## Papers in international peer-reviewed journals

### F. Ortu, F. Severino, A. Tamoni, C. Tebaldi (accepted) **A persistence-based Wold-type decomposition for stationary time series**. *Quantitative Economics.*

Abstract. This paper shows how to decompose weakly stationary time series into the sum, across time scales, of uncorrelated components associated with different degrees of persistence. In particular, we provide an Extended Wold Decomposition based on an isometric scaling operator that makes averages of process innovations. Thanks to the uncorrelatedness of components, our representation of a time series naturally induces a persistence-based variance decomposition of any weakly stationary process. We provide two applications to show how the tools developed in this paper can shed new light on the determinants of the variability of economic and financial time series.

### M. Marinacci, F. Severino (2018) **Weak time-derivatives and no-arbitrage pricing**.* Finance and Stochastics *22(4), 1007-1036. Accepted manuscript.

Abstract. We prove a risk-neutral pricing formula for a large class of semimartingale processes through a novel notion of weak time-differentiability that permits to differentiate adapted processes. In particular, the weak time-derivative isolates drifts of semimartingales and is null for martingales. Weak time-differentiability enables us to characterize no arbitrage prices as solutions of differential equations, where interest rates play a key role. Finally, we reformulate the eigenvalue problem of Hansen and Scheinkman (2009) by employing weak time-derivatives.

### F. Severino (2016)** Isometric operators on Hilbert spaces and Wold decomposition of stationary time series**. *Decisions in Economics and Finance* 39(2), 203-234. A preliminary version.

Abstract. The Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. It provides a moving average representation of the process under consideration in terms of uncorrelated innovations, whatever the nature of the process is. From an empirical point of view, this result enables to identify orthogonal shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem, that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail, employing the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. In particular, we characterize the innovation subspace by exploiting the adjoint operator. We also show that the isometry of the lag operator is equivalent to weak stationarity. Our methodology, fully based on operator theory, provides novel tools useful to discover new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. In such decompositions the orthogonality of innovations is ensured by construction since they are derived from the Abstract Wold Theorem.

## Book chapters

### D. Di Virgilio, F. Ortu, F. Severino, C. Tebaldi (2019) **Optimal asset allocation with heterogeneous persistent shocks and myopic and intertemporal hedging demand**. In book: *Behavioral finance: the coming of age* by *Itzhak Venezia, World Scientific*. A preliminary version.

## Working papers

### F. Severino: **Long-term risk with stochastic interest rates**

Abstract. Investors with heterogeneous trading horizons require compensation for the exposure to different risks. The no-arbitrage valuation over increasing horizons is described by the evolution of stochastic discount factors (SDFs). Each of them exhibits a multiplicative decomposition into deterministic growth term, permanent and transient component, provided by Hansen and Scheinkman (2009). In particular, the growth rate captures the deterministic discounting for risks that are relevant in the long term. When interest rates in the market are constant, the SDF growth rate coincides with the instantaneous rate. On the contrary, when rates of interest are stochastic, the SDF growth rate is given by the long-term yield of zero-coupon bonds, which is unsuitable for instantaneous no-arbitrage valuation.

We show how to reconcile the long-run properties of the SDF with the instantaneous rela- tions between returns and rates in stochastic-rate markets. In particular, we introduce a rate adjustment in pricing that isolates the short-term variability of rates. No-arbitrage prices are then factorized into rate-adjusted prices and a rate adjustment that is absent when interest rates are constant. Rate-adjusted prices employ constant yields to maturity for discounting future payoffs over time. The rate-adjusted SDF features the same long-term growth rate of the SDF in the market but has no transient component in its Hansen-Scheinkman decomposition. Therefore, rate-adjusted prices provide the proper valuation for long-term interest rate risk. Moreover, we show how this novel notion is fruitful for managing the interest rate risk related to fixed-income derivatives, life insurances and annuities.

### S. Cerreia-Vioglio, F. Ortu, F. Severino, C. Tebaldi: **Multivariate Wold decompositions**.

Abstract. The Wold decomposition of a weakly stationary time series extends to the multivariate case by allowing each entry of a weakly stationary vectorial process to linearly depend on the components of a vector of shocks. Since univariate coefficients are replaced by matrices, we propose a modelling approach based on Hilbert A-modules defined over the algebra of squared matrices. The Abstract Wold Theorem for Hilbert A-modules, that we prove, delivers two orthogonal decompositions of vectorial processes: the Multivariate Classical Wold Decomposition, which exploits the lag operator as isometry, and the Multivariate Extended Wold Decomposition, where a scaling operator is employed. The latter enables us to disentangle the heterogeneous levels of persistence of a weakly stationary vectorial process. Hence, the persistent components of the macro-financial variables into consideration are related to the overlapping of different sources of randomness with specific persistence. We finally provide a simple application to V AR models.

## Work in progress

### M. Madotto, F. Severino:** Heterogeneous awareness in financial markets**.

Abstract. The overlook of certain economic scenarios may result in unforeseen negative outcomes for economic agents. We consider a financial market with the structure of Kyle (1985) in which a fraction of investors is partially aware of the potential payoffs of a risky security. The disagreement on the possible future scenarios affects the information acquisition about the traded assets. In particular, partial awareness induces a distortion in the interpretation of signals, triggering a separating equilibrium. In such equilibrium, partially and fully aware investors act in opposite ways in response to intermediate signals. We show that both the presence of unawareness and its severity negatively affects the total number of informed traders. In particular, as unawareness rises, incentives to acquire information are transferred through the price to the fully aware investors. Such negative impact of unawareness on information acquisition increases the liquidity in the market. We then show that the misinterpretation of some signals caused by unawareness gives rise to novel price levels, while an increased severity of unawareness makes higher prices more likely.

### F. Ortu, F. Severino, C. Tebaldi:** Persistence-based Beveridge-Nelson decomposition**.

Abstract. Given an integrated process, the Beveridge-Nelson decomposition allows to isolate a permanent component (or random walk) from a cyclical one. We formalize the nature of the random walk in a functional analytical setting in which this component is a functional reached by taking a weak limit over time. Such functional shares many of the features of the Wiener process in a discrete-time setting. Hence, we provide a characterization of integrated process, based on their asymptotic behaviour in discrete time, as an alternative to the Functional Central Limit Theorem approach. In this framework we show that the convergence to the random walk can be obtained by taking the limit over increasingly wider time scales, by exploiting the persistence-based decomposition of Ortu, Severino, Tamoni and Tebaldi. After providing an Extended Beveridge-Nelson Decomposition, we describe several ways to distinguish weakly stationary from integrated processes by analysing the heterogeneous layers of persistence.