Weak discuss existence of singular solutions for Stokes and the Navier-Stokes equations in the half-space. We construct their solutions whose normal derivatives are unbounded for the Stokes and Navier-Stokes equations near boundary away from support of singular data. 

This presentation focuses on unbiased simulation methods for quantities associated with sample paths from stochastic differential equations. Unbiased simulation methods can be found by changing probability measures with appropriately selecting the Radon-Nikodym derivative processes. I propose an unbiased Monte-Carlo simulation method that can be used even when the Girsanov kernel for the change of probability measures is not bounded. Then, I illustrate its practical application through an example involving unbiased Monte Carlo simulation for pricing the continuously averaging arithmetic Asian options under the Black-Scholes model.

In this talk, we will introduce support properties of solutions to nonlinear stochastic reaction-diffusion equations driven by random noise ˙W : ∂tu = aijuxixj + biuxi + cu + ξσ(u) ˙W , (ω, t, x) ∈ Ω × R+ × Rd; u(0, ·) = u0, where aij , bi, c and ξ are bounded and random coefficients. The noise ˙W is spacetime white noise or spatially homogeneous colored noise satisfying reinforced Dalang’s condition. We present examples of conditions on σ(u) that guarantee the compact support property of the solution. In addition, we suggest potential generalization of these conditions. This is joint work with Kunwoo Kim and Jaeyun Yi.