We extend this model by assuming that the fundamental value evolves with time and that it is announced at a future random time. The goal of the paper is to study the structure of equilibrium, which is described by the optimal insider strategy and the competitive market prices given by the market makers. The market makers give prices via a pricing rule, that depends on the total demand and a certain price pressure. In this work we consider both the case of deterministic and stochastic price pressure. We provide necessary and sufficient conditions for the optimal insider strategy under general dynamics for the asset demands. Moreover, we study the behavior of the price pressure and the market efficiency. The presentation is based on joint works with: José Manuel Corcuera (U. Barcelona) and José Fajardo (Brazilian School of Public and Business Administration).

• Di Nunno, Giulia (2018). Kyle-Backs equilibrium model with a random time of information release. Show summary

  The continuous-time version of Kyle (1985)) developed by Back (1992)) is here studied. In Back’s model there is asymmetric information in the market in the sense that there is an insider having infor- mation on the real value of the asset. We extend this model by assuming that the fundamental value evolves with time and that it is announced at a future random time. First we consider the case when the release time of information is predictable to the insider and then when it is not. The goal of the paper is to study the structure of equilibrium, which is described by the optimal insider strategy and the competitive market prices given by the market makers. We provide necessary and sufficient conditions for the optimal insider strategy under general dynamics for the asset demands. Moreover, we study the behavior of the price pressure and the market efficiency. In particular we find that when the random time is not predictable, there can be equilibrium without market efficiency. Furthermore, for the two cases of release time and for classes of pricing rules, we provide a characterization of the equilibrium. The talk is based on joint works with: Jose Manuel Corcuera, Bernt Øksendal, Gergely Farkas

• Di Nunno, Giulia (2018). Kyle-Back’s model with a random horizon. Show summary

  The continuous-time version of Kyle (1985)) developed by Back (1992)) is here studied. In Back’s model there is asymmetric information in the market in the sense that there is an insider having infor- mation on the real value of the asset. We extend this model by assuming that the fundamental value evolves with time and that it is announced at a future random time. First we consider the case when the release time of information is predictable to the insider and then when it is not. The goal of the paper is to study the structure of equilibrium, which is described by the optimal insider strategy and the competitive market prices given by the market makers. We provide necessary and sufficient conditions for the optimal insider strategy under general dynamics for the asset demands. Moreover, we study the behavior of the price pressure and the market efficiency. In particular we find that when the random time is not predictable, there can be equilibrium without market efficiency. Furthermore, for the two cases of release time and for classes of pricing rules, we provide a characterization of the equilibrium.

• Di Nunno, Giulia (2018). Levy driven Volterra processes: approximation and integration. Show summary

  Volterra type processes appear in several applications ranging from turbulence, to energy finance, and biological modelling. In general, these processes are not semimartingales and a theory of stochastic integration with respect to such processes is in fact not standard. In this presentation, we consider L\’evy driven Volterra processes and we discuss some recent approaches based on fractional calculus. We study both semimartingale approximations to the Volterra process and approximation to the integral. As illustration, we detail the study of the so-called Gamma-Volterra processes, which is particularly popular in modelling. This is based on joint works with Andrea Fiacco, Erik Karlsen, Yuliya Mishura, Kostia Ralchenko

• Di Nunno, Giulia (2018). Malliavin Calculus and Applications to Finance.
• Di Nunno, Giulia (2018). On fully-dynamic risk-indifference pricing: time-consistency and other properties. Show summary

  Risk-indifference pricing is proposed as an alternative to utility indifference pricing, where a risk measure is used instead of a utility based preference. In this, we propose to include the possibility to change the attitude to risk evaluation as time progresses. This is particularly reasonable for long term investments and strategies. Then we introduce a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. The risk-indifference pricing system is studied from the point of view of its properties as a convex price system. We tackle questions of time-consistency in the risk evaluation and the corresponding prices. This analysis provides a new insight also to time-consistency for ordinary dynamic risk-measures. Our techniques and results are set in the representation and extension theorems for convex operators. We shall argue and finally provide a setting in which fully-dynamic risk-indifference pricing is a well set convex price system. The presentation is based on joint works with Jocelyne Bion-Nadal.

• Di Nunno, Giulia (2018). On the integration with respect to Volterra processes: fractional calculus and approximation.
• Di Nunno, Giulia (2018). On the integration with respect to Volterra type processes. Show summary

  Volterra processes appear in several applications ranging from turbulence, to energy finance, and biological modelling. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. We consider L\’evy driven Volterra processes and we discuss some some recent approaches and results within the framework of Malliavin calculus and fractional calculus. As illustration we consider specifically the so-called Gamma-Volterra processes, which is particularly popular in modelling. This is based on joint works with Andrea Fiacco, Erik Karlsen, Yuliya Mishura, Kostia Ralchenko, Josep Vives

• Di Nunno, Giulia (2018). Sandwich extensions of linear and convex operators and their applications. Show summary

  We propose some extension theorems for linear and convex operators from $L_p$ spaces to $L_p$ spaces that preserve both minoring and majoring stochastic bounds. We shall see how these extension theorems are applied within the context of pricing in mathematical finance. We discuss in fact the concept of a pricing system, which is in fact a family of such operators that has to preserve some reasonable conditions, including the time-consistency to guarantee that the fundamental economic criteria of the non-arbitrage principle is maintained. Various properties and form of pricing operators are presented. This talk is based on a series of works with various co-authors: Sergio Albeverio (U. Bonn), Jocelyne Bion-Nadal (CNRS Ecole Polytechnique), Inga B. Eide (now at Finanstilsynet), Yuri Rozanov (CNR Milano).

• Di Nunno, Giulia (2018). Stochastic calculus and control for systems driven by time-changed Levy noises.
• Di Nunno, Giulia (2018). Stochastic systems with memory, robustness and sensitivity. Show summary

  Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures. We study the case when the driving noises admit jumps provid- ing results on existence and uniqueness of strong solutions, estimates for the moments and the fundamental tools of calculus. We study the robustness of the solution to the change of noises. Specifically, we consider the noises with infinite activity jumps versus an adequately corrected Gaussian noise. In the case of Brownian driving noise, we consider 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 repre- sentation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our techniques make use of stochastic calculus via regularisations, Malliavin/Skorohod calculus and functional derivatives. The presentation is based on joint works with: D.R. Banos, F. Cordoni, L. Di Persio, H.H. Haferkorn, F. Proske, E.E. Røse.

• Di Nunno, Giulia (2017). Control of an economic with specialised sectors: a maximum principle approach for mean-field SDEs with time change. Show summary

  We consider an economy with specialised sectors modelled by a mean- eld SDE driven by time change Levy noises. Time change is a powerful technique for generating noises and providing exible models. In this study we fo- cus on time changed Brownian and Poisson ran- dom measures. We study the existence and uniqueness of the solution to a general mean- eld SDEs. Moreover we consider a mean field stochastic control problem for mean field controlled dynamics. In the application this corresponds to a centralised control for the economy under consideration. We present a necessary and sufficient maximum principle to solve the optimal control problem. For this we study existence and uniqueness of solutions to mean field BSDEs in the context of time change. This is based on joint work with Hannes Haferkorn

• Di Nunno, Giulia (2017). Dynamic risk indifference pricing. Show summary

  We deal with dynamic pricing from the seller's perspective in an incomplete market and we focus on risk indifference pricing. We propose a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. This is based on the concept of fully-dynamic risk measures which extends the one of dynamic risk measures by adding the actual possibility of changing the risk perspectives over time. Our framework fits well the study of both short and long term investments. In the dynamic framework we analyse whether the risk indifference criterion actually provides a proper convex price system. 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 measure so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. As it turns out, no-good-deal bounds also provide a method to select the risk measures that provide a proper fully-dynamic risk-indifference price system. The presentation is based on joint works with Jocelyne Bion-Nadal.

• Di Nunno, Giulia (2017). Dynamic risk indifference pricing. Show summary

  We deal with dynamic pricing of financial products in an incomplete market and we focus on risk indifference pricing. This can be seen as an alternative to the classical utility indifference pricing in which the performance is written in terms of evaluation of risks instead of utility. We propose a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. This is based on the concept of fully-dynamic risk measures which extends the one of dynamic risk measures by adding the actual possibility of changing the perspective on how to measure risk over time. Our framework fits well the study of both short and long term investments. Risk indifference pricing has been studied in the terms of how to find a solution and express the actual price under different assumptions on the underlying dynamics and information flows. Typically the price is evaluated at time of the initial investment t=0. We are interested in studying the pricing criterion in its time evolution in a setting free from specific choices of underlying dynamics. In the dynamic framework we analyse whether the risk indifference criterion actually provides a proper convex price system. 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 measure so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. As it turns out, no-good-deal bounds also provide a method to select the risk measures that provide a proper fully-dynamic risk-indifference price system. The presentation is based on joint works with Jocelyne Bion-Nadal.

• Di Nunno, Giulia (2017). Fully-dynamic risk indifference pricing. Show summary

  We deal with dynamic pricing of financial products in an incomplete market and we focus on risk-indifference pricing. This can be seen as an alternative to the classical utility-indifference pricing in which the performance is written in terms of evaluation of risks instead of utility. We propose a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. This is based on the concept of fully-dynamic risk measures which extends the one of dynamic risk measures by adding the actual possibility of changing the perspective on how to measure risk over time. Our framework fits well the study of both short and long term investments. Risk-indifference pricing has been studied from the point of view of how to find a solution under different assumptions on the underlying dynamics and information flows. Typically the price is also evaluated at the time of the initial investment t=0. We are interested in studying the risk-indifference pricing criterion in its time evolution in a setting free from specific choices of underlying dynamics. In the dynamic framework we analyse whether the risk-indifference criterion actually provides a proper convex price system. 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 measure so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. As it turns out, no-good-deal bounds also provide a method to select the risk measures that provide a proper fully-dynamic risk-indifference price system. Based on joint work with Jocelyne Bion-Nadal

• Di Nunno, Giulia (2017). Fully-dynamic risk-indifference pricing with no-good-deal bounds. Show summary

  We deal with the pricing of claims in an incomplete market and we focus on risk-indifference pricing techniques. We propose a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. This is based on the concept of fully-dynamic risk measures which extends the one of dynamic risk measures by adding the actual possibility of changing the risk perspectives over time. In this talk we shall explain what risk-indifference pricing is, convex risk measures and how these are going to be used. We shall use techniques of functional analysis mixed with probability theory to study these price operators and see if they fulfill the criteria to be proper convex prices. We restrict the presentation to the L2-framework where we can exploit the dynamic no-good- deal bounds to obtain a complete characterization of these dynamic prices. We shall recall that no-good-deal prices are those prices that guarantee that in the market there are no arbitrage opportunities and there are ”no deals that are too good to be true”. The presentation is based on joint works with Jocelyne Bion-Nadal

• Di Nunno, Giulia (2017). Introduction to Levy Processes and Applications to Finance.
• Di Nunno, Giulia (2017). Malliavin Calculus for Lévy processes and Time-Change.
• Di Nunno, Giulia (2017). Mathematics, Modelling, Time and Chaos.
• Di Nunno, Giulia (2017). On the integration with respect to Volterra processes: fractional calculus and approximation. Show summary

  In the first part we discuss a pathwise integration for Volterra processes driven by L\'evy noise or martingale noise based on fractional calculus. We obtain an integral with respect to a non-semimartingale process. Different from other recent approaches, this integration provides a framework for modelling where it is easy to keep track of the information flow. Then it can be attractive for applications, for example in finance and energy finance, where information is often linked to portfolio management and control. Then we study an approximation of the integrals introduced above based on the perturbation of the noise, which provides a semimartingale approximation of the integral. This turns to be interesting for simulations. This is based on joint work with Yuliya Mishura, Kostia Ralchenko, Erik H. Karlsen

• Di Nunno, Giulia & Isaksen, Karoline Kvellestad (2017, 08. mars). Det er vanskeligere for kvinner å komme seg opp og fram i matematikken. Intervju til: Professor Berit Stensønes (CAS gruppeleder 2016/17, NTNU), Professor Giulia Di Nunno (CAS gruppeleder 2014/15, UiO), Professor Knut Liestøl (Styreleder for Forskningsrådet BALANSE project, UiO), Professor Geir Ellingsrud (CAS Styreleder, UiO).  Forskning.no.
• Di Nunno, Giulia (2016). A Malliavin-Skorohod calculus in L^0 and L^1 for additive and Volterra-type processes.
• Di Nunno, Giulia (2016). Risk indifference pricing and dynamic no-good-deal bounds.
• Di Nunno, Giulia (2016). Sensitivity analysis in a market with memory. A join work with D.R.Banos, H.Haferkorn, f. Proske.
• Di Nunno, Giulia (2016). Sensitivity analysis in a market with memory. Work in collaboration with D.R. Banos, H. Haferkorn, F. Proske..
• Di Nunno, Giulia (2016). Series of Lectures on Levy Processes and Applications to Finance.
• Di Nunno, Giulia (2016). Stochastic systems with memory and jumps.
• Di Nunno, Giulia (2015). Dynamic no good deal bounds: linear and convex price systems.
• Di Nunno, Giulia (2015). Dynamic no good deal bounds: linear and convex price systems.
• Di Nunno, Giulia (2015). Intensive course: Malliavin Calculus for Levy Processes.
• Aarønæs, Lars; Benth, Fred Espen & Di Nunno, Giulia (2014, 01. oktober). Hvordan beregner vi framtida?. [Tidsskrift].  GLIMT - CAS Informasjonsblad.
• Di Nunno, Giulia (2014). A continuous auction model with insiders.
• Di Nunno, Giulia (2014). A continuous auction model with insiders and information release.
• Di Nunno, Giulia (2014). Optimal portfolio problems with price dynamics driven by time-changed Levy noises.
• Di Nunno, Giulia (2014). Optimal portfolios in markets driven by time-changed Levy noises.
• Di Nunno, Giulia (2014). Time-changed Levy processes and hedging formulae.
• Di Nunno, Giulia (2013). BSDEs driven by time-changed Levy noises and optimal control. Based on joint work with Steffen Sjursen.
• Di Nunno, Giulia (2013). BSDEs driven by time-changed Levy noises and optimal control.Based on joint work with Steffen Sjursen.
• Di Nunno, Giulia (2013). Backward stochastic differential equations with applications to dynamic risk measures. Series of 5 lectures.
• Di Nunno, Giulia (2013). Introduction to stochastic calculus and stochastic differential equations. Series of 8 lectures.
• Di Nunno, Giulia (2013). Market with memory: pricing and sensitivity analysis. Based on joint work with Frank Proske and David Banos.
• Di Nunno, Giulia (2013). Quadratic Hedging via Backward Stochastic Differential Equations with Jumps. Based on joint work with Asma Khedher and Michele Vanmaele.
• Di Nunno, Giulia (2013). Robustness of BSDEs and applications to quadratic hedging. Based on joint work with Asma Khedher and Michele Vanmaele.
• Di Nunno, Giulia (2013). Robustness of Quadratic Hedging Strategies in Finance via Backward Stochastic Differential Equations with Jumps. Based on joint work with Asma Khedher and Michele Vanmaele.
• Di Nunno, Giulia (2013). Robustness of quadratic hedging strategies to model risk. Based on joint work with Asma Khedher and Michele Vanmaele.
• Di Nunno, Giulia; Khedher, Asma & Vanmaele, Michèle (2013). Robustness of quadratic hedging strategies in finance via backward stochastic differential equations with jumps. Preprint series: Pure mathematics. 9. Full text in Research Archive.
• Sjursen, Steffen A. Søreide & Di Nunno, Giulia (2013). BSDES DRIVEN BY TIME-CHANGED LEVY NOISES AND OPTIMAL CONTROL. Preprint series (Universitetet i Oslo. Matematisk institutt). 1. Full text in Research Archive.
• Benth, Fred Espen; Di Nunno, Giulia; Khedher, Asma & Schmeck, Maren Diane (2012). Pricing of spread options on a bivariate jump market and stability to model risk. Preprint series (Universitetet i Oslo. Matematisk institutt). 2. Full text in Research Archive.
• Benth, Fred Espen; Di Nunno, Giulia; Khedher, Asma & Schmeck, Maren Diane (2012). Spread options and stability to model risk.
• Di Nunno, Giulia (2012). Aspects of Malliavin Calculus.
• Di Nunno, Giulia (2012). Sensitivity analysis and computation of the Greeks.
• Di Nunno, Giulia & Bion-Nadal, Jocelyne (2012). Dynamic no good deal bounds and pricing measures.
• Di Nunno, Giulia & Bion-Nadal, Jocelyne (2012). Dynamic no good deal pricing measures.
• Di Nunno, Giulia; L'Aurora, Edoardo; Moschetta, Marina; Proske, Frank Norbert & Ruiz-Banos, David (2012). Market with memory and sensitivity to the past.
• Di Nunno, Giulia; L'Aurora, Edoardo; Moschetta, Marina; Proske, Frank Norbert & Ruiz-Banos, David (2012). Market with memory: pricing and sensitivity analysis.
• Di Nunno, Giulia & Sjursen, Steffen A. Søreide (2012). Doubly stochastic Poisson random fields: from integral representations to BSDEs.
• Di Nunno, Giulia & Sjursen, Steffen A. Søreide (2012). Integral representations and BSDEs driven by doubly stochastic Poisson processes.
• Di Nunno, Giulia & Sjursen, Steffen A. Søreide (2012). On chaos representation and orthogonal polynomials for the doubly stochastic Poisson process. Preprint series (Universitetet i Oslo. Matematisk institutt). 1. Full text in Research Archive.
• Sjursen, Steffen A. Søreide & Di Nunno, Giulia (2012). On chaos representation and orthogonal polynomials for the doubly stochastic Poisson process.
• Di Nunno, Giulia & Bion-Nadal, Jocelyne (2011). Dynamic no-good-deal bounds and no-good-deal pricing measures.
• Di Nunno, Giulia & Bion-Nadal, Jocelyne (2011). Extension theorems for linear operators and dynamic no-good-deal pricing measures.

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