Optimal control of stochastic FitzHugh-Nagumo equation

This paper is concerned with existence and uniqueness of solution for the the optimal control problem governed by the stochastic FitzHugh-Nagumo equation driven by a Gaussian noise. First order conditions of optimality are also obtained

Novel approaches to the energy load unbalance forecasting in the Italian electricity market

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 ARMA-ARIMA model and the ARIMA-GARCH 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 one-day ahead energy load prediction, where the main focus is on guessing the right sign of the energy load unbalance

Optimal execution strategy in liquidity framework

A trader wishes to execute a given number of shares of an illiquid asset. Since the asset price also depends on the trading behaviour, the trader main aim is to find the execution strategy that minimizes the related expected costs. We solve this problem in a discrete time framework, by modeling the asset price dynamic as an arithmetic random walk with drift and volatility both modeled as Markov stochastic processes. The market impact is assumed to follow a Markov process. We found the unique execution strategy minimizing the implementation shortfall when short selling is allowed. This optimal strategy is given as solution of a forward-backward system of stochastic equations depending on conditional expectations of future values of model parameters. In the opposite case, namely when short selling is prohibited, we numerically obtain the solution for the associated Bellman equation that an optimal trading strategy must satisfy

A maximum principle for a stochastic control problem with multiple random terminal times

In the present paper we derive, via a backward induction technique, an ad hoc maximum principle for an optimal control problem with multiple random terminal times. We thus apply the aforementioned result to the case of a linear quadratic controller, providing solutions for the optimal control in terms of Riccati backward SDE with random terminal time

Delayed Forward-Backward stochastic PDE's driven by non Gaussian Lévy noise with application in finance

From the very first results, the mathematical theory of financial markets has undergone several changes, mostly due to financial crises who forced the mathematical-economical community to change the basic assumptions on which the whole theory is founded. Consequently a new mathematical foundation were needed. In particular, the 2007/2008 credit crunch showed the word that a new financial theoretical framework was necessary, since several empirical evidences emerged that aspects that were neglected prior to these years were in fact fundamental if one has to deal with financial markets. The goal of the present thesis goes in this direction; we aim at developing rigorous mathematical instruments that allow to treat fundamental problems in modern financial mathematics. In order to do so, the talk is thus divided into three main parts, which focus on three different topics of modern financial mathematics. The first part is concerned with delay equations. In particular, we will prove Feynman-Kac type result for BSDE's with time-delayed generator, as well as an ad hoc Ito formula for delay equations with jumps. The second part deal with infinite dimensional analysis and network models, focusing in particular on existence and uniqueness results for infinite dimensional SPDE's on networks with general non-local boundary conditions. The last part treats the topic of rigorous asymptotic expansions, providing a small noise asymptotic expansion for SDE with Lévy noise with several concrete application to financial models

Gaussian estimates on networks with dynamic stochastic boundary conditions

In this paper we prove the existence and uniqueness for the solution to a stochastic reaction\u2013diffusion equation, defined on a network, and subjected to nonlocal dynamic stochastic boundary conditions. The result is obtained by deriving a Gaussian-type estimate for the related leading semigroup, under rather mild regularity assumptions on the coefficients. An application of the latter to a stochastic optimal control problem on graphs, is also provided. Read More: http://www.worldscientific.com/doi/abs/10.1142/S021902571750001

Optimal control of mean field equations with monotone coefficients and applications in neuroscience

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution X=Xα of the stochastic mean-field type evolution equation in Rd dXt=b(t,Xt,L(Xt),αt)dt+σ(t,Xt,L(Xt),αt)dWt,X0∼μ(μ given),(1) under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control αt is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipschitz condition, and that the dynamics (2) satisfies an almost sure boundedness property of the form π(Xt)≤0. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipschitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle for (2), and numerically investigate a gradient algorithm for the approximation of the optimal control.TU Berlin, Open-Access-Mittel - 202