A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the pinning at expiration of stock options. We give the explicit solution to these optimization problems and in particular we provide a class of processes whose optimal barrier has the same form as the one of the Brownian bridge. These processes may be a possible alternative to the Brownian bridge in practice as they could better model real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain.


Introduction
In this paper, we study the optimal stopping-time problems related to an Itô diffusion, {X s }, for time s in [t, 1], modeling for example an investment gain, for which the terminal value, say at time s = 1, is a priori known. This could be the case of an insider trading [1][2][3] or of the pinning at expiration of stock options [4][5][6][7].
Roughly speaking, the class of stochastic processes subject of our study is defined by bringing the infinite horizon mean-reverting Ornstein-Uhlenbeck process with constant parameters to a finite horizon. We solve and provide explicit solutions to the optimal stopping-time problems associated with this class, which contains as particular case but it is not limited to, the optimal stopping time associated with the Brownian bridge dX s = − X s 1 − s ds + dB s , s ∈ [t, 1] , with X t = x .
In this case, it is known (see [8,9]) that the optimal stopping time of where {B t } is the standard Brownian motion. The mean of Z t converges to 0 and its variance converges to σ 2 /(2θ), as t goes to ∞, which is infinite if θ = 0. We want to adapt this model, which has infinite time horizon, to a finite time horizon. Without loss of generality, we assume the final time to be t = 1. We define the function b(s) := γ(s) − γ (1) , s ∈ [0, 1] , (1) that will be used below to make the time-change, where γ is the function that carries the view of the future in the model. The function b is assumed to be a non-negative decreasing continuous function, therefore differentiable almost everywhere. We map s in [0, 1] to t in [0, ∞] by t = − ln[b(s)/b(0)], we define X s := b(s)Z − ln[b(s)/b(0)] + γ(1) and we rewrite the Ornstein-Uhlenbeck process as Indeed, one can verify that As, by Lemma 1, represents the deviation of the market price to the final value in proportion to the deviation of the "right" value to the final value, at time s. As ] dB s , if we relabel the parameters by θ := α 2 and σ := 2/β 2 , we have that By multiplying by b(s) the two terms, we then obtain the equation for {X s }: starting at any time t in (0, 1). Please note that despite there are different ways to map [0, 1] to [0, ∞] making use of the function b, the chosen map is the most natural one. Indeed, with this choice we have the following equation for the expected value of X s − γ(1): i.e., the rate of change in logarithmic scale of E[X s − γ (1)] is proportional to the same rate of b(s).
We are interested in the optimization problem where τ ∈ [t, 1] is a stopping time. Here we assume that at the final time, which without loss of generality is normalized to s = 1, the market price coincides with the "right" value, i.e., X 1 = γ(1), as proved in Lemma 1.
which is the SDE of a Brownian bridge already treated in [9].
Hence, X 1 = γ(1) and for s ∈ [t, 1], Remark 2. Please note that by (5), the function γ uniquely determines the average value of the process {X s } which is generally known as the growth curve of the diffusion process.
Proof. By multiplying the two terms of Equation (2) by [b(t)/b(s)] 1+α 2 , which is not identically zero, this can be rewritten as Since the Itô derivative of (X we have that This implies the expression for X s in (4). Moreover, the formulas for the mean and the variance of X s can be directly derived from this expression and the Itô isometry.
Lastly, as the Var[X s ] → 0 when s converges to 1, and since {X s } is continuous, we obtain X 1 = γ (1). This completes the proof.
Moreover, observe that Var[X s ] depends continuously on α also when

The HJB Equation
In this section, we derive the Hamilton-Jacobi-Bellman (HJB) that allows discovery of a candidate solution for the optimal stopping problem given in (3).
Defining the bivariate Markov process {Y s } := {(s, X s )}, we have that its generator is given by the operator acting on any function f ∈ C 2 0 . The first differential operator comes from the first component of the process {Y s }, while the remaining two differential operators come from the Itô representation of the {X s } process given in (2). The introduction of the process {Y s } is required just to convert a non-homogeneous Markov process to a bivariate homogeneous one.
The HJB equation for the function V defined in (3) is given by max{LV(t, x), 0} = 0. Following [12], we can find a continuation set, {(x, t) : x < a(t) + γ(1)} for an unknown function a, where the function V is harmonic. The complement of this set is the stopping set, where trivially V(x, t) = x. Therefore, we have the following PDE system with t ∈ [0, 1] and where a : [t, 1] → R is the free boundary such that the stopping time is the optimal stopping time for the problem (3) The three boundary conditions in (8) are necessary, but not sufficient, conditions for V to be a candidate solution of the optimal stopping problem. Indeed, the first and second conditions are the HJB expressions in the continuation and stopping sets, respectively. The third equation is the smooth fit condition and the last one expresses the fact that minimum possible gain is γ(1), since the process {X s } will end up there almost surely when s = 1.
If we assume that given for an appropriate function f , then (8) can be rewritten as where y = (x − γ(1))/b(t). Indeed, from (9), it follows that .
which, after simplifying the term in f (y) and dividing by b (t), yields the equation in (10). The conditions in (10) come directly from the conditions in (8) and from (9). (10) admits a unique solution, the function F α , for any α ≥ 0 and for a unique β = β(α) which depends on α. The function F α is given by

Lemma 2. Equation
where the function h 1 is defined below in (17a).
Proof. If α = 0, the differential equation in (10) becomes which admits as general solution f (y) = c 1 e βy + c 2 e −βy , with c 1 , c 2 ∈ R. The boundary conditions in (10) give constraints on the parameters. Indeed, we have that Therefore, the solution assumes the form f (y) = e β(y−1) . Finally, the boundary condition f (1) = 1 implies that β = 1 and the solution is f (y) = e y−1 .
Assume now α > 0. By substituting f (y) =: h(−βy), with β > 0, and using x = −βy, we can rewrite the differential equation in (10) without boundary conditions by This can be rewritten as Finally, with a further substitution h(x) =: u(αx) exp[(αx) 2 /4] and z = αx, we obtain the so-called parabolic cylinder differential equation Two linear independent solutions of the parabolic cylinder differential equation arê where M(a, b, z) is the Kummer's function, see [13] [Chapter 13]. The functionsû 1 andû 2 are respectively an even and an odd function. They depend on the parameter α > 0; however, to keep notation as simple as possible, the dependence on the parameter will be dropped. For convenience, let us introduce other two independent solutions u 1 and u 2 defined by the following linear combinations ofû 1 andû 2 : where Γ denotes the gamma function. The expressions above are well defined for all α > 0. Please note that the function u 1 converges to 0 and u 2 diverges as |z| goes to ∞. Moreover, lim |z|→∞ e z 2 /4 z 1/α 2 u 1 (z) = 1 , lim |z|→∞ e −z 2 /4 z 1−1/α 2 u 2 (z) = √ π/2 . (16b) Using the relation between u and h, we have that and, by (16), it follows that for |z| → ∞ h 2 (z) ∼ √ π/2 e z 2 /2 z 1/α 2 −1 .
Going back to the differential equation in (10) without boundary conditions, we can write all its solutions by with c 1 , c 2 ∈ R. Applying the boundary conditions in (10), we have that Therefore, the solution assumes the form Finally, the boundary condition f (1) = 1 implies For every given α > 0, by Lemma A1 proved in the Appendix A, the equation Figure 1), then the differential equation with boundary conditions (10) admits the unique solution given by (20) with β = β(α). This completes the proof.

The Solution
In this section, we verify that the solution V to the HJB equation found in Section 3 is indeed solution to the optimal stopping problem (3). This is a needed verification as we remind that the HJB equation represents only a necessary condition for the solution to the optimal stopping-time problem, but it is not a sufficient condition.
Proof. Consider the process {V * (X s , s)}. By the Itô formula and the definition of V * , it follows that Let τ ∈ [t, 1] be a stopping time. Since V * (x, t) ≥ x, as shown in Lemma A2 proved in Appendix A, and the variable (1 + α 2 )(X s − γ(1))[b (s)/b(s)]1 [b(s),∞) (X s ) is non-positive, as the function b is non-negative and decreasing, and since { s t V * x (X r , r) −2b (r)b(r)/β 2 dB r } is a martingale, then, by the Optional Sampling theorem, Hence, since this is true for any stopping time τ, we have that V(x, t) ≤ V * (x, t).

Application
In this section, we show a brief application of the results of the previous sections. The process {X s } can model a pair trading process; see [14] for a survey or [15,16] for applications using the Ornstein-Uhlenbeck process. For our purposes we assume that we have a predicting model for the process trend that is given by with d > 0, which ensures that the function b is non-increasing for s ∈ [0, 1]. For fitting purposes one could choose a general function b of type b(s) = ∑ i c i (1 − s) d i , with restrictions on the coefficients to ensure the decreasing property. We use (23) to simplify the calculations. In addition, we assume γ(1) = 0 such that γ(s) = b(s). X 1 = 0 may represent the fact that at time s = 1 some information is publicly disclosed, such as the earning reports of the underlining stocks, that eliminates the pair difference. (2), we obtain that (24) Figure 2 shows the value function It can be interpreted as both the time to sell the underlying pair trading and the price of an American call option with null strike price derived on the underlying pair trading process.

Discussion
The relevance of Theorem 1 is in that it proves the existence of a class of Itô diffusion processes for any specified (decreasing) optimal stopping boundary function b and gives the explicit expression of the corresponding value function (3). This provides a flexible model for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain, when the investor owns, or decides to include, additional information on the future trend of the underlining stock.
As an example, we can find the class of processes that have the same optimal stopping barrier of the standard Brownian bridge, i.e., with γ(t) := β(α) √ 1 − t. By choosing d = 1/2 in (23) we get the following Itô representation, for α ≥ 0, It follows that multiplying by a constant factor (1 + α 2 )/2 the drift in the Itô representation of the Brownian bridge the optimal barrier has the same shape as the barrier of the Brownian bridge up to a factor equal to β(α)/β (1). For α ≥ 0, α = 1, the process {X s } in (25) is not a Brownian bridge as, by Lemma 1, it is equal to This class of processes is already known in the literature under the name of α-Wiener bridges, see for example [17], even if technically they are not bridges. Indeed, they cannot be generated, for α = 1, by conditioning a gaussian Itô diffusion to be equal to 0 at time 1.
By the result above, they can be characterized by the fact that the associated optimal barrier is identical, modulo a factor, to the one of the Brownian bridge. Figure 3 gives a sample of simulations for different values of α ≥ 0. This class provides a catalogue of alternative diffusion processes to the Brownian bridge which in practice could offer a better fitting to the data. The light grey areas represent one standard deviation of X s above and below its expected value (null in the simulations). The black solid curves represent the optimal stopping boundaries.
In the literature, the α-Wiener bridges have already been used in economic settings. They appeared from the first time in [18] to model the arbitrage profit associated with some future contracts in absence of transaction costs. Then in [19,20] they were used to describe the fundamental component of an exchange rate process. For more information we refer the reader to [21] and references therein.