Riesz representation and optimal stopping with two case studies

Riesz representation and optimal stopping with two case studies

\snmSören Christensen,label=e1]sorenc@chalmers.se [    \snmPaavo Salminenlabel=e2]phsalmin@abo.fi Research supported in part by a grant from Svenska kulturfonden via Stiftelsernas professorspool, Finland[ Chalmers University of Technology and Göteborg University Sören Christensen
Department of Mathematical Sciences
Chalmers University of Technology and
Göteborg University
SE-412 96 Göteborg
Sweden
\printeade1
Paavo Salminen
Faculty of Science and Engineering
Åbo Akademi University
FIN-20500, Åbo
Finland
\printeade2
Abstract

In this paper we demonstrate that the Riesz representation of excessive functions is a useful and enlightening tool to study optimal stopping problems. After a short general discussion of the Riesz representation we concretize, firstly, on a -dimensional and, secondly, a space-time one-dimensional geometric Brownian motion. After this, two classical optimal stopping problems are discussed: 1) the optimal investment problem and 2) the valuation of the American put option. It is seen in both of these problems that the boundary of the stopping region can be characterized as a unique solution of an integral equation arising immediately from the Riesz representation of the value function. In Problem 2 the derived equation coincides with the standard well-known equation found in the literature.

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Riesz representation and optimal stopping {aug}

class=AMS] \kwd[Primary ]60G40 \kwd60J25, 62L15 \kwd[; secondary ]60J30

geometric Brownian motion, convex set, resolvent kernel, duality, integral representation for excessive function, optimal investment problem, American option, integral equation

1 Introduction

An optimal stopping problem (OSP) can be formulated as follows: Find a function (value function) and a stopping time (optimal stopping time) such that

(*)

where is a strong Markov process taking values in is the time horizon of the problem, is the set of all stopping times in the natural filtration of with values in , and the function (reward function) is often assumed to be non-negative and continuous. In case, in (*1) we define

Notice that we use boldface letters to denote non-random vectors and matrices.

Optimal stopping problems arise naturally in many different areas, such as stochastic calculus (maximal inequalities), mathematical statistics (sequential analysis), and mathematical finance (pricing of American-type derivatives and real options), for these applications and further references, see, e.g.. the monographs [45] and [41]. An explicit solution for optimal stopping problems is often hard to find. Most examples are such that the underlying process is one-dimensional, often a diffusion process, and the time horizon is infinite, see e.g. [43, 4, 16] and the references therein. In contrast, the class of explicit examples with a multidimensional underlying process or with finite-time horizon are very limited. In this article, we describe a solution method for such problems based on the Riesz representation of the excessive functions. Notice that finite-time horizon problems with one-dimensional underlying process may be seen as two-dimensional where, in fact, we use the space-time process as the underlying.

More precisely, we consider the classical problem of optimal timing for an irreversible investment decision under the assumption that the revenue and cost factors follow (possibly correlated) geometric Brownian motions. It is furthermore assumed that the cost factors consist of many different sources making the model more realistic. For the mathematical formulation, see the expression for the value function in (3.1) in Section 3.

This problem has been studied extensively over the last decades, see, e.g., [30, 37, 23, 34, 21, 22, 12], and the references therein. However, no explicit description of the optimal stopping set is known so far to the best of our knowledge. We remark that in [23] a closed form solution was presented under certain conditions on the parameters and the optimal stopping time was claimed to be a hitting time of a halfspace. Unfortunately, it turned out that this closed form solution is only valid in trivial degenerated cases if the dimension is greater than one, see [12] and [34]. Because the structure of the reward function is additive and not multiplicative, there is no hope for such an easy solution in dimensions .

Our contribution hereby is to give an implicit description of the stopping region via an integral equation which has the boundary curve of the stopping region as a unique solution. It is seen in Section 3.6 that the equation has a fairly simple form especially in the two-dimensional case. We also present an ad hoc numerical metod for solving the integral equation.

The optimal investment problem in one dimension and with finite horizon is equivalent with the optimal stopping problem for finding the price of an American put option. To characterize the exercise boundary analytically and to develop numerical algorithms for finding it explicitly is an important and much studied topic in mathematical finance with the origin in McKean [31]. We refer to Peskir and Shiryayev [41] pp. 392-395 for a discussion with many references. Our main object of interest is an integral equation for the exercise boundary derived at the beginning of the 1990s in the papers by Kim [27], Jacka [24], and Carr, Jarrow and Myneni [8]. See also Myneni [33], Karatzas and Shreve [26], Peskir and Shiryayev [41], and Pham [42], Lamberton and Mikou [29] for the problem with underlying jump diffusions. The uniqueness of the solution of the equation was proved by Peskir [39] using a delicate stochastic analysis involving local times on curves. The method presented in this paper results to the same equation and we offer here a proof for the uniqueness based on the uniqueness of the representing measure in the Riesz representation of the value function.

To briefly motivate our approach, recall that a non-negative, measurable function is called -excessive for if the following two conditions hold:

Assuming that has continuous sample paths and the reward function is lower semicontinuous and positive satisfying the condition

it can be proved that the value function exists and is characterized as the smallest -excessive majorant of see Theorem 1 p. 124 in Shiryayev [45]. Moreover, if is continuous, the optimal stopping time is then known to be the first entrance time into the set

(1.1)

called the stopping region. For the finite time horizon problem, analogous results hold for the space-time process since herein the first co-ordinate can also be seen as a (deterministic) Markov process. To utilize these basic theoretical facts to solve explicit problems of interest, we need a good description of -excessive functions. Such a description – the Riesz representation – is discussed in the following section with emphasis on geometric Brownian motion. From Section 3 onward the paper is organised as follows. In Section 3 we study the optimal investment problem. Section 4 is on American put option and the paper is concluded with an appendix where proofs of some more technical results are given.

2 The Riesz representation of excessive functions

Our basic tool in analyzing and solving OSP is the Riesz representation of excessive functions according to which an excessive function can be written as the sum of a potential and a harmonic function. For thorough discussions of the Riesz representation and related matters in a general framework of Hunt processes, see Blumenthal and Getoor [5] and Chung and Walsh [14]. A more detailed representation of excessive functions is derived in the Martin boundary theory which, in particular, provides representations also for the harmonic functions, see Kunita and Watanabe [28] and Chung and Walsh [14] Chapter 14. For applications of the Riesz and the Martin representations in optimal stopping, see Salminen [43], Mordecki and Salminen [32], Christensen and Irle [12], and Crocce and Mordecki [15]. We also remark that in Christensen et al. [13] an alternative representation of excessive functions via expected suprema is utilized to characterize solutions of OSPs and, moreover, the connection with the Riesz representation is studied.

2.1 Multi-dimensional geometric Brownian motion

Let

be a -dimensional Brownian motion started from such that for

It is assumed that the non-negative definite matrix with is non-singular. A -dimensional geometric Brownian motion is a diffusion in with the components defined by

where for The differential operator associated with is of the form

where The (row) vector and the matrix are called the parameters of

To be able to apply the Riesz representation on this process should satisfy some regularity conditions. Firstly, we note that is a standard Markov process, see [5] p. 45. Secondly, has a resolvent kernel given by

(2.1)

where and is a transition density of The following proposition shows that the transition density may be taken with respect to a measure such that the corresponding resolvent is self-dual, i.e., in duality with itself relative to this means that relationship (2.2) below holds (see, e.g., [14] p. 344). Note that, since turns out to be absolutely continuous with respect to the Lebesgue measure, it is a matter of standardization to choose the Green kernel with respect to or with respect to the Lebesgue measure. In the following, we consider with respect to and use the notation

where satisfies some appropriate measurability and integrability conditions.

Proposition 2.1.

A -dimensional geometric Brownian motion as introduced above is self-dual, i.e. for all nonnegative measurable functions and it holds that

(2.2)

where is the measure on with the Lebesgue density

, , and denotes transposition.

Proof. See Section A1 in the appendix.

From the self duality it follows that Hypothesis B in [28, p. 498] holds. Notice also that since the dual resolvent kernel is identical with the resolvent kernel of the process associated with the dual resolvent is a standard Markov process identical in law with Consequently, the following (strongest) form of the Riesz representation theorem holds.

Theorem 2.2.

Let be a locally integrable -excessive function for a -dimensional geometric Brownian motion Then can be represented uniquely as the sum of a (non-negative) -harmonic function and an -potential . For the potential there exists a unique Radon measure depending on and on such that for all

(2.3)

Moreover, if is an open set having a compact closure in then is -harmonic on if and only if

We remark that the uniqueness of follows from the fact that is a self dual standard process (see, e.g., [28, Proposition 7.11 p. 503]). The statement about -harmonicity on can be deduced from ibid. Proposition 11.2 p. 513. Recall also that a non-negative measurable function is called -harmonic on if for all

(2.4)

where

In general, it is often difficult to find explicit expressions for the harmonic function in the Riesz decomposition presented in Theorem 2.2. The following proposition gives an easy condition under which vanishes.

Proposition 2.3.

Let be a bounded -excessive function for a -dimensional geometric Brownian motion . Then in the integral representation of given in Theorem 2.2.

Proof.

Take a sequence of compact subsets of such that as . Then, due to the boundedness of and the -harmonicity of , it holds that

In case is smooth enough the representing measure  can be obtained by applying the differential operator on This is made precise in the next

Proposition 2.4.

Let be a bounded -excessive function for a -dimensional geometric Brownian motion such that and let be a convex set with on . Furthermore, assume that is locally bounded around . Then the representing measure for on in the integral representation (2.3) is absolutely continuous with respect to the measure and is given for by

Proof. See Section A2 in the appendix.

2.2 Space-time geometric Brownian motion

We now consider a one-dimensional geometric Brownian motion in space-time, that is, the two-dimensional process with the state space The differential operator associated with is

(2.5)

We remark that the definition of an -excessive function for can be written in the form

where denotes the expectation operator associated with The resolvent kernel of can be defined as follows

(2.6)

where the transition density is taken with respect to the speed measure Notice that the kernel is also defined for

Proposition 2.5.

Let be an -excessive function of locally integrable on with respect to where denotes the Lebesgue measure. Then there exists a unique Radon measure on such that for

(2.7)
Proof.

It is proved in the appendix, see Section A3, that there exists a (dual) resolvent kernel such that for non-negative and measurable and

(2.8)

It can be checked then that Hypothesis (B) in Kunita and Watanabe [28, p. 498] holds. Consequently, see ibid Theorem 2 p. 505, has the Riesz representation

(2.9)

where is a Radon measure, is a harmonic function and the integration is over the set consisting of the points in the state space for which is a potential. The representation of in (2.9) as the sum of a potential and a harmonic function is unique. Moreover, the representing measure is unique, cf. [28, p. 503] . To deduce (2.7) notice firstly that since is a potential for all This follows readily from the definition of where it is stated that for Secondly, applying the Martin boundary theory (we omit the details) it can be proved that the harmonic function in (2.9) has the representation

(2.10)

where is a Radon measure on Also here the representing measure is uniquely determined by Combining (2.9) and (2.10) yields (2.7). ∎

Remark 2.1.

The proof of the duality does not use any particular properties of geometric Brownian motion (see Appendix). Consequently, the uniqueness of the representing measure in (2.7) holds for general space-time one-dimensional diffusions.

For -excessive functions that are smooth enough, we can describe the form of the measure more explicitly. The following result is useful generalization of [44, Proposition 2.2] based on Alsmeyer and Jaeger [3].

Proposition 2.6.

Let be a bounded -excessive function for on such that and are continuous on , and is absolutely continuous as a function of the second argument. Then the representing measure for on in the representation (2.7) is absolutely continuous with respect to the Lebesgue measure on and is given by

Proof.

By [17, Teorema 8.2] (see [18, Theorem 8.2] for an English translation) it holds that for all continuous functions with compact support in we have

Therefore, we have to prove that for

(2.11)

exists and is equal to . From [3, Corollary 2.2] it is seen that Itô’s formula can be applied to obtain -a.s.

Using a stopping argument and taking expectations yield the existence of the limit in (2.11) and, hence, the claim is proved. ∎

Remark 2.2.

The regularity assumptions in Proposition 2.4 are fairly strong and sometimes difficult to check. However, these are possible to relax by applying other extensions of the Ito formula (without the local time terms).

2.3 Basic idea of using the Riesz representation for solving optimal stopping problems

There is a wide range of different approaches for solving OSPs. Many of these are based on considering candidates for the value function of the form

(2.12)

for candidate sets and associated first hitting times , and then finding properties of the true value function that characterize one candidate set as the optimal stopping set. This idea can then be translated into a free-boundary problem, as described extensively in the monograph [41]. One of the major technical problems in using this approach is that a priori the candidate functions are typically not smooth on the boundary of , so that it is not straightforward to apply tools such as Itô’s formula or Dynkin’s lemma.

Our idea for treating OSPs using the Riesz representation theorem can basically be described as follows (for the infinite time horizon): Using the general results presented earlier in this section, we first show that the value function can be written in the form

for some known function . Now, in contrast to (2.12), we characterize the unknown stopping set by considering candidates for the value function of the form

(2.13)

and then identify one candidate set as the optimal stopping set. From a technical point of view, these candidate solutions are easy to handle, since the strong Markov property immediately yields that a variant of Dynkin’s formula holds true for all , see Lemma 3.6 below.

To show the applicability of this approach for treating concrete problems of interest, we concentrate in this article on two case studies, namely the multidimensional optimal investment problem with infinite time horizon in Section 3 and the American put problem in Section 4. Notice that the latter problem can be viewed as the optimal investment problem under a finite time horizon with .

3 Optimal investment problem

In this section we concentrate on one of the most famous OSPs in continuous time with multidimensional underlying process: the optimal investment problem, which goes back to [30]. The value function associated with the optimal investment problem is given by

(3.1)

Here, we assume the time horizon to be infinite, i.e. , , is a weight vector, and is a -dimensional geometric Brownian motion as defined in Subsection 2.1 with the indices and running from 0 to As discussed in [34] and [12], we furthermore assume that to guarantee the value function to be finite and the optimal stopping time not to be infinite a.s.

3.1 Problem reduction

First, by the explicit dependence of on the starting point, we see that for all , where we write . Therefore, we may take . Because the reward function is homogeneous, it is standard to reduce the dimension of the problem, see e.g. [34]. To recall this briefly, notice that for all and all it holds that

where are geometric Brownian motions under the measure given by

To be more explicit, under has drift and volatility . Consequently, we may, without loss of generality, take (a positive constant).

To summarize, we consider the optimal stopping problem

(3.2)

that is, an optimal stopping problem of the form (*1) with

where, for notational convenience, the problem (3.2) is formulated for under (instead of the transformed process ). A typical assumption in the literature for this problem is that for , see [23, 37]. This guarantees that the optimal stopping time is a.s. finite. Since some arguments can be shortened (see e.g. Lemma 3.1), we also use this assumption throughout Section 3.

3.2 Preliminary results

We first collect some elementary results of the optimal stopping set as defined in (1.1). Similar results and lines of argument can also be found in [37, 38].

Lemma 3.1.
  1. is a subset of

  2. is a closed convex set.

  3. is south-west-connected, that is if , then so is for all , where we understand componentwise.

Proof.

For note that if , then the reward for immediate stopping in is 0; since, obviously, , cannot be in the optimal stopping set. Furthermore, if , then is -subharmonic in a neighborhood of , so that at it is also not optimal to stop. Moreover, since and are continuous (for the latter claim, see, e.g., [5] p. 85) it follows that cf. (1.1), is closed. For convexity, take , and let be a stopping time. Then we have

which proves that , i.e., is convex. For , note that for and , it holds for all stopping times

where we used for the second inequality that, by our assumption on the drift, i.e., for all and since , the process is a nonnegative supermartingale. This proves that . ∎

3.3 Integral representation of the value function

By the general theory of -excessive functions described in Section 2, we know that the value function has the representation

(3.3)

where is an -harmonic function. This is our starting point for solving the optimal stopping problem (3.2). We check first that the value function has enough regularity. This is formulated in the next lemma.

Lemma 3.2.

It holds that and . Furthermore, is locally bounded around , i.e. for each , there exists such that is bounded on

Proof. See Section A4 in the appendix.

Using now the results obtained in Subsection 2.1 we obtain the explicit form of the integral representation of the value function.

Theorem 3.3.

For all it holds that

(3.4)

where

Proof.

First note that in the representation (3.3), the measure vanishes on since is -harmonic on the continuation set. Since is bounded, so is . Using this fact together with Lemma 3.2 and Lemma 3.1, Propositions 2.3 and 2.4 are applicable and yield that and on

But since on , we obtain

which gives the result. ∎

Evaluating (3.4) at we obtain the following corollary.

Corollary 3.4.

For all

(3.5)

Note that a description of the stopping boundary in (3.5) is very natural and similar results arise in many other treatments of optimal stopping problems, in particular with a finite time horizon, see, e.g., the examples given [41].

3.4 Uniqueness of the solution of the integral equation

In the previous section we have found the identity (3.5) which can be seen as an equation for the unknown boundary of the stopping set. When analyzing this equation from a purely analytical point of view, there does not seem to be much hope that this equation would characterize the manifold uniquely. However, using a probabilistic reasoning based on our integral representation, we now show that is indeed uniquely determined by (3.5). More precisely we prove

Theorem 3.5.

Let be a nonempty, south-west connected, convex set such that

and assume that for all it holds

(3.6)

Then .

In the proof of this theorem, we frequently make use of the following well-known version of Dynkin’s formula for functions of form (3.3). The proof is an easy application of the strong Markov property and can be found in Section A5 in the appendix.

Lemma 3.6.

Let be a measurable function and

Then for each stopping time and each

(3.7)
Proof of Theorem 3.5.

Write for

where We proceed in four steps:

  1. for all :
    Let . Using Lemma 3.6 and assumption (3.6) we obtain for all

    where we applied the continuity by monotone convergence (since is bounded and is continuous for all ) of and in the second equality and Dynkin’s formula to obtain the last equality.

  2. for all :
    For the inequality holds by step 1. Now, let and write . Using again Lemma 3.6 yields

    since is the value function.

  3. :
    Let and write . Then by step 1, Lemma 3.6, and step 2

    On the other hand, by Lemma 3.6 applied to , i.e. to ,

    Subtracting yields that . Since this equation holds for all and on , we obtain

    (3.8)

    Now, if there would exist , by the closeness of , the boundary of the rectangular solid

    has a positive surface area in . By the southwest connectedness of , it holds that and , in contradiction to (3.8). This proves .

  4. :
    Let be the optimal stopping time, that is, Then for all it holds by step 2 and Lemma 3.6 that

    Since we obtain by step 1 that , so that

    hence as above for all , i.e., .

Remark 3.1.

In the previous proof, we use a similar structure as, e.g., in the proof of the uniqueness in [39].

3.5 Solution of the investment problem in case

To understand how Theorem 3.5 can be used to solve OSP (3.2) explicitly, we consider the case and In other words, we consider the problem connected to pricing a perpetual American put with strike price in a Black-Scholes market. We refer to [31] for an early treatment of the problem. Hence, the underlying process is the geometric Brownian motion with drift parameter (the risk neutral interest rate) and volatility Recall (see Borodin and Salminen [6]) that the (symmetric) Green kernel with respect to the speed measure is given by