Shigeo Kusuoka

On law invariant coherent risk measures

The idea of coherent risk measures has been introduced by Artzner, Delbaen, Eber and Heath [1]. We think of a special class of coherent risk measures and give a characterization of it. Let (Ω, ℱ, P) be a probability space. We denote L ∞(Ω, ℱ, P) by L ∞. Following [1], we give the following definition.

A Remark on default risk models

We study some mathematical models on default risk. First, we study a “standard model” which is an abstract setting widely used in parctice. Then we study how the hazard rates changes, if we change a basic probability measure. We show that the usual assumptions on hazard rates hold in a standard model, but do not hold in general if we change a basic measure. Finally we study a filtering model.

Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus

The author gives a new numerical computation method of Expectation of Diffusion Processes, which is an improvement of a results in [3].

The Poisson kernel for certain degenerate elliptic operators

Abstract Let L = 1 2 ∑ k = 1 d V k 2 + V 0 be a smooth second order differential operator on R n written in Hormander form, and G be a bounded open set with smooth noncharacteristic boundary. Under a global condition that ensures that the Dirichlet problem is well posed for L on G and a nondegeneracy condition at the boundary (precisely: the Lie algebra generated by the vector fields V0, V1,…, Vd is of full rank on the boundary) then the harmonic measure for L starting at any point in G has a smooth density with respect to the natural boundary measure. Estimates on the derivatives of this density (the Poisson kernel) similar to the classical estimates for the Poisson kernel for the Laplacian on a half space are given.

Homogenization on nested fractals

SummaryWe study the homogenization problem on nested fractals. LetXt be the continuous time Markov chain on the pre-nested fractal given by puttingi.i.d. random resistors on each cell. It is proved that under some conditions,

A mechanical model of Brownian motion in half-space

\alpha ^{ - n} X_{t_E^n t}

Gaussian K-scheme: justification for KLNV method

converges in law to a constant time change of the Brownian motion on the fractal asn→∞, where α is the contraction rate andtE is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martingale method.

Term structure and SPDE

AbstractThe authors apply a new simulation method of diffusion processes to finance problems and show that this new method realizes extremely fast calculation.