## Visiting address

Niels Henrik Abels husMoltke Moes vei 35 (map)

0851 OSLO

Norway

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UiO
UiO
University of Oslo
Department of Mathematics
The Faculty of Mathematics and Natural Sciences

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Last modified Jan. 10, 2020 4:10 PM by Michele Giordano

**Abstract**

During the winter season, contamination of runway surfaces with snow, ice, or slush causes potential economic and safety threats for the aviation industry. The presence of these materials reduces the available tire–pavement friction needed for retardation and directional control. Therefore, pilots operating on contaminated runways need accurate and timely information on the actual runway surface conditions. In order to validate existing models for assessing runway conditions, or to create models from machine learning, it is necessary to estimate runway friction. We show how this can be done using flight data from airplanes. Data such as longitudinal acceleration, airspeed, ground speed, flap settings, engine speed and brake pressures are sampled at least each second during landings. However, converting these samples into reliable estimates of runway friction is far from trivial, and we discusses some of the challenges with this.

Last modified Jan. 13, 2020 2:59 PM by Michele Giordano

**Abstract**

In this research, we consider a stochastic differential equation with drift having singularity in zero and an additive noise being a fractional Brownian motion with an arbitrary Hurst parameter. Under some assumptions, existence, uniqueness and positivity of the solution are proved, finiteness of the moments of all orders (including the negative ones) is obtained. As examples, we provide fractional generalisations of the CIR and CEV processes and, using the finiteness of their negative moments, prove convergence of the corresponding backward Euler-type schemes.

Separately, we discuss possible extensions of the obtained results including replacing the fractional Brownian motion with general Holder-continuous Volterra process.

Last modified Jan. 13, 2020 9:03 AM by Michele Giordano

**Abstract**

The theory of regularization by noise is concerned with discovering the regularizing effects that stochastic or deterministic noise may have on ordinary and partial differential equations. In this presentation I will give a short review of recent results in this theory, as well as some applications.

In particular, I will begin to show a simple strategy to prove the regularizing effect of differential equations using the concept of averaging operators given as deterministic Riemann integral of a function along an irregular path. A specific example of such averaging operator can be obtained from the local time of a stochastic processes. Using these operators we construct a solution theory for equations driven by a non-linear Young integral. I will then show some applications of these techniques towards the regularization of ill-posed and singular ODEs and PDEs perturbed by noise.

Last modified Jan. 7, 2020 9:20 AM by Michele Giordano

**Abstract**

We investigate the fractional Vasicek model described by the stochastic Langevin equation driven by fractional Brownian motion with known Hurst parameter. We study the maximum likelihood estimators for unknown drift parameters. We derive their asymptotic distributions and prove their asymptotic independence.

Last modified Jan. 9, 2020 11:39 AM by Michele Giordano

**Abstract**

The goal of this paper is to study a stochastic game connected to a system of forward-backward stochastic differential equations (FBSDEs) involving delay and noisy memory. We derive sufficient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in the game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives. This kind of equation has not been studied before. The maximum principles give conditions for determining the Nash equilibrium of the game. We use this to derive a closed form Nash equilibrium for an economic model where the players maximize their consumption with respect to recursive utility.

Last modified Jan. 13, 2020 9:09 AM by Michele Giordano

**Abstract**

In the present paper, a decomposition formula for the call price due to Alòs is transformed into a Taylor type formula containing an infinite series with stochastic terms. The new decomposition may be considered as an alternative to the decomposition of the call price found in a recent paper of Alòs, Gatheral and Rodoi\v{c}i\'{c}. We use the new decomposition to obtain various approximations to the call price in the Heston model with sharper estimates of the error term than in the previously known approximations. One of the formulas obtained in the present paper has five significant terms and an error estimate of the form $O(\nu^{3}(\left|\rho\right|+\nu))$, where $\nu$ and $\rho$ are, respectively, the vol-vol an the correlation in the Heston model. Another approximation formula contains seven more terms and the error estimate is of the form $O(\nu^4 \left(1 + |\rho|\nu\right))$. For the uncorrelated Heston model ($\rho=0$), we obtain a formula with four significant terms and an error estimate $O(\nu^6)$. Numerical experiments show that the new approximations to the call price perform especially well in the high volatility mode.

Last modified Jan. 12, 2020 10:50 AM by Michele Giordano

**Abstract**

We study a stochastic game between two palyers, based on a forward stochastic Volterra integral equation (FSVIE) and two backward stochastic Volterra integral equations (BSVIEs). All those processes are driven by a time changed Lévy noise. We use a nonanticipating derivative to prove both a necessary both a necessary (Pontryagin) and a sufficient (Mangasarian) maximum principle for an optimal control problem. We propose a formulation both for the zero sum and the non zero sum games and give some examples in both cases.

Last modified Jan. 13, 2020 9:06 AM by Michele Giordano

**Abstract**

Predicting the amount of wind power generation with high accuracy represents a priority for any market participant or power balancing authority. One of the main ingredients when deriving these predictions are the wind speed data. The current research in progress aims at refining our understanding of wind speed measurements and developing a method to find those which are most relevant to the wind power generation. In practice, we perform this study on the wind farms in Southern California market (also known as CAISO SP15). For this region, we build random fields by time series of gridded historical forecasts and actual wind speed measurements provided by the aviation reports (METAR). As none of these values are taken from where the wind farms are, we use kriging techniques in order to direct our field in space and time to the exact farm locations. In this process, an important challenge is brought by the fact that the actual measurements are instantaneous values in time and they can very often be zero-valued. The main benefits of this work is that (1) it will highlight when the wind speeds will not allow the mills to produce electricity and (2) it will provide a way to forecast wind speeds at wind farm locations.

Last modified Jan. 12, 2020 4:10 PM by Michele Giordano

**Abstract**

We price European type options having forward contracts in the commodity market as underlying, which we model by the HJM approach with stochastic volatility operator. As stochastic volatility models are cost demanding in terms of Monte Carlo simulations, in a first step we train a neural network as function of the parameters of the HJM model plus the features of the option, such as strike and time to maturity, in order to approximate the option price functional. Once trained the neural network with data obtained by cost demanding simulations of an SPDE, we get a deterministic function capable to price options. In a second step, we use this function together with options market data to calibrate the HJM model and get the forward curve parameters. Since only the second step is using market data, and it is based on a deterministic function that is the neural network, no simulations are required and the approach allows to fast parameters calibration. Indeed, the first step being off-line, this does not need to be trained frequently, while the second step is less cost demanding and can be run, e.g., daily, in order the HJM model to reflect the current market data.

Last modified Jan. 6, 2020 2:25 PM by Michele Giordano

**Abstract**

We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space $D[0,T]$. Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann--Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process. Joint work with Kostiantyn Ralchenko and Sergiy Shklyar.

P.O box 1053

Blindern

0316
OSLO

Norway

+47 22 85 58 88

ekspedisjonen@math.uio.no