Optimal Investment with Stopping in Finite Horizon

In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to study a manager's decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon.


Introduction
Optimal stopping problems, a variant of optimization problems allowing investors freely to stop before or at the maturity in order to maximize their profits, have been implemented in practice and given rise to investigation in academic areas such as science, engineering, economics and, particularly, finance. For instance, pricing American-style derivatives is a conventional optimal stopping time problem where the stopping time is adapted to the information generated over time. The underlying dynamic system is usually described by stochastic differential equations (SDEs). The research on optimal stopping, consequently, has mainly focused on the underlying dynamic system itself. In the field of financial investment, however, an investor frequently runs into investment decisions where investors stop investing in risky assets so as to maximize their expected utilities with respect to their wealth over a finite time investment horizon. These optimal stopping problems depend on underlying dynamic systems as well as investors' optimization decisions (controls). This naturally results in a mixed The initial motivation of this paper comes from our recent studies on choosing an optimal point at which an investor stops investing and/or sells all his risky assets (see Choi-Koo-Kwak (2004) and Henderson-Hobson (2008)). The objective is to find an optimization process and stopping time so as to meet certain investment criteria, such as, the maximum of an expected utility value before or at the maturity. This is a typical problem in the area of financial investment. However, there are fundamental difficulties in handling such optimization problems. Firstly, our investment problems, which are different from the classical American-style options, involve optimization process over the entire time horizon. Secondly, it involves the portfolio in the drift and volatility terms so that the problem including multi-dimensional financial assets is more realistic than those addressed in finance literature (see Capenter (2000)). Therefore, it is difficult to solve these problems either analytically or numerically using current methods developed in the framework of studying American-style options. In our model, the corresponding HJB equation of the problem is formulated into a variational inequality of a fully nonlinear equation. We make a dual transformation for the problem to obtain a new free boundary problem with a linear equation. Tackling this new free boundary problem, we establish the properties of the free boundary and optimal strategy for the original problem.
The remainder of the paper is organized as follows. In Section 2, the mathematical formulation of the model is presented, and the corresponding HJB equation is posed. In Section 3, a dual transformation converts the free boundary problem of a fully nonlinear PDE to a new free boundary problem of a linear equation but with a complicated constraint (3.15). In Section 4, it is a further idea that we simplify the constraint condition in (3.15) to obtain a new free boundary problem with a simple condition (4.5). Moreover, we show that the solution of problem (4.5) must be the solution of problem (3.15). Section 5 devotes to the study for the free boundary of problem (4.5). In Section 6, we go back to the original problem (2.6) to show that its free boundary is decreasing and differentiable. Moreover we present its financial meanings. Section 7 concludes the paper. The manager operates in a complete, arbitrage-free, continuous-time financial market consisting of a riskless asset with instantaneous interest rate r and n risky assets, The risky asset prices S i are governed by the stochastic differential equations where the interest rate r, the excess appreciation rates µ i , and the volatility vectors σ i are constants, W is a standard n-dimensional Brownian motion. In addition, the covariance matrix Σ = σ ′ σ is strongly nondegenerate.
A trading strategy for the manager is an n-dimensional process π t whose i-th component, where π i,t is the holding amount of the i-th risky asset in the portfolio at time t. An admissible trading strategy π t must be progressively measurable with respect to {F t } such that X t ≥ 0. Note that where π 0,t is the amount invested in the money. The value of the wealth X t evolves according to In addition, short-selling is allowed.
The manager controls assets with initial value x. The manager's dynamic problem is to choose an admissible trading strategy π t and a stopping time τ to maximize his expected utility of the exercise wealth: where r > 0 is the interest and K is a positive constant (e.g., a fixed salary), is the utility function.

HJB equation
Applying dynamic programming principle, we get the following Hamilton-Jacobi-Bellman (HJB) equation Suppose that V (x) is strictly increasing and strictly concave, i.e., ∂ x V > 0, ∂ xx V < 0. Note that the gradient of π ′ Σπ with respect to π ∇ π (π ′ Σπ) = 2Σπ, where a 2 = µ ′ Σ −1 µ. Now we find a condition under which the free boundary exists. A simple calculation shows It follows that then (2.7) holds for any x > 0, the solution to problem (2.6) is U (x + K) at all. If then (2.7) is impossible for any x > 0. Therefore, in this case, the solution to problem (2.6) satisfies (2.10) We summarize the above results into the following theorem.
Theorem 2.1 In the following cases, problem (2.6) has trivial solution.
(2) If (2.9) holds, the solution to problem (2.10) is the solution to problem (2.6) as well.
Recalling (2.8) and (2.9), in the following we always assume that In the case of (2.11), there exists the free boundary.

Dual transformation
Define a dual transformation of V (x, t) (see Pham (2009) If ∂ x V (·, t) is strictly decreasing, which is equivalent to the strict concavity of V (·, t) (We will show this fact in Section 7), then the maximum in (3.1) will be attained at just one point which is the unique solution of x=I(y,t) = V (I(y, t), t) − yI(y, t). , Now we derive the terminal condition for v(y, T ). Note that Next, we determine the upper bound y 0 for y. In fact, V (x, t) = 1 γ (x + K) γ in the neighborhood of x = 0, so the upper bound is (3.13) In addition, we need to determine the value v(y 0 , t). By (3.8), we also have v(y 0 , t) = V (0, t) + y 0 · 0 = 1 γ K γ . (3.14) Combining (3.9) and (3.12)-(3.14), we obtain is very complicated. In the following section, we simplify this condition.
Remark: The equation in (3.15) is degenerate on the boundary y = 0. According to Fichera's Theorem (see Oleinik-Radkevie (1973)), we must not put the boundary condition on y = 0.

Simplifying the complicated constraint condition
By the y coordinate, Deriving ∂ y v from the first equality in (4.2) yields This is the simplified constraint condition. We assume that u(y, t) satisfies Moreover, we split the domain Q y into two parts, denote (see Fig. 1) In order to prove this theorem, we first show the following two lemmas.
Lemma 4.1 For any (y, t) ∈ Q y , we have Proof: Equation (4.8) follows from the definition (4.6) directly. Also, in CR y Differentiating (4.10) to y yields where ∂(CR y ) is the boundary of CR y .
Theorem 5.1 The Problem (4.5) has a unique solution u ∈ W 2,1 p,loc (Q y ) ∩ (Q y \{y = 0}), and Proof: According to the existence and uniqueness of W 2,1 p,loc (Q y ) ∩ (Q y \{y = 0}), the solution for system (4.5) can be proved by a standard penalty method (see Friedman (1975)). Here, we omit the details. The first inequality in (5.1) follows from (4.5) directly, and now we prove the second inequality in (5.1). Denote where A > 0 to be determined later on. We first show that W (y, t) is a supersolution to problem (4.5). In fact, So, W (y, t) is a supersolution to problem (4.5). Hence, the second inequality in (5.1) holds.
Financial meanings: At time t, the manager should continue to invest according to (2.5) if x > g(t), while the investor should stop investment if x < g(t).