Abstract:
We consider an agent who wants to liquidate an asset with unknown drift. The agent believes that the drift takes one of two given values and has initially an estimate for the probability of either of them. As time goes by, the agent observes the asset price and can therefore update his beliefs about the probabilities for the drift distribution. We formulate an optimal stopping problem that describes the liquidation problem, and we demonstrate that the optimal strategy is to liquidate the first time the asset price falls below a certain time-dependent boundary. Moreover, this boundary is shown to be monotonically increasing, continuous and to satisfy a nonlinear integral equation. 1. Introduction This paper treats the problem of optimal timing for an irreversible sale of an indivisible asset under incomplete information about its drift. The asset price is assumed to follow a geometric Brownian motion with unknown drift, and an agent who decides to sell at time receives at this time the amount . The objective of the agent is to choose a liquidation time for which the expected value of the (discounted) asset price is maximised. Such problems are important for all types of investors with insufficient knowledge of the future trend of an asset. In the case with complete information about the model parameters of , the corresponding optimal liquidation problem is trivial. Indeed, if the drift is larger than the interest rate, then on average the asset price grows faster than money in a risk-free bank account, and the agent should keep the asset as long as possible. Similarly, a drift smaller than the interest rate implies that the agent should liquidate the asset immediately, and instead deposit the money in the bank. However, we remark that the assumption of complete information about the parameters of is quite strong. While the volatility of an asset, at least in principle, can be estimated instantaneously by observing the price fluctuations over an arbitrarily short time period, the drift is notoriously difficult to estimate from historical data. In fact, to achieve a decent accuracy in the estimate for the drift, one typically needs observations of the process from hundreds of years. Instead, we allow for incomplete information by modelling the drift as a random variable which is not directly observable for the agent. Initially, the agent’s beliefs about the drift are summarised by a probability distribution. As time goes by, however, he observes the asset price process, and based on these observations his beliefs may change. Naturally, if the asset price

Abstract:
The discrete sell or hold problem (DSHP), which is introduced in \cite{H12}, is studied under the constraint that each asset can only take a constant number of different values. We show that if each asset can take only two values, the problem becomes polynomial-time solvable. However, even if each asset can take three different values, DSHP is still NP-hard. An approximation algorithm is also given under this setting.

Abstract:
This paper is concerned with an optimal stock selling rule under a Markov chain model. The objective is to find an optimal stopping time to sell the stock so as to maximize an expected return. Solutions to the associated variational inequalities are obtained. Closed-form solutions are given in terms of a set of threshold levels. Verification theorems are provided to justify their optimality. Finally, numerical examples are reported to illustrate the results.

Abstract:
Assume that we want to shell an asset with unknown drift but known that the drift is a two value random variable, and the initial distribution can be estimated. As time goes by, this distribution is updated and base on the probability of the drift takes the small one gives us the stopping rule. Research results show that the optimal strategy to sell the asset is if the initial probability that the drift receives a small value greater than a certain threshold then liquidates the asset immediately, otherwise the asset holder will wait until the probability of the drift receives a small value passing a certain threshold, it is the optimal time to liquidate the asset.

Abstract:
In this paper, we consider the problem to determine the optimal time to sell an asset that its price conforms to the Black-Schole model but its drift is a discrete random variable taking one of two given values and this probability distribution behavior changes chronologically. The result of finding the optimal strategy to sell the asset is the first time asset price falling into deterministic time-dependent boundary. Moreover, the boundary is represented by an increasing and continuous monotone function satisfying a nonlinear integral equation. We also conduct to find the empirical optimization boundary and simulate the asset price process.

Abstract:
We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift parameter by allowing an arbitrary probability distribution to characterise the uncertainty about the drift parameter. Filtering theory is used to describe the evolution of the posterior beliefs about the drift once the price process is being observed. An optimal stopping time is determined as the first passage time of the posterior mean below a monotone boundary, which can be characterised as the unique solution to a non-linear integral equation. We also study monotonicity properties with respect to the prior distribution and the asset volatility.

Abstract:
We study the effect of drift in pure-jump transaction-level models for asset prices in continuous time, driven by point processes. The drift is as-sumed to arise from a nonzero mean in the efficient shock series. It follows that the drift is proportional to the driving point process itself, i.e. the cumulative number of transactions. This link reveals a mechanism by which properties of intertrade durations (such as heavy tails and long memory) can have a strong impact on properties of average returns, thereby poten-tially making it extremely difficult to determine long-term growth rates or to reliably detect an equity premium. We focus on a basic univariate model for log price, coupled with general assumptions on the point process that are satisfied by several existing flexible models, allowing for both long mem-ory and heavy tails in durations. Under our pure-jump model, we obtain the limiting distribution for the suitably normalized log price. This limiting distribution need not be Gaussian, and may have either finite variance or infinite variance. We show that the drift can affect not only the limiting dis-tribution for the normalized log price, but also the rate in the corresponding normalization. Therefore, the drift (or equivalently, the properties of dura-tions) affects the rate of convergence of estimators of the growth rate, and can invalidate standard hypothesis tests for that growth rate. As a rem-edy to these problems, we propose a new ratio statistic which behaves more

Abstract:
Rapid energy releases (RERs) in the solar corona extend over many orders of magnitude, the largest (flares) releasing an energy of $10^{25} $J or more. Other events, with a typical energy that is a billion times less, are called nanoflares. A basic difference between flares and nanoflares is that flares need a larger magnetic field and thus occur only in active regions, while nanoflares can appear everywhere. The origin of such RERs is usually attributed to magnetic reconnection that takes place at altitudes just above the transition region. Here we show that nanoflares and smaller similar RERs can be explained within the drift wave theory as a natural stage in the kinetic growth of the drift wave. In this scenario, a growing mode with a sufficiently large amplitude leads to stochastic heating that can provide an energy release of over $10^{16} $J.

Abstract:
It is a common belief that the behavior of shareholders depends upon the direction of price fluctuations: if prices increase they buy, if prices decrease they sell. That belief, however, is more based on ``common sense'' than on facts. In this paper we present evidence for a specific class of shareholders which shows that the actual behavior of shareholders can be markedly different.

Abstract:
In this paper we consider a modification of the classical Merton portfolio optimization problem. Namely, an investor can trade in financial asset and consume his capital. He is additionally endowed with a one unit of an indivisible asset which he can sell at any time. We give a numerical example of calculating the optimal time to sale the indivisible asset, the optimal consumption rate and the value function.