Hassan Emamirad

Chaotic solution for the Black-Scholes equation

The Black-Scholes semigroup is studied on spaces of continuous functions on (0,∞) which may grow at both 0 and at ∞, which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces Y s,τ := {u ∈ C((0,∞)) : lim x→∞ u(x) 1 + xs = 0, lim x→0 u(x) 1 + x−τ = 0} with norm ‖u‖Y s,τ = sup x>0 ∣ ∣ ∣ u(x) (1+xs)(1+x−τ ) ∣ ∣ ∣ 1, τ ≥ 0 with sν > 1, where √ 2ν is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.

Hypercyclicity in the scattering theory for linear transport equation

We show how the hypercyclicity of the transport semigroup can intervene in the scattering theory to characterize the density property of the Lax and Phillips representation theorem and conversely, how the existence of the wave operators of the scattering theory can be used for recovering the hypercyclicity of the absorbing transport group in some weighted L1 spaces.

Relation between scattering and albedo operators in linear transport theory

Abstract In spite of the fact that the scattering and albedo operators are defined in a different setting in the linear transport theory, there is a close resemblance between them. In this paper we try to explore this analogy. The scattering operator comes from the evolutionary linear transport system: and it involves a comparison of the dynamics U(t) of the above system with the dynamics U0(t) of the collisionless transport system, the so called advection system:

Relationship Between the Albedo and Scattering Operators for the Boltzmann Equation with Semi‐transparent Boundary Conditions

The albedo and scattering operators are central objects in the time-dependent transport theory. Their mutual relationship has recently been established by Arianfar and Emamirad for the case of transparent boundaries. In this paper, we extend the result to general boundary conditions. To allow for this extension, the scattering theory for a transport-like equation is generalized to include partially reflecting boundary conditions. The existence of the wave and scattering operators is directly inferred from the properties of the evolution operators that are determined, in turn, by the physics of collisions within and at the boundaries of the scattering domain.

Dirichlet-to-Neumann semigroup acts as a magnifying glass

The first aim of this paper is to illustrate numerically that the Dirichlet-to-Neumann semigroup represented by P. Lax acts as a magnifying glass. In this perspective, we used the finite element method for discretizing of the correspondent boundary dynamical system using the implicit and explicit Euler schemes. We prove by using the Chernoff’s Theorem that the implicit and explicit Euler methods converge to the exact solution and we use the (P1)-finite elements to illustrate this convergence through a FreeFem++ implementation which provides a movie available online. In the Dirichlet-to-Neumann semigroup represented by P. Lax the conductivity

Regularized Scalar Operators

\gamma

The liouville equation in L1 spaces

γ is the identity matrix