194 research outputs found
On two-sided controls of a linear diffusion
siirretty Doriast
Global CÂą regularity of the value function in optimal stopping problems
We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary
Bayesian sequential testing of the drift of a Brownian motion
We study a classical Bayesian statistics problem of sequentially testing the
sign of the drift of an arithmetic Brownian motion with the - loss
function and a constant cost of observation per unit of time for general prior
distributions. The statistical problem is reformulated as an optimal stopping
problem with the current conditional probability that the drift is non-negative
as the underlying process. The volatility of this conditional probability
process is shown to be non-increasing in time, which enables us to prove
monotonicity and continuity of the optimal stopping boundaries as well as to
characterize them completely in the finite-horizon case as the unique
continuous solution to a pair of integral equations. In the infinite-horizon
case, the boundaries are shown to solve another pair of integral equations and
a convergent approximation scheme for the boundaries is provided. Also, we
describe the dependence between the prior distribution and the long-term
asymptotic behaviour of the boundaries.Comment: 28 page
- …