This paper considers the problem of optimal timing for selling of an asset under incomplete information about its drift. The asset price is assumed to follow a geometric Brownian motion X with unknown drift, and an agent who decides to sell at time t for which the expected value of the discounted asset price is maximized. Of course, when information is completed, the corresponding optimal liquidation problem is trivial. Indeed, if the drift of the asset price 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, if the drift is smaller than the interest rate, it implies that the agent should liquidate the asset immediately and instead deposit the money in the bank.
In this paper we consider the incomplete information problem by modelling the drift as a random variable which takes two values. Then we reduce the two-dimensional problem into a one-dimensional optimal stopping problem.
There are many studies that have used various methods to solve the optimal stopping time problem. For instance, the authors in  gave the general theory of optimal stopping time and considered a set of optimal stopping time problems in areas such as mathematical statistics, mathematical finance, and financial engineering.
The optimal stopping time problem with deterministic drift is considered in  in various cases. To get the estimation of the parameters we use theory in  and  and use these estimated parameters to solve realistic optimal stopping problem. This paper is a continuation of the study in the paper   and  . However, the problem is considered in the case that the last time is infinite so the method and the result are different from the previous results.
2. The Problem and Its Solution
We assume that the asset price process is modeled by a geometric Brownian motion (see  ) as follows
where is a standard Brownian motion and this process is assumed to be independent of on a probability space . We also assume that the drift is a random variable which can take two values and satisfying the condition , where is constant interest rate, and is initial price. Assuming the property owner wants to sell his property but does not know the rate of increase of the price is or and he only knows that at the initial time the probability distribution of the events and as follows (Table 1).
The purpose of the property holder is to sell it for maximum expected returns and the purpose of this problem is the same. Mathematically we denote be the σ-field generated by the process X and property holders the choose -stopping time with such that the supremum
is achieved at a certain stopping time.
For , let is the conditional probability that the
Table 1. The initial distribution of the drift.
drift receives small value at time t and therefore .
From Theorems 7.12 and 9.1 in  , the price process is modeled by the following stochastic differential equation
and the belief probability process satisfy equation
where and is a P-Brownian motion defined by
To reduce the dimension of the problem we define a new process W by
and a new probability measure Q with the Radon-Nikodym derivative as defined
with respect to measure P where is chosen such that and . Girsanov’s Theorem so that Z is a Brownian motion under measure Q.
We define a new process . Application of Ito’s formula we obtain
Expressing of X in terms of Z gives
Then, under the Q measure, both and are geometric Brownian motions. Moreover, the σ-field generated by Z and X is coincided.
Now to build the calculations on the new measure we define the following process
Proposition 1. We have
where is a stopping time adapts to the filter .
We have and therefore
Consider the process
Using Ito’s formula, we obtain
So almost sure. Thus we have:
This proves the Proposition. ,
From the above Proposition the value function in the measure Q given by
We see that larger is the less likely that the gain will increase upon continuation. This suggests that there exists a point such that the stopping time
is optimal in the problem (7).
By optimal stopping theory, the pair is the solution of the following free boundary problem
where is infinitesimal operator and the condition is added to define .
This follows that F is the solution of the differential equation
One may now recognize this differential equation as the Cauchy-Euler equation and it has characteristic equation:
We find that
Therefore g(x) has two solutions thus the general solution can be written as
where and are constants which are determined later. The three conditions (9)-(11) can be used to determine and uniquely.
The condition gives
and the condition gives us .
The condition is equivalent to .
Substitute and from (16) into above equation we have
We have equation to determine B as follow
Theorem 2.1. The equation (16) has unique solution .
Consider function in . We find that
and since . So equation has a solution B in .
We see that
by choosing and therefore .
Thus is increasing function in . This implies there exists unique B satisfies the problem. ,
with B > 1 is unique solution off the following equation
and defined in (16) and stopping time is optimal of (7).
Let B be the unique solution to (17), and define function G by
Then we have , and if and only if .
The process satisfies the following stochastic differential equation
Assume that the solution B of (17) satisfies the condition . Then the drift of Y is always negative, and therefore Y is a super-martingale and is martingale.
Let is a stopping time. We have inequality
This follows .
And if , we see that the inequalities in (18) are equalities.
It follows that . ,
Corollary 2.2. Let . The value function is given by:
Moreover, stopping time is optimal of problem (1).
Theorem 2.3. Value function V is decreasing in .
When . If , we have
If the function
is increasing function so follows and if we have
this gives that V is decreasing in and is increasing function in .
We complete the proof. ,
In this section we simulate the price process, Posterior probability process and the threshold for selling the asset.
The parameters are
Let are the solutions of the following equation
with the parameters are given in (19), equation (20) becomes:
or and we have and .
And B is the solution of
We use Matlab software to solve Equation (21) and get B = 1.2798.
In the Figure 1, we see that the posterior probability process cannot pass through the red threshold, so the holder of the property will not be able to sell it. However, since the time in simulation is finite, the property owner will sell the asset at the end that is . And in the Figure 2 below the posterior probability shows that the fall in prices is very fast so the optimal stopping time is very close to the initial time.
Figure 1. The stopping time and corresponding asset price in a simulation ;
Figure 2. The stopping time and corresponding asset price in a simulation ;
This paper studies the optimal strategy to liquidate an asset when it is uncertain whether the asset price is rising or falling (high or low drift). To solve the problem we have to change the initial measure to the new measure and under this measure the price process is martingale. We also have to solve a nonlinear equation to find the threshold of the probability that the drift receives the smaller value. The results show that the value function is decreasing in the initial probability that the drift is low. Simulation results are consistent with proven theory.