Abstract |
The primary purpose of this thesis is to accomplish the combination and co-existence of mathematics
in the field of physics and finance in order to approach dynamical systems guided by uncertainty.
This is a combination of research works ([14], [13]). Let start with the motivation of [14]. The Cahn-
Hilliard/Allen-Cahn stochastic model belongs to the general category of the dynamical systems, and
specifically represents an evolutionary equation of the form reaction-diffusion. Actually it is a combination
of Cahn-Hilliard and Allen-Cahn equation similar to [135] with the presence of multiplicative spacetime
white noise with respect to Walsh, [226]. As a modified version of Cahn-Hilliard and Allen-Cahn
expresses both the energy interactions between the particles of a surface, of the form particle/particle,
and simultaneously, the dynamics of the adsorption-desorption phenomenon of these molecules, to (adsorption)
and from (desorption) the aforementioned surface, respectively, [135]. Then and based on the
results of the study of [16] we apply Malliavin calculus on the Cahn-Hilliard/Allen-Cahn stochastic equation
with unbounded noise diffusion. The path has as follows. At first we turn the differential form of
Cahn-Hilliard/Allen-Cahn into a weak version. Then by using an appropriate truncated function, this
weak formulation converts into a piecewise approximation, un of the solution of the Cahn-Hilliard/Allen-
Cahn equation, u. Then applying Picard iteration scheme we prove the existence and uniqueness of
un. This part indicates that the primary sequence un is well-defined and acceptable for Malliavin calculus.
Equivalently, we apply the Malliavin operator on un and establish in this way it’s existence and
uniqueness , i.e., the existence, uniqueness and the regularity version of the Malliavin derivative on the
approximated weak formulation. Then we examine that a certain norm of this derivative is almost surely
positive. This case establishes the absolute continuity of the sequence un. Then the solution (as a random
variable) of Cahn-Hilliard/Allen-Cahn equation has a density, that is under some non-degeneracy
condition on the noise diffusion coefficient we prove that the law of the solution un is absolutely continuous
with respect to the Lebesgue measure on R. However, the regularity version of un indicates that u
is localized well-defined and that the Malliavin operator on u holds on. Then, through convergement of
un for n → ∞ the same results follow for u. Note that the previous approach are based on the research
work of [50].
The next motivation of this thesis, i.e., the work of [13], is a part of the one phase Stefan problem.
Particularly it is a one-dimensional dynamics model describing the total density of sell and buy limit
orders of an asset with spread. In addition this model satisfies a stochastic Stefan problem for the Heat
equation in presence of space-time white noise. Then and under the time-dependent change of variables
with respect to the space-variable in the liquid phase of the model, we act on this transformed equation
a reflection measure in order to constrain the function of density strictly positive. This is Stefan problem
for reflected stochastic partial differential equation driven by multiplicative space-time white noise similar
to Walsh, [226]. The first goal here, as a preprint paper, is to accomplish stochastic existence of global
solutions of the evolution model by using the technique of the work [16], and of [112]. The second goal is
to determine the financial environment illustrating by a stochastic Stefan problem for the Heat equation
with space-time white noise. More precise, this equation satisfies a specific class of financial markets,
that is markets structured by electronic trading platforms. In addition the financial instruments consist
on the presence of noise and are available for trading through the electronic trading platforms with limit
order book sense. Then the respectively market participants are those which have the goal to achieve
returns by using the aforementioned financial instruments through electronic trading platforms of limit
order books.
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