46 research outputs found

### On the computation of the nth power of a matrix

In this note we discuss the problem of finding the nth power of a matrix
which is strongly connected to the study of Markov chains and others
mathematical topics. We observe the known fact (but maybe not well known) that
the Cayley-Hamilton theorem is of key importance to this goal. We also
demonstrate the classical Gauss elimination technique as a tool to compute the
minimum polynomial of a matrix without necessarily know the characteristic
polynomial

### Ergodic Theorems for discrete Markov chains

Let $X_n$ be a discrete time Markov chain with state space $S$ (countably
infinite, in general) and initial probability distribution $\mu^{(0)} =
(P(X_0=i_1),P(X_0=i_2),\cdots,)$. What is the probability of choosing in random
some $k \in \mathbb{N}$ with $k \leq n$ such that $X_k = j$ where $j \in S$?
This probability is the average $\frac{1}{n} \sum_{k=1}^n \mu^{(k)}_j$ where
$\mu^{(k)}_j = P(X_k = j)$. In this note we will study the limit of this
average without assuming that the chain is irreducible, using elementary
mathematical tools. Finally, we study the limit of the average $\frac{1}{n}
\sum_{k=1}^n g(X_k)$ where $g$ is a given function for a Markov chain not
necessarily irreducible

### A novel approach to construct numerical methods for stochastic differential equations

In this paper we propose a new numerical method for solving stochastic
differential equations (SDEs). As an application of this method we propose an
explicit numerical scheme for a super linear SDE for which the usual Euler
scheme diverges.Comment: 2 figure

### On Neumann superlinear elliptic problems

In this paper we are going to show the existence of a nontrivial solution to
the following model problem, \begin{equation*} \left\{\begin{array}{lll}
-\Delta (u) = 2uln(1+u^2)+\frac{|u|^2}{1+u^2}2u+u(sin(u)-cos(u)) \mbox{a.e. on
} \Omega \frac{\partial u}{\partial \eta} = 0 {a.e. on} \partial \Omega.
\end{array} \right. \end{equation*} As one can see the right hand side is
superlinear. But we can not use an Ambrosetti-Rabinowitz condition in order to
obtain that the corresponding energy functional satisfies (PS) condition.
However, it follows that the energy functional satisfies the Cerami (PS)
condition

### Existence result for a Neumann problem

In this paper we are going to show the existence of a nontrivial solution to
the following model problem,
$\{\begin{array}{lll} - \Delta (u) =
2uln(1+u^2)+\frac{|u|^2}{1+u^2}2u+usin(u) {a.e. on} \Omega
\frac{\partial u}{\partial \eta} = 0 {a.e. on} \partial \Omega. \end{array}
\}$
As one can see the right hand side is superlinear. But we can not use an
Ambrosetti-Rabinowitz condition in order to obtain that the corresponding
energy functional satisfies (PS) condition. However, it follows that the energy
functional satisfies the Cerami (PS) condition

### Landesman-Laser Conditions and Quasilinear Elliptic Problems

In this paper we consider two elliptic problems. The first one is a Dirichlet
problem while the second is Neumann. We extend all the known results concerning
Landesman-Laser conditions by using the Mountain-Pass theorem with the Cerami
$(PS)$ condition

### An explicit and positivity preserving numerical scheme for the mean reverting CEV model

In this paper we propose an explicit and positivity preserving scheme for the
mean reverting CEV model which converges in the mean square sense with
convergence order $a(a-1/2)$.Comment: 7 page

### On explicit numerical schemes for the CIR process

In this paper we generalize an explicit numerical scheme for the CIR process
that we have proposed before. The advantage of the new proposed scheme is that
preserves positivity and is well posed for a (little bit) broader set of
parameters among the positivity preserving schemes. The order of convergence is
at least logarithmic in general and for a smaller set of parameters is at least
$1/4$. Next we give a different explicit numerical scheme based on exact
simulation and we use this idea to approximate the two factor CIR model.
Finally, we give a second explicit numerical scheme for the two factor CIR
model based on the idea of the second section.Comment: 24 pages, 1 figure

### An elementary approach to the option pricing problem

Our goal here is to discuss the pricing problem of European and American
options in discrete time using elementary calculus so as to be an easy
reference for first year undergraduate students. Using the binomial model we
compute the fair price of European and American options. We explain the notion
of Arbitrage and the notion of the fair price of an option using common sense.
We give a criterion that the holder can use to decide when it is appropriate to
exercise the option. We prove the put-call parity formulas for both European
and American options and we discuss the relation between American and European
options. We give also the bounds for European and American options. We also
discuss the portfolio's optimization problem and the fair value in the case
where the holder can not produce the opposite portfolio.Comment: 17 page

### Construction of positivity preserving numerical schemes for multidimensional stochastic differential equations

In this note we work on the construction of positive preserving numerical
schemes for systems of stochastic differential equations. We use the semi
discrete idea that we have proposed before proposing now a numerical scheme
that preserves positivity on multidimensional stochastic differential equations
converging strongly in the mean square sense to the true solution