Numerical Modeling of Stochastic Differential Problems with Applications

Abstract

The interest in the study of stochastic differential equations has grown considerably in recent years. The main reason for this interest is that stochastic differential equations

are a potential tool for modeling evolutionary problems, especially when the dynam- ics are affected by stochastic perturbations. Of particular importance is the study of

numerics related to these problems since, in the literature, there are a few stochastic differential equations whose solution is known explicitly. Therefore, the spirit of this thesis is to analyze numerical methods for stochastic differential equations and how these can be used for the description of real-life problems. After an introduction to stochastic differential problems and some models in which these are used, we will then move on to some recalls of numerical methods, known in the literature. Subsequently, the focus will be on new research results obtained, divided into three essential parts. In

the first part, based on the well-known idea of collocation for Volterra Integral Equa- tions, we obtained continuous numerical methods, which allow us to know the solution

not only at the grid points of the numerical discretization but throughout the entire

integration interval. Research on this front continued by pointing out how these contin- uous extensions can be applied to obtain a good estimate of the local truncation error.

In fact, as is also known in the deterministic context, this is a first building-block for the development of a variable step-size algorithm, useful especially in the integration

of stiff problems. The second part, instead, focused on geometric numerical integra- tion for stochastic Hamiltonian problems. Unlike deterministic Hamiltonian problems,

where energy is conserved over time, stochastic Hamiltonian problems of Itô type and driven by the additive Wiener process satisfy the trace equation, i.e. the expected

value of the Hamiltonian function grows linearly over time. Interest in this study de- rives from some results on stochastic Runge-Kutta methods developed by K. Burrage

et al. in 2012. These methods, in fact, have a significant error that increases with increasing stochastic noise. Therefore, through a perturbative analysis, the reason for this behaviour was analyzed, concluding that the preservation of the main features of stochastic Hamiltonian problems does not occur directly for any discretization of time. The research was then extended to stochastic Hamiltonian problems with multiplicative

noise, first obtaining a characterization of the behaviour of the mean value of the Hamil- tonian and then showing that first-order approximations to such systems are unable to

maintain such behaviour. In the last part of this thesis, two different models were ana- lyzed in which stochastic differential equations (including stochastic partial differential

equations) occur. Specifically, first, the stochastic FitzHugh-Nagumo system for signal propagation in nerve cells was analyzed, in which the voltage variable is the solution of a one-dimensional partial derivative differential equation of a parabolic type with a cubic nonlinearity driven by additive space-time white noise. Splitting methods for temporal integration will then be developed for that model, showing that such schemes admit strong convergence order 1{4. Next, the analysis shifted to modeling the spread

out of fake news through the stochastic SIR model, which is widely used in the epidemi- ological context for the spread of an epidemic. In particular, interest was placed on the

stiffness property of the differential problem by pointing out that in a given population, the more stiff the problem, the faster the transit of fake news. Numerical evidence, which will demonstrate the effectiveness of the theoretical results, will be provided in the development of the thesis. [edited by Author]