Novel approaches to the energy load unbalance forecasting in the Italian electricity market
 Luca Di Persio^{1}Email authorView ORCID ID profile,
 Alessandro Cecchin^{2} and
 Francesco Cordoni^{3}
https://doi.org/10.1186/s133620170035y
© The Author(s) 2017
Received: 2 August 2016
Accepted: 9 February 2017
Published: 21 February 2017
Abstract
In the present paper we study the statistical properties of the Italian daily electricity load market, by mean of different statistical methods, such, e.g., the exponential smoothing model, the ARMAARIMA model and the ARIMAGARCH model, also providing results about the goodness of each of the proposed approaches. Moreover, we show how the aforementioned models behave if exogenous regressors, as the day of the week or the temperature, are additionally taken into account. Analysed methods are then exploited to perform the oneday ahead energy load prediction, where the main focus is on guessing the right sign of the energy load unbalance.
Keywords
energy markets statistical forecasting spot prices random dynamics time series analysis1 Introduction
Energy Markets are financial markets that deal with the demand and supply of energy as well as to what concern financial derivatives structured on them. One of the most relevant characteristics of energy market frameworks relies on the fact that energy in general cannot be stored efficiently. Latter fact is just one of the main reasons why such particular type markets turn to be highly complicated both from theoretical, empirical and numerical point of view, see, e.g., [1] and references therein. During last years an increasing attention has been devoted particularly with the aim to obtain robust forecast methods capable to provide effective estimates about the production and consumption of, e.g., oil, gas, electricity and performances of the methods related to their productions, e.g., solar cells, oil plants, wind turbines, etc. Latter goals have to deal with the need to take into account both natural and social variables, such, e.g., weather conditions, firm needs, urban energy consumption, energy transportation, possible ways to storage electricity, as in the case of batteries, etc. Last but not least, almost every country has its particular type of energy market, with its own regulations, controlling bodies, etc. The sum of the aforementioned factors results in a financial framework which is also strongly characterized by high volatility levels, whose behaviour can be effectively studied exploiting the approach used in, e.g., [2], and references therein, or using more sophisticated approximation techniques, as those highlighted in [3], and references therein.
The latter implies that a general theory to treat energy markets, even if we restrict ourselves just to particular forecasting settings, is not a realistic goal and ad hoc studies have to be done in order to obtain effective results.
The present work aims at addressing a first fundamental task in energy trading in the Italian energy market, that is the forecast of energy load. Latter challenge is of crucial relevance particularly from the financial point of view by the side of agencies that produce and sell energy on the market under the supervision of the Italian Power Exchange (IPEX), managed by the Gestore del Mercato Elettrico (GME in Italian), which is the exchange for electricity and natural gas spot trading in Italy. In particular IPEX comprises two spot markets, namely the Day Ahead (in Italian: Mercato del Giorno Prima, or MGP) and the Intra Day (in Italian: Mercato Infragiornaliero, or MI). The latter, which since the 10th of February, 2015, is divided into five components, or sessions, which are called MI1, MI2, MI3, MI4, MI5.
Because there are not any efficient and economical sustainable ways to store electricity, power systems need to be constantly balanced between production and consumption, see, e.g., [4]. The latter implies that an accurate next day imbalance forecast has to be derived if one wants to obtain profit as well as to avoid losses caused by wrong imbalance in sign, whence the need to have concrete methods able to predict next days such a sign.
To the best of our knowledge, just few results have been already developed concerning last issue, moreover most of them, if not all, do not address the Italian market, but rather USA markets, as in the case of activities related to the California Energy Commission, or the Germany market, particularly with respect to the recently launched plan called Energiewende or energy turnaround.
We therefore intend to address this crucial topic in a series of paper, where we aim at giving a characterization as extensive as possible of the problem of energy load forecasting within the Italian energy market framework.
The present paper is so structured as follows: in Section 2 we give an overview of the Italian energy markets functioning and of its main peculiarities, addressing also the main problem that motivates our study; in Section 3 we provide a quick overview of the theoretical foundation of the method we later exploit in our analysis; eventually, in Section 4 we study the aforementioned problem with respect to a concrete real case.
2 The Italian energy markets functioning
The Italian power exchange market (IPEX) (Mercato elettrico italiano) is a free system that allows producers and consumers to enters into hourly contracts for buying or selling electricity. Such a market is divided into two main markets: future market and spot market, which are themselves divided into different sessions. We would like to underline that such a market implies a particular treatment of its financial basis since its nature is rather different from, e.g., the one characterising the usual derivatives/options/assets scenarios, see, e.g., [5, 6], and references therein.
In the future markets (FM) participants buy and sell bilateral contracts for delivery of energy on a specified future date. In such a contracts the parties are obliged to sell and buy the agreed amount of energy.
In the spot market (SM) any market operator has as a counterpart the transport system operator (TSO) (Gestore del Mercato Elettrico). Spot market is divided into the day ahead market (mercato del giorno prima) (MGP), the intra day market (mercato infragiornaliero) (MI), which since February 2015 is composed of five sessions (MI1, MI2, MI3, MI4 and MI5), and the Mercato per il Servizio di Dispacciamento (MSD).
The day ahead market
In the MGP negotiations for energy trading for any hour of the next day take place. This market is based upon an implicit auction mechanism, each player submits his bids composed by a quantity and a price representing the maximum price at which he is willing to buy energy or the minimum price at which he is willing to sell energy.
At closure of each session, one for every hour of the day, all the offers are processed and either accepted or refused according to the System Marginal Price, that is, bids, resp. asks, are ordered from the lowest to the highest price, resp. from the highest to the lowest price, the equilibrium price and total exchanged energy are determined by the intersection of the two curves.
The intra day market
The MI is composed of five sessions, namely MI1, MI2, MI3, MI4 and MI5. MI1 and MI2 take place the day before the actual delivery, whereas MI3, MI4 and MI5 occur the same day of the actual delivery. In each session every operator can modify his program of injection or withdrawal of energy. Also MI sessions follows the same exact rules of price formation of MGP.
The dispatching services market
Dispatching guarantees the overall equilibrium between production and loading and thus ensures the correct functioning of the national electric grid. In Italy the dispatching system is managed by Terna S.p.A., which is the owner of the high voltage national transmission network. Terna, in order to guarantee the proper functioning of the electrical network, has to deal with the congestion resolution activities between the different market areas, the creation of the reserve of energy and the realtime balance between production and consumption.
In MSD Terna obtains the necessary reserves to the dispatching service by acting as central counterparty in negotiations with operators enabled to the dispatching service. In this market all accepted bids are remunerated at the price presented, according to the paid as bid method. The offers of purchase in MSD are also called downward, meaning that such offers will be accepted if it is necessary to reduce the amount of energy generated, while offers to sell are called upward.
2.1 The problem of the unbalance forecasting
As already mentioned, the continuous balance between production and consumption of energy is a fundamental task in order to guarantee the correct functioning of the whole electrical network. This balance is guaranteed by Terna through MSD. In order to create the necessary energy reserves to balance the grid, Terna needs to know as precisely as possible the production of different plants.
While for traditional sources plants, such as coal, gas and other fossil fuels, is a relatively simple task to predict the next day production, for nonprogrammable renewable sources plants, such as wind and solar, this forecast is very difficult task, with many factors affecting the final outcome.
In order to ease the work of Terna to balance the network, all the producers from traditional sources and programmable renewable sources, such as for instance some types of hydroelectric energy, are required to provide to Terna the exact production plan for the next day; in the event that these programs are not observed, then the actor has to pay a penalty. Producers of nonprogrammable renewable sources that do not meet the scheduled production, however, incur in penalties or rewards depending on the relative sign between their unbalance and the unbalance of the macrozone in which the plant is located.
In Italy there are two different macrozones for balancing purposes: the northern macrozone consists of all the regions of Northern Italy, including Emilia Romagna, whereas the southern macroone consists of all other regions. The aggregate zonal unbalance is the algebraic sum, changed in sign, of the amount of energy procured by Terna in MSD at a given time in a given macrozone. When the aggregate unbalance is positive means that the energy produced is greater than the energy scheduled and then most of the offers accepted in MSD were downward; when the aggregate unbalance is negative the opposite happens.
Imbalance price settlement mechanism
Positive actor unbalance: actor receives  Negative actor unbalance: actor pays  

Positive network unbalance  \(\min\{ P_{\mathrm{MGP}}, P_{\mathrm{MSD}}^{\downarrow}\}\)  \(\min\{P_{\mathrm{MGP}}, P_{\mathrm{MSD}}^{\downarrow}\}\) 
Negative network unbalance  \(\max\{P_{\mathrm{MGP}}, P_{\mathrm{MSD}}^{\uparrow} \}\)  \(\max\{P_{\mathrm{MGP}}, P_{\mathrm{MSD}}^{\uparrow}\}\) 
3 Statistical methods
 The exponential smoothing model :

Let us first consider the exponential smoothing (ES), see, e.g., [7–9], which is mainly based on predictive procedures built starting from an exponentially weighted average of past observations. The general model involves a state vector \(y_{t} = (l_{t},b_{t},s_{t},\dots,s_{tm+1})\), where \(l_{t}\) represents the level of the series, \(b_{t}\) represents the growth and \(s_{t}\) is the seasonal component, coupled with a state space equation of the form\(\epsilon_{t}\) being a centred Gaussian noise with finite variation \(\sigma^{2}_{t}\), and for some suitable functions, to be choose according to the model one wishes to fit, w, r, f and g, see, e.g., [10] or also [8]. We remark that in what follows we will not choose a particular ES model to be fitted to data, but rather we run a routine in order to choose the best performing ES model.$$\textstyle\begin{cases} X_{t} = w(y_{t1}) + r(X_{t1}) \epsilon_{t}, \\ y_{t} = f(y_{t1}) + g(X_{t1}) \epsilon_{t}; \end{cases} $$
 The ARMAARIMA model :

When one concerns the study of time series, the AutoRegressive Moving Average (ARMA) models play a central role because they are capable of describe weakly stationary stochastic processes with a rather restricted set of assumptions, being mainly based on the use of two polynomials: the first one takes into account the autoregressive character of the data set, while the second takes into account the moving average. It is worth to mention that such a method results as a combination of the Moving Average method (MA) together with an AutoRegressive (AR) one. In particular, denoting by \(X_{t}\) the unknown value of the series of interest at time t, which is in fact treated as random variable, a pth order Auto Regressive method (\(\operatorname{AR}(p)\)) is defined as followswhere \(\epsilon_{t}\) is a general random noise, while the coefficients \(\phi_{1},\dots,\phi_{p}\) are the AR (or regression) coefficients. In the most simple case the noise \(\epsilon_{t}\) is assumed to be Gaussian, however more general type of random disturbance can be considered.$$X_{t}  \sum_{k=1}^{p} \phi_{k} X_{tk} = \epsilon_{t} , $$Concerning the moving average component of the ARMA model, it is defined by the Moving Average method of order q, which will be indicated by \(\operatorname{MA}(q)\), and it is defined as follows\(\epsilon_{t}\) still being the noise, or uncertainty, affecting our observations, or elements of the time series we are studying, which is not necessarily of Gaussian type. As before X is the process that we would like to forecast, on the basis of previous observations, while \(\theta_{1},\dots,\theta_{q}\) are the moving average parameters.$$X_{t} = \epsilon_{t} + \sum_{k=1}^{q} \theta_{k} \epsilon_{tk} , $$Merging the methods already introduced, it is possible to define the AutoRegressive Moving Average method of order \((p,q)\), indicated by \(\operatorname{ARMA}(p,q)\), and defined as follows$$ X_{t}  \sum_{k=1}^{p} \phi_{k} X_{tk} = \epsilon_{t} + \sum _{k=1}^{q} \theta_{k} \epsilon_{tk} . $$(1)A further step is represented by the AutoRegressive Integrated Moving Average (ARIMA) method, which allows to take into account time series which are not of stationary type. In particular an ARIMA model of order \((p,d,q)\), indicated by \((\operatorname{ARIMA}(p,d,q))\) where d is the degree of differencing, namely it represents how much the time series we are dealing with is far from being stationary, is defined as followswhere \(\nabla X_{t} = (1B)X_{t}\) is the lag 1 differencing operator, and B is the backward shift operator defined by \(B^{n} X_{t} := X_{tn}\), considering the use of the following short notations$$\phi(B) \nabla^{d}X_{t} = \theta(B) \epsilon_{t} , $$We refer the interested reader to, e.g., [8], Section 3.4, for further details.$$\begin{aligned} &\phi(B) = 1 \phi_{1} B^{1}  \cdots \phi_{p} B^{p} , \\ &\theta(B) = 1 + \theta_{1} B^{1} + \cdots+ \theta_{q} B^{q} . \end{aligned} $$
 The ARMAGARCH model :

The ARMAGARCH model models the mean equation via an \(\operatorname{ARMA}(p,q)\) model, see Eq. (1), whereas the random noise components, represented by \(\epsilon_{t}\), are modelled with a Generalized Autoregressive Conditional Heteroskedastic (GARCH), model. In particular the GARCH model of order \((p,q)\), indicated by \((\operatorname{GARCH}(p,q))\), defines the residual \(\epsilon_{t}\) appearing in equation (1) as followsfor some positive coefficients \(\alpha_{k}, \beta_{k} \geq0\). The latter approach allows to overcome one of the main issue affecting the ARMA process, namely the fact that the mean equation cannot take into account for heteroskedastic effects of the time series process, as, e.g., happens for the so called fat tails.$$\begin{aligned} &h_{t} = \epsilon_{t} \sigma_{t} , \\ &\sigma^{2}_{t} = \alpha_{0} + \sum _{k=1}^{q} \alpha_{k} h_{tk}^{2} + \sum_{k=1}^{p} \beta_{k}^{2} \sigma^{2}_{tk} , \end{aligned} $$
 ARMAGARCH method with exogenous variables :

The ARMAGARCH models can be enriched by considering also the role played by the so called exogenous variables. In particular exogenous variables can be added to the ARMA component, as well as to the GARCH model one. However, for the sake of brevity, here we only consider the case of exogenous variables added to the ARMA process, the case related to the GARCH one being analogous.
We thus define the AutoRegressive Moving Average method with exogenous variables (\(\operatorname{ARMAX}(p,q,r_{1},\dots,r_{n})\)), as followswhere \(r_{i}\)’s are the orders of the exogenous variables \(Y^{1},\dots ,Y^{n}\) and$$\phi(B)X_{t} = \theta(B)\epsilon_{t} + \sum _{k=1}^{n} \psi^{k}(B) Y^{k}_{t} , $$and we refer to [8], Section 3.4.9, for a detailed introduction to modelling with exogenous variables.$$\psi^{k}(B) = \psi^{k}_{0} + \psi^{k}_{1} B + \cdots+ \psi^{k}_{r_{i}} B^{r_{i}} , $$
4 Estimating the model
In what follows we will apply the methods introduced in Section 3 to analyse the net hourly energy load imbalance in the Italian northern macrozone, where the positive, resp. negative, sign is to be intended as explained in Section 1. We recall that the Italian northern macrozone is composed by all regions from northern Italy, including Emilia Romagna.
First we study the hourly time series of energy load, applying time series method recalled in Section 3 to outline which of them better perform according to a concrete criterion that will be specified later on. Then, we will consider the task of forecasting the next day imbalance sign. We would like to underline that we are mainly interested not in the prediction of the exact amount of the zonal imbalance, but rather in predicting the right sign of imbalance, since the latter is the main factor affecting the energy trading in Italy, nowadays. The latter characteristic is due to the imbalance mechanics on which the MSD is based, see Table 1 and Section 2.1, for details. Last but not least, the forecast of the next day imbalance sign for the Italian energy market constitutes the main novelty of the present work.
Therefore, motivated by previous facts, we perform our study on the time series of the effective unbalance at a given hour. In particular, in what follows we focus our attention on one of the most challenging hour of the day, namely we consider the time series at 2 p.m. Nevertheless, it is worth to mention that different hours, albeit being characterized by their specific peculiarities, can be analogously treated exploiting our approach. It is worth to mention that our analysis will be performed with respect to different time windows, showing how changing the number of days taken into consideration may slightly affect the overall fit of the model. We stress that the problem of choosing the right time window is really a hard task in energy markets, mainly because it is affected by a large number of seasonal components. The latter implies that, going too far in the past, may only lead to an increase of the overall instability of the model. Furthermore, since the national regulation of the Italian energy market has been changed several times during last years, one has to take past values with particular care on the chosen period in order to avoid to treat non homogeneous numbers resulting as the output of different regulatory settings.
Once we have calibrated all the aforementioned models, we exploit the associated AIC values to chose the one that has better performed. Eventually, the same procedure is repeated over different time windows. It is worth to mention, as briefly said above, that choosing the right time windows to be used to perform the statistical analysis turns to be a rather difficult task. In the framework of energy market, latter problem is even more complicated, playing, at the same time, a more relevant role. In fact, energy related time series often exhibit different seasonal components, being the energy load correlated to the day of the week as much as the season on is considering. Besides, it can happen that some exogenous variables may affect the data only in a given season, being irrelevant during the others. Concerning the latter issues, we refer the interested reader to [8], Section 3.2.4.
We list the resulting AIC values for all the exploited models, with respect to particularly relevant time windows.
Resulting AIC values for considered models over different time windows
ES  ARIMA  ARIMAX  ARMAGARCH  ARMAXGARCH 

2,229.917  1,992.997  1,985.405  16.417  16.405 
2,239.055  2,002.135  1,998.176  16.535  16.504 
2,237.519  2,000.600  1,998.778  16.525  16.375 
2,238.399  2,001.083  1,999.042  16.537  16.445 
2,238.138  2,001.106  1,999.415  16.556  16.516 
AIC value for each statistical model at different time of the day
Hour  ES  ARIMA  ARMAGARCH  ARMAXGARCH 

h:3:00  2,080.565  1,836.58  15.418  15.405 
h:10:00  2,171.46  1,914.24  16.147  16.417 
h:18:00  2,219.794  1,955.092  16.486  16.462 
h:22:00  2,466.305  1,917.174  15.833  15.837 
4.1 On the forecast for the next day unbalance
As mentioned at the beginning of the current Section, we are mainly interested in the forecast of the next day energy load, with particular attention to the overall sign of the imbalance in the macrozone, rather than to what concerns the exact quantity of imbalanced energy. Moreover, our main goal is to obtain accurate short time previsions of the right sign of next days imbalance. The latter is due to the specific mechanism behind the trading strategies explained in Section 2.1, see in particular Table 1. For a general treatment of forecasting within different energy markets and/or with respect to different scopes, we refer to, e.g., [2, 11–13], while we refer to, e.g., [11, 13] for the study of the longtime horizon forecast and to [14], and references therein, for the treatment of related computational issues.
In particular, our forecasting procedure is structured as follows: we chose an appropriate positive threshold and, if the forecasted value is in absolute value higher than the threshold, then we enter the market according to the predicted unbalance sign, otherwise, namely, as it is most likely to happen, the outcome is too close to zero implying a high probability to unbalance in the wrong direction, we do not enter the market. We underline that the aforementioned threshold can be chosen according to different parameters, for instance we considered the estimated volatility of the time series.
Different methods: overall performance
Right  Wrong  Not played  Overall played  Overall right  

ES  106  57  246  39.8%  65% 
ARIMA  28  25  356  13%  52.8% 
ARMAGARCH  63  44  302  26%  58.8% 
ARMAXGARCH  95  44  270  34%  68.3% 
Table 4 shows on one side the expected results, while on the other it highlights some peculiar features. In fact, as expected from results reported in Section 4, the ARIMA model is poorly performing, being indeed the worst when dealing with the daily time series as well as considering the overall performance. Viceversa, again as we expect, the ARMAGARCH model performs rather well, with a slight improvement when one also considers the exogenous variables. What turns to be an unexpected result concerns the ES model performance, since, according to the AIC criterion, see Table 2, it is by far the worst performing one. Nevertheless, if one addresses the problem of forecasting it can be seen that the ES model outperforms the ARMAGARCH model without exogenous variables, being also, even if by a few percentage, better than the ARMAGARCH with exogenous variables. In particular, it appears that the ES model and the ARMAXGARCH model perform similarly in predicting the next day sign, being the main difference among the two represented by the number of times one enters the market which is greater for the ES, most probably because it overestimates the next day outcome, leading to a higher number of plays.
5 Conclusion and further developments
The present work constitutes a first step towards the solution of the highly difficult task of next day energy imbalance forecast. Such an ambitious goal is affected by several issues due to many different reasons, such, e.g., the difficulty to find statistically good data, which means that time series are rather often dirtied by regulatory changes, exogenous noises, sensors faults, etc. Besides, a rather important issue, which is typically not considered, will play a fundamental role in the next future, namely the one concerning the problem of optimal allocation/transportation of energy resources/products, a particularly difficult task that is intrinsically linked to the solution of stochastic optimal problems stated on networks, and whose solution has been the subject of an increasing number of researches during recent years, not only within the energy market framework, see, e.g., [3, 15–17] and references therein.
Moreover, different factors can affect energy loads, and we have considered just the day of the week and temperature. Even if the latter appears, from standard literature, to be the most relevant, nevertheless recent studies have shown that also different factors could play a crucial role, as for the case of renewable energies. In particular, considering renewable energies lead to at least two nontrivial problems: exact values for such an exogenous parameter are not always available, moreover the exact value of its next day production is rather tricky to be predicted with enough accuracy. Previous reasons suggest a very careful and detailed study, that goes beyond the aims of the present work, but whose results will be of great relevance since the production of renewable energy plays a fundamental role in the zonal unbalance, that is why we will address this key task in a future work. The second being how to chose the most relevant renewable energy with respect to its impact on energy loads, since the effect of different renewable energies may vary a lot from region to region.
We outline that the latter point cannot be neglected in order to develop a solid method to predict future energy loads, indeed such a subject will be the main focus of our future works. Last but not least, we would like to underline that the most of the computational part which has been developed so far with respect to the type of problems we have analysed in the present paper, is mainly based on Monte Carlo type techniques. Such a type of numerical approach is particularly ineffective for our purposes, because of its slow rate of convergence and poor accuracy, at least compared to more sophisticated methods as the ones based on the Polynomial Chaos Expansion approach, see, e.g., [14] and references therein.
Declarations
Acknowledgements
The authors gratefully acknowledge BeFree s.r.l. for financial support and for providing data.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Di Persio L, Perin I. An ambit stochastic approach to pricing electricity forward contracts: the case of the German energy market. J Probab Stat. 2015;2015:626020. MathSciNetGoogle Scholar
 Di Persio L, Frigo M. Gibbs sampling approach to regime switching analysis of financial time series. J Comput Appl Math. 2016;300:4355. MathSciNetView ArticleMATHGoogle Scholar
 Marinelli C, Di Persio L, Ziglio G. Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps. J Funct Anal. 2013;264(12):2784816. MathSciNetView ArticleMATHGoogle Scholar
 Aïd R. Electricity derivatives. Berlin: Springer; 2015. View ArticleMATHGoogle Scholar
 Cordoni F, Di Persio L. Backward stochastic differential equations approach to hedging, option pricing, and insurance problems. Int J Stoch Anal. 2014;2014:152389. MathSciNetMATHGoogle Scholar
 Cordoni F, Di Persio L. Invariant measure for the Vasicek interest rate model in the HeathJarrowMortonMusiela framework. Infin Dimens Anal Quantum Probab Relat Top. 2015;18(3):1550022. MathSciNetView ArticleMATHGoogle Scholar
 Mills TC. Time series techniques for economists. Cambridge: Cambridge University Press; 1991. MATHGoogle Scholar
 Weron R. Modeling and forecasting electricity loads and prices: a statistical approach. New York: Wiley; 2007. Google Scholar
 Gardner E. Exponential smoothing: the state of the art. J Forecast. 1985;4:128. View ArticleGoogle Scholar
 Hyndman RJ, Akram M, Archibald BC. The admissible parameter space for exponential smoothing models. Ann Inst Stat Math. 2008;60(2):40726. MathSciNetView ArticleMATHGoogle Scholar
 Feinberg E, Genethliou D. Load forecasting. In: Applied mathematics for restructured electric power systems. New York: Springer; 2005. p. 26985. View ArticleGoogle Scholar
 Ghelardoni L, Ghio A, Anguita D. Energy load forecasting using empirical mode decomposition and support vector regression. IEEE Trans Smart Grid. 2013;4(1):54956. View ArticleGoogle Scholar
 Liu K, et al. Comparison of very shortterm load forecasting techniques. IEEE Trans Power Syst. 1996;11(2):87782. View ArticleGoogle Scholar
 Bonollo M, Di Persio L, Pellegrini G. Polynomial chaos expansion approach to interest rate models. J Probab Stat. 2015;2015:369053. MathSciNetGoogle Scholar
 Barbu V, Cordoni F, Di Persio L. Optimal control of stochastic FitzHughNagumo equation. Int J Control. 2016;89(4):74656. MathSciNetView ArticleMATHGoogle Scholar
 Benazzoli C, Di Persio L. Default contagion in financial networks. Int J Math Comput Simul. 2016;10:1127. Google Scholar
 Di Persio L, Ziglio G. Gaussian estimates on networks with applications to optimal control. Netw Heterog Media. 2011;6(2):27996. MathSciNetView ArticleMATHGoogle Scholar