Abstract: In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein–Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.
Abstract: In this article, we consider a regulated market and explore the default events. By using a so-called reflected Ornstein-Uhlenbeck process with
two-sided barriers to formulate the price dynamics, we derive the expression
on the conditional default probability. In the cases of single observation
and multiple observations, the conditional default probabilities are explicitly
expressed in terms of the inverse Laplace transforms. Finally, we present a
numerical simulation associated with the conditional default probability.
Abstract: In this paper, we consider a class of reflected stochastic differential
equations (abbr. SDEs) and we are particularly interested in some
integral functionals of the solutions to the equations. We explicitly derive
the Laplace transforms of those integral functionals, which are subsequently
applied for the financial arguments. Here we consider a regulated market, in
which the price dynamics is driven by a reflected SDE.We will calculate the
conditional default probability under such price dynamics, and meanwhile
we also give the pricing on some digital options. Finally, for practical purpose, an illustration for the numerical inversion of the Laplace transforms is presented in the Appendix.
Abstract: In this paper, we study first passage times of (reflected) Ornstein-Uhlenbeck processes over compound Poisson-type boundaries. In fact, we extend the results of first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes in Perry et al. (J. Appl. Prob. 41: 1059-1070, 2004) to those of (reflected) Ornstein-Uhlenbeck case.
Abstract: Reflected Brownian motion has been played an important role in economics, finance, queueing and many other fields. In this paper, we present the explicit spectral representation for the hitting time density of the reflected Brownian motion with two-sided barriers, and give some detailed analysis on the computational issues. Numerical analysis reveals that the spectral representation is more appealing than the method of numerical Laplace inversion. Two applications are included at the end of the paper.
Abstract: We consider a portfolio optimization problem in a defaultable market. The investor can dynamically choose a consumption rate and allocate his/her wealth among three financial securities: a defaultable perpetual bond, a default-free risky asset and a money market account. Both the default risk premium and the default intensity of the defaultable bond are assumed to rely on some stochastic factor which is described by a diffusion process. The goal is to maximize the infinite horizon expected discounted log utility of consumption. We apply the dynamic programming principle to deduce a Hamilton-Jacobi-Bellman (HJB) equation. Then an optimal Markov control policy and the optimal value function is explicitly presented in a verification theorem. Finally a numerical analysis is presented for illustration.
Abstract: This paper introduces dynamic models for the spot foreign exchange rate with capturing both the rare events and the time-inhomogeneity in the fluctuating currency market. For the rare events, we use a compound Poisson process with log-normal jump amplitude to describe the jumps. As for the time-inhomogeneity in the market dynamics, we particularly stress the strong dependence of the domestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on thetime-varying sovereign ratings in the currency market. The time-varying ratings are formulated by a continuous-time finite-state Markov chain. Based on such a spot foreign exchange rate dynamics, we then study the pricing of some currency options. Here we will adopt a so-called regime-switching Esscher transform to identify a risk-neutral martingale measure. By determining the regime-switching Esscher parameters we then get an integral expression on the prices of European-style currency options. Finally, numerical illustrations are given.
