• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.4
• /
• pp.835-844
• /
• 2010

In this paper we study the $H{\grave{a}}jeck-R{\grave{e}}nyi$ type inequality and strong law of large numbers for asymptotically quadrant sub-independent(AQSI) sequences. We also prove the integrability of supremum for AQSI sequences.

• Journal of the Korean Mathematical Society
• /
• v.43 no.2
• /
• pp.357-371
• /
• 2006

Let $X\;=\;(X_t,\;t{\in}[0, T])$ be a generalized Brownian motion(gBm) determined by mean function a(t) and variance function b(t). Let $L^2({\mu})$ denote the Hilbert space of square integrable functionals of $X\;=\;(X_t - a(t),\; t {in} [0, T])$. In this paper we consider a class of nonlinear functionals of X of the form F(. + a) with $F{in}L^2({\mu})$ and discuss their analysis. Firstly, it is shown that such functionals do not enjoy, in general, the square integrability and Malliavin differentiability. Secondly, we establish regularity conditions on F for which F(.+ a) is in $L^2({\mu})$ and has its Malliavin derivative. Finally we apply these results to compute the price and the hedging portfolio of a contingent claim in our financial market model based on a gBm X.

• East Asian mathematical journal
• /
• v.38 no.5
• /
• pp.643-647
• /
• 2022

Let F² denote the space of entire functions f on ℂ that are square integrable with respect to the Gaussian measure $dG(z)={\frac{1}{\pi}}{e^{-{\mid}z{\mid}^2}}$, where dA(z) = dxdy is the ordinary area measure. The Fock-Dirichlet space $F^2_{\mathcal{D}}$ consists of all entire functions f with f' ∈ F². We estimate Taylor coefficients of functions in the Fock-Dirichlet space. The Fock-Nevanlinna space $F^2_{\mathcal{N}}$ consists of entire functions that possesses just a bit more integrability than square integrability. In this note we prove that $F^2_{\mathcal{D}}=F^2_{\mathcal{N}}$.

• Korean Journal of Mathematics
• /
• v.8 no.1
• /
• pp.73-86
• /
• 2000

We study the Sobolev spaces to the generalized Sobolev spaces $H^s_{\mathcal{G}}$ of exponential type based on the Silva space $\mathcal{G}$ and investigate its properties such as imbedding theorem and structure theorem. In fact, the imbedding theorem says that for $s$ > 0 $u{\in}H^s_{\mathcal{G}}$ can be analytically continued to the set {$z{\in}\mathbb{C}^n{\mid}{\mid}Im\;z{\mid}$ < $s$}. Also, the structure theorem means that for $s$ > 0 $u{\in}H^{-s}_{\mathcal{G}}$ is of the form $$u={\sum_{\alpha}\frac{s^{{|\alpha|}}}{{\alpha}!}D^{\alpha}g{\alpha}$$ where $g{\alpha}$'s are square integrable functions for ${\alpha}{\in}\mathbb{N}^n_0$. Moreover, we introduce a classes of symbols of exponential type and its associated pseudo-differential operators of exponential type, which naturally act on the generalized Sobolev spaces of exponential type. Finally, a generalized Bessel potential is defined and its properties are investigated.

• Bulletin of the Society of Naval Architects of Korea
• /
• v.24 no.1
• /
• pp.1-8
• /
• 1987

It is shown that irregular frequencies in the source and doublet distribution method, can be eliminated if the Green function associated with Kelvin's source of pulsating strength, is modified only in the region inside the body at the level of the undisturbed free surface. The system of the resulting Green integral equation is augmented without loss of the square-integrable property of its kernel so hat the discretisation yield N linearly independent equations for N unknown variables. From the solution, the potential and velocity at any point on the wetted surface of a surface-piercing body can be found using the properties of the double layer composed of the source and normal doublet distribution.

• Communications of the Korean Mathematical Society
• /
• v.19 no.4
• /
• pp.661-681
• /
• 2004

We research properties of the space of measurable functions square integrable with weight exp$2\nu $\mid$x$\mid$$, and those of the space of Fourier hyperfunctions. Also we show that the several embedding theorems hold true, and that the Fourier-Lapace operator is an isomorphism of the space of strongly decreasing Fourier hyperfunctions onto the space of analytic functions extended to any strip in $C^n$ which are estimated with the aid of a special exponential function exp($\mu$｜x｜).