A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.

In this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.

The attractive feature of the singularity method for steady Stokes flows is that the hydrodynamic forces acting on the particle can be calculated by the total strength of distributed singularities. For unsteady Stokes flows, however we have to derive hydrodynamic forces acting on a solid body in terms of the strengths of both unsteady Stokeslets as well as unsteady potential dipoles if mass and force sources are both taken into consideration. Since the hydrodynamic force formulation results in a Volterra integral equation of the first kind, and the strengths are numerically approximated by means of the Lubich convolution quadrature method (CQM) in this study. As far as the numerical solutions of time-domain integral formulations of the unsteady Stokes equations are concerned, this paper requires only the Laplace-domain instead of the time- domain fundamental solutions of the governing equations. The stability and accuracy of the proposed method are verified through some well selected numerical examples. In total we include two examples presenting the accuracy of Lubich CQM, and another two examples for calculating general hydrody-namic forces of a sphere in oscillating or non-oscillating unsteady Stokes flows. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions instead to limit to the oscillating flow assumption.

Let $\mathbb{G}$ be a step-two nilpotent group of $\text{H}$ -type with Lie algebra $\mathfrak{G}\,=\,V\,\oplus \,\text{t}$ . We define a class of vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ on $\mathbb{G}$ depending on a real parameter $k\,\ge \,1$ , and we consider the corresponding $p$ -Laplacian operator ${{L}_{p,\,k}}u\,=\,di{{v}_{X}}\left( {{\left| {{\nabla }_{X}}u \right|}^{p-2}}{{\nabla }_{X}}u \right)$ . For $k\,=\,1$ the vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$ ; for $\mathbb{G}$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator ${{L}_{p,\,k}}$ and as an application, we get a Hardy type inequality associated with $X$ .

We consider a class of fractal subsets of ${{\mathbb{R}}^{d}}$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.