89 research outputs found

### Do All Integrable Evolution Equations Have the Painlev\'e Property?

We examine whether the Painleve property is necessary for the integrability
of partial differential equations (PDEs). We show that in analogy to what
happens in the case of ordinary differential equations (ODEs) there exists a
class of PDEs, integrable through linearisation, which do not possess the
Painleve property. The same question is addressed in a discrete setting where
we show that there exist linearisable lattice equations which do not possess
the singularity confinement property (again in analogy to the one-dimensional
case).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA

### Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

We perform a classification of the Lie point symmetries for the
Black--Scholes--Merton Model for European options with stochastic volatility,
$\sigma$, in which the last is defined by a stochastic differential equation
with an Orstein--Uhlenbeck term. In this model, the value of the option is
given by a linear (1 + 2) evolution partial differential equation in which the
price of the option depends upon two independent variables, the value of the
underlying asset, $S$, and a new variable, $y$. We find that for arbitrary
functional form of the volatility, $\sigma(y)$, the (1 + 2) evolution equation
always admits two Lie point symmetries in addition to the automatic linear
symmetry and the infinite number of solution symmetries. However, when
$\sigma(y)=\sigma_{0}$ and as the price of the option depends upon the second
Brownian motion in which the volatility is defined, the (1 + 2) evolution is
not reduced to the Black--Scholes--Merton Equation, the model admits five Lie
point symmetries in addition to the linear symmetry and the infinite number of
solution symmetries. We apply the zeroth-order invariants of the Lie symmetries
and we reduce the (1 + 2) evolution equation to a linear second-order ordinary
differential equation. Finally, we study two models of special interest, the
Heston model and the Stein--Stein model.Comment: Published version, 14pages, 4 figure

### On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits

In this paper, we construct grammian-like quasideterminant solutions of a
non-Abelian Hirota-Miwa equation. Through continuum limits of this non-Abelian
Hirota-Miwa equation and its quasideterminant solutions, we construct a cascade
of noncommutative differential-difference equations ending with the
noncommutative KP equation. For each of these systems the quasideterminant
solutions are constructed as well.Comment: 9 pages, 1 figur

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