Optimal Stopping of a Brownian Bridge with an Uncertain Pinning Time

02/26/2019 ∙ by Kristoffer Glover, et al. ∙ 0

We consider the problem of optimally stopping a Brownian bridge with an uncertain pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we consider a general prior distribution of the pinning time and allow the stopper to update their belief about this time through sequential observations of the process. Structural properties of the optimal stopping region are shown to be qualitatively different under different priors, however we are able to provide a sufficient condition for a one-sided stopping region. Certain gamma and beta distributed priors are shown to satisfy this condition and these cases are subsequently considered in detail. In the gamma case we reveal the remarkable fact that the optimal stopping problem becomes time homogeneous and is completely solvable in closed form. In the beta case we find that the optimal stopping boundary takes on a square-root form, similar to the classical solution with a known pinning time. We also consider a two-point prior distribution in which a richer structure emerges (with multiple optimal stopping boundaries). Furthermore, when one of the values of the two-point prior is set to infinity (such that the process may never pin) we observe that the optimal stopping problem is also solvable in closed form.

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1. Introduction

The problem of stopping a Brownian bridge so as to maximise the expected value upon stopping has a long and rich history in the field of optimal stopping. It was first considered in [14] and solved explicitly in [39]. It can also be seen as a special case of the problem studied in [23]. The problem is quite remarkable in that it is one of the very few finite-horizon optimal stopping problems to yield an explicit solution. The same problem (amongst others) was also solved in [21] using an alternative (Markovian) approach and other notable work on this problem includes [22], who provided yet another alternative proof of the result of [39]. Further, [9] extended the work of [21] to consider the problem of maximising the expected spread between the value of the process between two stopping times. Once more the authors were able to obtain explicit solutions to this more complicated problem.

More recent work has introduced uncertainty in the pinning level of the Brownian bridge. For example, [20] considered the problem of stopping a Brownian bridge with an unknown pinning point so as to maximise the expected value upon stopping. The authors allowed for a general prior distribution of the unknown pinning point but revealed a rich structure of the optimal stopping region even in the simple case of a two-point distribution. Similar optimal trading problems were also considered in [13] and [31], who used an exponential randomized Brownian bridge to model asset price dynamics under a trader’s subjective market view. Once more a rich solution structure was found by these authors, with disconnected continuation/exercise regions.

In contrast to uncertainty in the pinning level, the present paper considers uncertainty in the time at which the Brownian bridge pins. Such uncertainty for a Brownian bridge has recently been introduced in [7], who considered a model of a Brownian bridge on a random time interval to model the flow of information about a company’s default. The authors outlined the basic properties of these processes along with many useful results. However no optimal stopping problems were considered. To the best of our knowledge, this is the first paper to consider the problem of stopping a Brownian bridge with an uncertain pinning time.

Exploiting the results of [7]

, we formulate the optimal stopping problem for a general prior distribution and, like previous studies, find that the structural properties of the optimal stopping region are highly dependent on the chosen prior. However, we do provide a sufficient condition which ensures only a single optimal stopping boundary exists. Certain gamma and beta distributed priors are examined in further detail and various properties of the solution in these cases are presented. We also document the remarkable property that for a specific gamma distributed prior the stopper’s conditional estimate of the pinning time is such that the conditional dynamics of the underlying Brownian bridge becomes time homogeneous. Finally, we consider the case of a two-point prior distribution in which the bridge will either pin at time 1 with probability

or at time with probability . The two-point prior is also of practical interest since the case when corresponds to a prior in which the stopper believes the process to be either a standard Brownian bridge (with probability ) or a driftless Browninan motion (with probability ).

It is well known that the Brownian bridge appears as the large population limit of the cumulative sum process when sampling without replacement from a finite population (see [38]). As such, and as was noted in [39] and [21], the problem of maximising a stopped Brownian bridge can be thought of as a continuous analog of the following urn problem. balls are drawn randomly and sequentially from an urn containing an equal number () of red and black balls. If every red ball wins you a dollar and every black ball loses you a dollar, and you could stop the game at any time, what would your optimal strategy be so as to maximise your expected profit? The classical problem considered in [39] corresponds to the situation in which the number is known. The problem in the current article, however, corresponds to a situation where the stopper does not know with certainty, but has a prior belief about its value and updates this belief as balls are subsequently drawn. Intuitively, if more balls of a given colour have already been drawn, then drawing an additional ball of the same colour would suggest that the total number of balls in the urn is larger than previously believed. It is important therefore that such learning be incorporated into the optimal stopping strategy, complicating the problem somewhat.

Brownian bridges also play a key role in many areas of statistics and probability theory and have found use in many applications across numerous fields. In addition to its appearance in the large population limit of the cumulative sum process mentioned above, it also appears in the Kolmogorov-Smirnov test for the equality of two distributions. In finance, they have arisen in the modelling of the so-called stock pinning effect (see

[2, 27, 3]), the modelling of arbitrage dynamics (see [10, 33]), and as the equilibrium price dynamics in a classical model of insider trading (see [4, 29] and more recently [5]).

Outside of the Brownian bridge, there has also been numerous examples in the extant literature considering optimal stopping problems with incomplete information about the underlying stochastic process. For example, optimal liquidation problems with an unknown drift were studied in [18] and [19], and with an unknown jump intensity in [34]. In the context of American-style option valuation, the effect of incomplete information on optimal exercise was also considered in [15], [24] and [41]. All of the above examples consider incomplete information about the parameters of a time-homogenous process. Here, however, the underlying dynamics are time inhomogeneous.

The remainder of this article is structured as follows. In Section 2 we formulate the optimal stopping problem under the assumption of a general prior distribution for the pinning time and investigate various structural properties of the solution. The cases of a gamma, beta, and two-point distributed prior are studied in further detail in Sections 3, 4 and 5, respectively. Finally, the article is concluded in Section 6 by considering the limiting case of the two-point prior in which one of the pinning times goes to infinity.

2. Problem formulation and filtering assumptions

1. Let be a Brownian bridge that pins to zero (without loss of generality) at some strictly positive time . The Brownian bridge is therefore known to solve the following stochastic differential equation

(1)

for and where is a standard Brownian motion defined on a probability space . We assume that is unknown to the optimal stopper but that they are able to glean information about its true value through continuous observation of the process . As such, we adopt a Bayesian approach in which the stopper has some prior belief about the pinning time, denoted , and updates this belief (via Bayes) over time. We further assume that is independent of under and that the process is absorbed at zero at (i.e., for all ). Such random horizon Brownian bridges have recently been studied in detail by [7], where it was shown (in Corollary 6.1) that the completed natural filtration generated by satisfied the usual conditions. It was also shown (in Proposition 3.1) that is a stopping time with respect to this filtration. Hence we let the filtration used by the stopper be the filtration generated by , denoted . We will further assume that the prior distribution

has a finite first moment to avoid certain technical issues. However we will return to this issue in Section

6 where we consider a specific example of a non-integrable pinning time .

2. The problem under investigation is to find the optimal stopping strategy that maximises the expected value of upon stopping, i.e.

(2)

where the supremum is taken over all -stopping times. In fact, since for all we can take stopping times without loss of generality (since the expected payoff would be the same otherwise). Moreover, since by assumption, the stopping time must also be integrable.

3. Next, recall that under the Bayesian approach the stopper will update their belief about the pinning time given continuous observation of the process . The details of this updating has recently been provided in [7] which motivates the following result.

Proposition 2.1.

Let satisfy . Then

(3)

for any and where

(4)
Proof.

The proof in the case of is given in the proof of Corollary 4.1 in [7]. To extend these arguments to a nonzero starting value we exploit the results of conditioning Brownian motion at time on the knowledge of its value at both an earlier and later time (cf. [16, pp. 116-117]) given by

where . Therefore, setting we arrive at the desired density, which can be used in place of the case in the proof of Corollary 4.1 in [7]. ∎

With this result in place we can obtain the dynamics of adapted to as follows.

Proposition 2.2.

For , the dynamics of can be written as

(5)

where is a -Brownian motion and

which can be expressed as

(6)

For , we have .

Proof.

Given the dynamics in (1) we define and the process

for . To show that is also an -Brownian motion we have the following arguments

and clearly . Hence we have

for . Next, to determine the required expression for we would like to use the results of Proposition 2.1 after setting . Unfortunately, for this particular choice of the required integrability condition is not satisfied in general, nor for our specific choices of made below. However, it can be shown that (3) is still valid for by applying Proposition 2.1 to a truncated version of this function and then passing to the limit. Specifically, let for some and assume that is such that . We can apply Proposition 2.1 to and take the limit as to yield

obtaining the desired expression for and completing the proof.∎

4. It is clear from its definition that (since ) and it can also be shown to have the following intuitive property.

Proposition 2.3.

Given as defined in (6) we have that is increasing for and decreasing for for any .

Proof.

Straightforward differentiation of (6) yields

where the definition of covariance has been used in the third equality above. Further, since and are both monotonically decreasing functions of , the conditional covariance above must be positive proving the claim. ∎

  • Proposition 2.3 demonstrates the intuitive result that a movement of away from zero gives information that the process is more likely to pin at a later time, i.e. that is larger and hence is smaller. In other words, learning about the unknown pinning time produces a decreased pinning force as moves away from zero. However, the term in the drift of (5) will result in an increased pinning force as deviates from zero. The overall affect of these two competing contributions to the drift is not clear in general and indeed we will find that different priors can result in vastly different behaviour of the function and hence the optimal stopping strategy in (2).

We also observe the following general properties of the function , which will be seen in the specific examples considered later. Firstly, Proposition 2.3 implies that the drift of the SDE in (5) satisfies the (sublinear) growth condition for some positive constant . Therefore it is known that (5) admits a weak solution (cf. Proposition 3.6 in [28, p. 303]). Secondly, since is not integrable at but is, it can be seen that at time points for which has a strictly positive density. Otherwise, . Thirdly, in the cases where , the drift function will be seen to have a non-zero (but finite) limit as . This limit appears difficult to analysis in general from (6) but can be seen clearly in the examples considered in Sections 3 and 4. A consequence of this non-zero limit is that a discontinuity appears in at (since is even, hence odd).

5. To solve the optimal stopping problem in (2) we embed it into a Markovian framework were the process starts at time with value . Hence the problem becomes

(7)

where are stopping times with respect to defined by

(8)

While (8) tells us the conditional dynamics of the process up to the random pinning time , it does not tell us when this time will actually occur. Hence the horizon of the optimal stopping problem in (7) is random and this must be incorporated into our analysis appropriately. To do so we assume from now on (without loss of generality) that the process is killed at time (and sent to some cemetery state if desired). Since the process can visit zero many times before eventually being killed there, we also observe that killing is elastic at zero, in the sense that the process is killed there only at some ‘rate’. The killing rate in the case when admits a continuous density is given by the following result.

Proposition 2.4.

If the distribution of admits a continuous density function with respect to Lebesgue, denoted by and with , then the infinitesimal killing rate for the process is given by

(9)

and denotes the Dirac delta function of at zero.

Proof.

Identifying as the (random) lifetime of the process it is well known (see, for example, [30, p. 130]) that the infinitesimal killing rate is given by

(10)

To obtain the probability required above we note that, under the assumption that has a continuous density with respect to Lebesgue, Theorem 3.2 in [8] states that the compensator of the indicator process admits the representation

(11)

where is as defined in (9) and denotes the local time at zero of the process up to time . It thus follows that

(12)

Using the identity (12) in (10) therefore yields

and the claim is proved.∎

The result above indicates that for continuous priors we have elastic killing of the process at zero and hence we should expect a jump in the -derivative of the value function across zero (cf. [11, p. 123]). We will delay further discussion of this feature to Sections 3 and 4 when we consider specific examples of a continuous prior. If the prior is not continuous then we expect the killing rate to be more involved and we defer this discussion to Section 5 when considering the specific example of a two-point prior.

6. Next, from (7) it is evident that for all . As such, we define the continuation region and the stopping region . These regions are of importance in the general theory of optimal stopping (see [37]) and, as we will see later, the structure of the optimal stopping region depends crucially on the prior distribution . For example, it can take the form of a simple one-sided boundary in the case of certain gamma and beta distributed priors (Sections 3 and 4) but can have a disconnected stopping region in the case of a simple two-point prior (Section 5). We will consider each of these cases in more detail below, but before this we discuss briefly some general properties of the optimal stopping region.

First, from (8) and an application of the optional sampling theorem we have that for any given stopping time

(13)

We note from (13) that, since , it will not be optimal to stop when is negative, hence . In other words, since it is known that the process will eventually pin (yielding a payoff of zero), it would not be optimal to stop and receive a negative payoff before this time. Given this fact, it is therefore evident that if a single optimal stopping boundary were to exist it could not be of the form , i.e. a single lower boundary. The following result provides a sufficient condition for the existence of a one-sided upper boundary.

Proposition 2.5.

(Condition for a one-sided upper stopping boundary). Assume that in (6) is such that the SDE in (5) admits a unique strong solution. If is non-increasing in then there exists a single upper stopping boundary such that .

Proof.

Letting we observe that, under the existence of a unique strong solution to (5), the trajectories of and do not cross before . Hence, expression (13) and an assumption that is non-increasing in , imply that

which after taking the supremum over stopping times yields

Therefore, implies that , and hence we can conclude that also; completing the proof. ∎

  • We emphasize that Proposition 2.5 requires knowledge that (5) admits a unique strong solution, which appears difficult to determine in general (only existence of a weak solution can be guaranteed by the sub-linear growth of the drift). The non-Lipschitz nature of the drift function at means that standard arguments to establish existence and uniqueness of a strong solution cannot be applied. However, existence of a unique strong solution can be established in the specific cases considered in Section 3 (a gamma distribution) and in Section 5 (a two-point distribution). Moreover, in Section 4 (a beta distribution), where a strong solution cannot be established, the existence of a one-sided boundary can be seen directly from the verification arguments provided (see Theorem 4.5).

7. To close this section we briefly review the solution to the classical Brownian bridge problem with a known pinning time which will be used in our subsequent analysis. When is known and fixed the stopping problem (7) has an explicit solution (first derived in [39] and later in [21]) given by

(14)

where

denotes the standard cumulative normal distribution function and

with the unique positive solution to

(15)

which is approximately 0.839924. Further, the optimal stopping strategy is given by for all and hence the optimal stopping region is given by

(16)

with denoting the continuation region.

  • It is intuitive that if in (7), then , where is given in (14), and consequently , where is given in (16). Formally, this can be seen by considering the value function in (7) if the true value of was revealed to the stopper immediately (at ). Denoting this value by it is clear that due to . Furthermore, since the set of stopping times when knowing the pinning time is larger than when not knowing the pinning time it is clear that and the stated inequality follows.

3. The case of a gamma distributed prior

1. It is perhaps most obvious to consider an exponentially distributed prior for

, however it appears that explicit computation of the function in (6) for such distributions is not possible. A related distribution for which can be computed explicitly however is a gamma distribution when for positive integers . Note that this distribution is supported on the semi-infinite interval , but that the pinning time is still integrable with . We also note that when the gamma distribution with

reduces to a chi-squared distribution of

degrees of freedom, i.e. . Therefore, this case encompasses chi-squared distributions with odd degrees of freedom, i.e.  for . For these distributions we have the following result.

Proposition 3.1.

Let and with and (for ) such that

(17)

The function in (6) can be calculated explicitly as

(18)

with

(19)

where is the modified Bessel function of the second kind (of order ). Hence the drift function in (5) is given by

(20)

where sgn is defined as

(21)
Proof.

In order to compute (6) with the density (17) we must evaluate the integral

with corresponding to the integral in the denominator of (6) and to the integral in the numerator. Letting , the integral above reduces to

(22)

We were not able to find an explicit computation of the above integral for arbitrary . However, it can be seen that if is a non-negative integer then the term can be expanded into integer powers of and we can apply the following known integral identity (cf. [17, p. 313])

(23)

valid for . Letting we can identify , , , and using and , respectively, to perform the integration we obtain the stated expression. ∎

Corollary 3.2.

When for the function in (18) becomes time independent and given by

(24)
Proof.

Setting in (19) reveals that

(25)

upon noting that , which produces the desired result. ∎

  • Corollary 3.2 reveals the remarkable property that when the dynamics in (5) are time homogeneous and dependent only on the sign of . As such, a movement of the process away from zero increases the stopper’s expected pinning time, and hence decreases the expected pinning force (via the term), by just enough to offset the increased pinning force due to the process being further way from zero (via the term).

We also observe from (19) that (since is increasing for ) and hence we conclude from (18) that , as expected from the strictly positive density of at for all . We also observe that the drift function in (20) has a discontinuity at since . However, despite this discontinuity, the SDE in (5) can be seen to have a unique strong solution since and the drift is bounded (cf. [42]). Finally, we note that in the case when , the SDE in (5) is often referred to as bang-bang Brownian motion or Brownian motion with alternating drift. Moreover, this process has arisen previously in the literature in the study of reflected Brownian motion with drift. In fact, the drawdown of a Brownian motion with drift (i.e., the difference between the current value and its running maximum) has been shown to be equal in law to the absolute value of bang-bang Brownian motion (see [25]). We also note here the recent work of [35] who consider discounted optimal stopping problems for a related process with a discontinuous (broken) drift.

2. Turning now to the optimal stopping problem in (7), we find that the time-homogeneity in the case when allows us to solve the problem in closed form. For all other values of considered in Proposition 3.1 the problem is time-inhomogeneous and must be solved numerically. We therefore restrict our attention to the () case and expose the solution there in full detail. The other cases are left for the subject of future research.

To derive our candidate solution to (7) when we observe from (24) that the function is non-increasing and thus we should expect a one-sided stopping region (from Proposition 2.5). Further, noting that has a continuous density, and computing the killing rate from (9), we see that , a constant. Therefore the optimal stopping problem becomes time-homogeneous and moreover we expect the optimal stopping strategy to be of the form for some constant to be determined. Under this form of stopping strategy the general theory of optimal stopping (see, for example, [37]) indicates that the value function and optimal stopping boundary should satisfy the following free-boundary problem.

(26)

where denotes the infinitesimal generator of . Recall that the derivative condition at represents smooth pasting and the derivative condition at is due to the elastic killing of the process at zero (cf. [11, p. 29]). Problem (26) can be solved explicitly to yield the following candidate for the optimal stopping value

(27)

where the optimal stopping threshold is given by .

Figure 1 plots the value function in (27) and the associated optimal stopping boundary for various values of . Note the kink in the value function at zero. We can also see that is decreasing and hence is also decreasing. This is consistent with the fact that and hence as increases the process is expected to pin sooner and the option value to stop smaller. We also observe that which is consistent with the fact that it would never be optimal to stop in this limit as the process would become a standard Brownian motion (which never pins).

x V beta b

Figure 1. On the left: The candidate value function for the case of given by (27) for (solid line) and for (dashed lines). Lower curves correspond to higher values of . On the right: The optimal stopping boundary .

3. We conclude this section with the verification that the candidate value function in (27) is indeed the solution to the optimal stopping problem.

Theorem 3.3.

(Verification). The value function defined in (27) coincides with the function defined in (7) with . Moreover, the stopping time is optimal.

Proof.

Setting to simplify the notation, we first note that the problem is time homogeneous and hence for an arbitrary stopping time ,

(28)

where the last equality above is due to (12) upon setting and and recalling that . Secondly, an application of the local time-space formula (cf. [36]), given that is continuous across but not across , yields

where is a local martingale and is a decreasing process since . Thirdly, combining (28) with the above equality we see that

(29)

where the last inequality follows from the optional sampling theorem upon noting that is bounded and hence is a true martingale. Consequently, taking the supremum over all admissible stopping times in (29) yields

To establish the reverse inequality note that, since and , both inequalities in (29) become equalities for . Thus

completing the proof. ∎

4. The case of a beta distributed prior

1. Another natural prior to consider is that of a beta distribution for . Such a distribution allows for the pinning time to occur on a bounded interval which is often the case in real life applications of Brownian bridges. Once more, while explicit computation of the function does not appear possible under a general beta distribution for arbitrary and , it is possible to obtain an explicit expression when (for any ) and when . Therefore, in this section we make the standing assumption that and . The case is also of particular interest since corresponds to the well-known arcsine distribution (see [26]). We further note that, without loss of generality, the beta distribution defined over can be taken (rather than over ) since the scaling and could be used otherwise. Finally to aid with the interpretation of our results, we recall that the unconditional expectation of can be calculated as and hence the expected pinning time is decreasing in with and .

Proposition 4.1.

Let and for , such that

(30)

The function in (6) can thus be computed explicitly as

(31)

where denotes Tricomi’s confluent hypergeometric function (cf. [40]).

Proof.

To compute under this distribution we must evaluate the integral

with corresponding to the integral in the denominator of (6) and to the integral in the numerator. Letting , the integral above reduces to

(32)

which can be computed explicitly by noting the following integral representation of Tricomi’s confluent hypergeometric function (see [1, p. 505])

valid for . Identifying , and we thus have

(33)