Hyeong-Ohk Bae

TEMPORAL AND SPATIAL DECAY RATES OF NAVIER-STOKES SOLUTIONS IN EXTERIOR DOMAINS

We obtain spatial-temporal decay rates of weak solutions of incompressible flows in exterior domains. When a domain has a boundary, the pressure term yields difficulties since we do not have enough information on the pressure term near the boundary. For our calculations we provide an idea which does not require any pressure information. We also estimated the spatial and temporal asymptotic behavior for strong solutions.

Time-asymptotic interaction of flocking particles and an incompressible viscous fluid

We present a new coupled kinetic-fluid model for the interactions between Cucker–Smale (C–S) flocking particles and incompressible fluid on the periodic spatial domain . Our coupled system consists of the kinetic C–S equation and the incompressible Navier–Stokes equations, and these two systems are coupled through the drag force. For the proposed model, we provide a global existence of weak solutions and a priori time-asymptotic exponential flocking estimates for any smooth flow, when the kinematic viscosity of the fluid is sufficiently large. The velocity of individual C–S particles and fluid velocity tend to the averaged time-dependent particle velocities exponentially fast.

APPLICATION OF FLOCKING MECHANISM TO THE MODELING OF STOCHASTIC VOLATILITY

In this study, we present a new stochastic volatility model incorporating a flocking mechanism between individual volatilities of assets. Collective phenomena of asset pricing and volatilities in financial markets are often observed; these phenomena are more apparent when the market is in critical situations (market crashes). In the classical Heston model, the constant theoretical mean of the square of the volatility was employed, which can be assumed a priori. Our proposed model does not assume this mean value a priori, we instead use the flocking effect to continuously update the theoretical mean value using the local weighted average of individual volatility values. To perform this function, we use the Cucker–Smale flocking mechanism to calculate the local mean. For some classes of interaction weights such as all-to-all and symmetric coupling with a positive lower bound, we show that the fluctuations of the square process of volatility are uniformly bounded, such that the overall dynamics are mainly dictated by the averaged process. We also provide several numerical examples showing the dynamics of volatility.

EXISTENCE OF STRONG MILD SOLUTION OF THE NAVIER-STOKES EQUATIONS IN THE HALF SPACE WITH NONDECAYING INITIAL DATA

We construct a mild solutions of the Navier-Stokes equations in half spaces for nondecaying initial velocities. We also obtain the uni- form bound of the velocityeld and its derivatives.

Analyticity and asymptotics for the Stokes solutions in a weighted space

Abstract We first show the analyticity of Stokes operator in a weighted space L p γ ( R + 3 ) . We also show that the Hodge decomposition holds on Lpγ(R+3). Then, we estimate the asymptotic behavior of the Stokes solutions in space and time directions. For 1 ∫ R 3 + | u (x,t)| q ω(x) γq dx 1/q ⩽ C t (3/2)(1/r−1/q)+1/2−γ/2 , where γ is a constant with 0⩽γ

Regularity for the Navier–Stokes equations with slip boundary condition

For the Navier-Stokes equations with slip boundary conditions, we obtain the pressure in terms of the velocity. Based on the representation, we consider the relationship in the sense of regularity between the Navier-Stokes equations in the whole space and those in the half space with slip boundary data.

A SURVEY ON AMERICAN OPTIONS: OLD APPROACHES AND NEW TRENDS

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European op- tion can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up un- til now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These ap- proaches typically provide numerical or approximate analytic methods to �nd the price and the boundary. Topics included in this survey are early approaches(treesnite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, an- alytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochas- tic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

Option pricing and Greeks via a moving least square meshfree method

We apply a meshfree method using the fast moving least squares approximation to option pricing, particularly for the purpose of obtaining high-order Greeks. The method is shown to be accurate and efficient in obtaining prices and Greeks of European, Asian and Barrier options. We also include a complicated Equity Linked Security (ELS) from the Korean OTC market, as a real-world example.

Perturbed integral equation method on determining unknown geometry in fluid flow

We consider an inverse problem arising in fluid flow. An algorithm to find the shape of a body in uniform flow is proposed when the tangential velocity on its boundary is given a priori. The fluid flow is assumed to be inviscid, incompressible and irrotational.The essential idea to develop our algorithm is the boundary modification process toward the solution shape with the help of the perturbed integral equations. The perturbed integral equations are derived from the boundary perturbation. We also give examples exhibiting the reliability for our proposed algorithm.

ANALYTICITY FOR THE STOKES OPERATOR IN BESOV SPACES

We first show the analyticity of Stokes operator in Besov spaces B a p;q(R n). Then, we estimate the asymptotic behavior of the Stokes solutions. We also show the Hodge decomposition.