## 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]