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Mirror and Synchronous Couplings of Geometric Brownian Motions

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.


Introduction
Let the process B = (B t ) t≥0 be a fixed standard Brownian motion and consider a standard Brownian motion V = (V t ) t≥0 on the same probability space. For any starting points x, y ∈ R, define the coupling time τ (V ) to be the first time the processes x + B and y + V meet. It is obvious that the synchronous coupling V = B maximises the coupling time as it makes it infinite almost surely (assuming x = y). Note further that the coupling time τ (V ) for any Brownian motion V cannot be smaller than the first time one of the processes x + B and y + V reaches level (x + y)/2. In the case of the mirror coupling V = −B, this random time actually equals τ (V ) and the coupling inequality becomes an equality. In particular, for any fixed T ≥ 0, the extremal Brownian motion in the optimisation problem, (1) minimise (resp. maximise) P(τ (V ) > T ) over all Brownian motions V , is given by the mirror (resp. synchronous) coupling, uniformly over all finite time horizons.
It is natural to investigate the following closely related problem for geometric Brownian motion: minimise the coupling time of the processes dX t = σ 1 X t dB t and dY t (V ) = σ 2 Y t (V ) dV t over all Brownian motions V on a given filtered probability space. The aim here is to maximise the probability of the event that X and Y (V ) couple before a given fixed time T . Since the processes X and Y (V ) are, at any time t, given by explicit deterministic functions of B t and V t respectively, the discussion above might suggest that mirror coupling of B and V should be optimal. Furthermore, since X and Y (V ) are martingales, the Dambis-Dubins-Schwarz representation of the difference X − Y (V ) intuitively suggests that the two processes will meet as early as possible if the coupling of the Brownian motions B and V is chosen so that the instantaneous volatility of X − Y (V ) is as large as possible. Equivalently put, the minimal coupling time should be achieved by the Brownian motion V which maximises (at every moment in time) the instantaneous quadratic Since the (random) Lebesgue density of the covariation measure d[B, V ] s on [0, ∞) is always between −1 and 1, it follows that the mirror coupling V = −B should be optimal. However, as we shall see, both of these intuitive arguments turn out to be false in general.
This paper investigates the problems of minimising and maximising the coupling time of two general (i.e. possibly non-martingale) geometric Brownian motions (GBMs) using a finite time, infinite time and ergodic average criteria. In the finite time horizon case we study the analogue of Problem (1) for GBMs and give a necessary and sufficient condition on the value function for the mirror (resp. synchronous) coupling to be optimal. This leads to an if-and-only-if condition on the parameters of the GBMs, which characterises the suboptimality (and hence optimality) of the mirror (resp. synchronous) coupling for any finite time horizon. In contrast to the intuitive arguments given above, this condition implies that mirror (resp. synchronous) coupling can be suboptimal in Problem (1) for GBMs even if the geometric Brownian motions are martingales.
This raises a natural question: is the exponential tail of the mirror (resp. synchronous) coupling optimal or, put differently, is the coupling efficient in the sense of [3]? We show that the mirror (resp. synchronous) coupling is efficient if and only if it is optimal, and hence may be inefficient.
In the case where the coupling is suboptimal, the proof of the aforementioned equivalence suggests the conjecture that the synchronous (resp. mirror) coupling is efficient in the minimisation (resp. maximisation) problem.
The stationary and infinite time horizon (for some "discount" rate q > 0) problems are given as the analogues of Problem (1) with P(τ (V ) > T ) replaced by respectively. It is clear that in the case of Brownian motion, the mirror (resp. synchronous) coupling is optimal according to both of these criteria. In this paper we prove that, unlike in the finite time horizon case, the same holds for all (possibly non-martingale) geometric Brownian motions. In particular this implies that the mirror coupling, which may be inefficient (i.e. has a thicker exponential tail than the optimal coupling), nevertheless minimises both the Laplace transform of the tail probability for any "discount" rate q and its ergodic average. Our proofs are based on Bellman's principle.
An application in mathematical finance of the coupling problems considered in the present paper can be described as follows. Assume that the performance of a portfolio manager is assessed at some fixed future time (e.g. one year from now) with respect to a benchmark security (e.g. some equity index), which evolves as a geometric Brownian motion X. Put differently, the remuneration of the manager depends on whether her portfolio, which evolves as Y (V ), exceeds the benchmark X in normalised terms. Assume also that the manager's mandate stipulates that, over the same time horizon, her portfolio may not exceed a pre-specified amount of realised variance. Both of these assumptions are realistic and are used extensively in practice, since the investor wants to beat the index but cannot tolerate arbitrary amounts of volatility in the mean time (e.g. investors like pension funds routinely stipulate such realised variance conditions). Imagine now a situation where the manager has a given amount of time, say T , before the evaluation of her performance, but is behind the benchmark by a certain amount. The question of how to trade in such a way The mirror coupling and the synchronous coupling of Brownian motions and related processes have attracted much attention in the literature. For example the classical book [7] and paper [8] introduce the mirror couplings of Brownian motions and diffusion processes (see also book [10] for the general theory of coupling). In [5] it is established that the mirror coupling is not the only maximal coupling, although it is the unique maximal coupling in the family of Markovian (also known as immersed) couplings. In [2] it is proved that the tracking error of two driftless diffusions is minimised by the synchronous coupling of the driving Brownian motions. In [6] generalised mirror coupling and generalised synchronous coupling of Brownian motions are introduced; the former minimises the coupling time and maximises the tracking error of two regime-switching martingales, whereas the latter does the opposite. Articles [1], [3], and [9] discuss various applications of the mirror coupling of reflected Brownian motions and other processes. In particular in [3], the notion of efficiency of a Markovian coupling, also used in the present paper, is studied in the context of the spectral gap of the generator of a Markov process.
The remainder of the paper is organised as follows. Section 2 describes the setting and basic notation, which is used throughout. Section 3 establishes the optimality of the mirror and synchronous couplings in the infinite time horizon (Section 3.1, Theorem 1) and stationary (Section 3.2, Proposition 5) problems. In Section 4 we characterise the optimality of the mirror and synchronous couplings over a finite time horizon (Section 4.1, Theorem 8) and analyse the efficiency of the two couplings (Section 4.2, Theorem 9). Appendix A contains a well-known lemma from stochastic analysis, which enables us to apply Bellman's principle.

Setting and notation
Let (Ω, F, (F t ) t≥0 , P) be a filtered probability space which is rich enough to support a standard be the set of all (standard) (F t )-Brownian motions on this probability space.
Let X = (X t ) t≥0 and Y (V ) = (Y t (V )) t≥0 be geometric Brownian motions, satisfying stochastic differential equations The Brownian motion B is fixed throughout and V is any element of the set V, defined in (2).
Define the coupling time of the two processes in (3) as

Stationary and infinite time horizon problems
3.1. Infinite time horizon problems. For any q > 0, we consider the following two problems: A simple integration by parts yields )/r for any nonnegative random variable τ and r > 0. Therefore Problems (qInf) and (qSup) are equivalent to finding Note also that if e q is an exponential random variable with E(e q ) = 1/q, independent of the filtration (F t ) t≥0 , then Problems (qInf) and (qSup) are equivalent to minimising and maximising P(τ (V ) > e q ) over V ∈ V, respectively. The following theorem holds.
Theorem 1. A solution to Problem (q±) is (for any q > 0) given by Remark 1. (i) Observe that by Theorem 1, the mirror coupling V (+) = −B solves Problem (qInf) and the synchronous coupling V (−) = +B is the solution to Problem (qSup).
(ii) Note that the solution depends neither on the parameters in (4) nor on the discount rate q.
3.1.1. Proof of Theorem 1. Observe that, due to the symmetry in Problem (q±), we may assume without loss of generality that the starting points x, y in (3)-(4) satisfy (x, y) ∈ D, where the set D ⊂ R 2 is given by Fix q > 0 and define the following function, closely related to the right-hand side in Problem (q±): The proof of Theorem 1 is in two steps: we first establish sufficient conditions for a function Ψ : D → R + implying that ±Ψ is equal to the right-hand side in Problem (q±) (Lemmas 2 and 3), and then prove that Ψ (±) in (7) satisfies these conditions (Lemma 4). Throughout the For any measurable function Ψ : D → R + and Brownian motion V ∈ V, consider the process (here and in the rest of the paper we denote s ∧ t := min(s, t)). Then the following lemma (a suitable version of Bellman's principle) holds.
for all V ∈ V (I {·} denotes the indicator of the event {·}). Since U (∓B, Ψ) is a martingale, for V (±) = ∓B this inequality becomes an equality and the lemma follows.
Our next task is to establish a verification lemma for Problem (q±). Let D • be the interior (in R 2 ) of the set D defined in (6). For any twice differentiable function f ∈ C 2,2 (D • ) we define the function L (±) f by the formula where (x, y) ∈ D • and f x , f y , f xx , f yy and f xy denote the partial derivatives of f . For any function Ψ : D → R + , such that Ψ ∈ C 2,2 (D • ), and Brownian motion V ∈ V, the local martingale Lemma 3. Assume the following hold: Proof. The definition of X and Y (V ) in (3) and Lemma 10 in the Appendix imply Itô's lemma, the assumptions in Lemma 3 and definition (8) for all (x, y) ∈ D and V ∈ V. Since X, Y (V ) and 1 ± C are non-negative processes and, by assumption (4), we have σ 1 σ 2 > 0, the integrand in the representation of ±U (V, Ψ) is non-positive, making ±U (V, Ψ) a P x,y -supermartingale. This representation, together with assumption (III), implies that U (∓B, Ψ) is a P x,y -martingale.
Note the following equivalence: It is clear that under condition (11) Theorem 1 holds. Lemmas 2 and 3 imply that in order to establish Theorem 1 in general, it is sufficient to prove that, when (11) fails, the function Ψ (±) : D → R + in (7) satisfies the assumptions of Lemma 3. More precisely, the following lemma holds.  (7), if for some (x, y) ∈ D • we have P x,y (τ (∓B) = ∞) < 1.
Proof. Under the assumption of the lemma, the following representation holds: and σ ± and µ are defined in (5). Since, by assumption, the condition on the right-hand side in (11) is not satisfied, the equality σ ± = 0 implies µ > 0, making k ± a well-defined real number.
Formula (12) follows from the fact that τ (∓B) has the same law as the first-passage time of the Brownian motion with drift (σ ± B t + µt) t∈[0,∞) over the level log(x/y). The Laplace transform of this random time is given in [4, p. 295] and amounts to the right-hand side of (12).
The formula in (12) and some simple calculations imply that for (x, y) ∈ D • the following holds: and Hence assumption (II) of Lemma 3 is also satisfied. The equalities in (13) and the definition in (10) of the local martingale M (V, Ψ (±) ) imply that the integrands in the stochastic integrals are bounded processes and therefore square integrable. Hence M (V, Ψ (±) ) is a P x,y -martingale for all (x, y) ∈ D and V ∈ V and assumption (III) of Lemma 3 also holds.

Stationary problems. Note first that Fubini's theorem and the Dominated Convergence
Theorem imply the existence of the limit: Hence the stationary problems from the introduction can be rephrased as: find V inf ∈ V and A solution to these problems, independent of the values of the parameters of the geometric Brownian motions in (3), is given in the following proposition. Note in particular that, unlike in the finite time horizon case, no new phenomena arise when the ergodic average criterion is used (i.e. the solution is completely analogous to the infinite time horizon case). Proof. As in Section 3.1.1 we may assume that, due to symmetry, the starting points of X and Y (V ) satisfy (x, y) ∈ D (see (6)). By (3) and the definition of τ (V ) in Section 2 we have where µ is defined in (5) and the convention inf ∅ = ∞ is used. If x = y we have τ (V ) = 0 for all V ∈ V and Proposition 5 follows. So we can assume (x, y) ∈ D • in the rest of the proof.
In the case µ ≤ 0 we return to the formulation of Problems (SInf) and (SSup) above. Observe that Theorem 8(b) below yields the optimal couplings that minimise and maximise the probability P(τ (V ) > t) for every t ≥ 0. Since the couplings are independent of t, they also minimise and maximise the stationary criteria in Problems (SInf) and (SSup), which concludes the proof.
Remark 2. The proof of Proposition 5 relies in an obvious way on Theorem 8(b) below. We would like to stress that there is no circularity in this argument since Proposition 5 is not used in Section 4. Stationary problems are considered in Section 3 rather than later on in the paper, because the structure of the solution is the same as that of the infinite time horizon problems.

4.1.
Finite time horizon problems. Retain the setting and notation from Section 2. For any T > 0, consider the following problems:

As in Section 3, we can reduce Problem (T±) to the case where diffusions in (3) start at (x, y) ∈ D,
where D is given in (6). Define the set E := D × [0, T ] and recall that the value function for Problem (T±) is defined by Based on the results in Section 3, one might expect that ±Φ (±) , where Φ (±) (x, y, t) := P x,y (τ (∓B) > t) , would be the value function for Problem (T±). In order to investigate this, we define the function where (x, y, t) ∈ E • and f x , f y , f t , etc. denote the partial derivatives of f . For any sufficiently smooth function Φ : E → R + and any Brownian motion V ∈ V, we define the local martingale The following proposition provides the key ingredient in the proof of Theorem 8 below.
is a P x,y -submartingale for any V ∈ V and (x, y) ∈ D (proof as in Lemma 3). For any t ∈ [0, T ], the boundary conditions in assumption (i) imply Hence, for any (x, y) ∈ D and V ∈ V, the submartingale property yields the inequality As in Lemma 2, this establishes the implication (note that, unlike Lemma 2, in this case we do not need, and in fact do not have, the continuity of Φ on E).
The continuity of Φ xy implies that there exists r > 0, such that Φ xy is strictly negative on the ≤ T P x,y -a.s. and P x,y (τ 2 ) > 0 (there is a slight abuse of notation in the definition of τ (±) 2 as it is assumed that the process Y (±B), defined in (3), is driven by the Brownian motion ±B as indicated, but started at the random time τ Define the processṼ (±) = (Ṽ (±) t ) t∈[0,∞) by the following formula:
We will now apply Proposition 6 to study the question of whether ±Φ (±) , defined in (17), is the value function for Problem (T±).
Proof. The explicit formula for the distribution of the running maximum of a Brownian motion with drift (see e.g. [4, p. 250]) yields the following representation of the function in (17): where, for any z ≥ 0 and s > 0, we define Simple (but tedious) calculations using this representation yield the properties required in assumptions (i)-(iii) of Proposition 6. Indeed, note that the partial derivatives h These formulae and the representation in (20) imply the formula for Φ (±) xy (x, y, t), as well as assumptions (i) and (ii) of Proposition 6. The martingale property of the process in (18) (i.e. assumption (iii) in Proposition 6) follows by Itô's isometry from the fact that both functions xΦ (±) x (x, y, t) = h (±) z (log(x/y), t) and yΦ (±) y (x, y, t) = −h (±) z (log(x/y), t) are bounded on E. This completes the proof of the lemma.
We are now ready to prove that the mirror (resp. synchronous) coupling of the driving Brownian motions in (3) is not necessarily optimal in Problem (T+) (resp. (T−)). In Theorem 8, we give a necessary and sufficient condition for the function ±Φ (±) , defined (17), to be the value function for Problem (T±).
Theorem 8. Recall that µ and σ ± are given in (5). Then the following holds for any positive time horizon and distinct starting points: (a) If µ > 0 and σ ± = 0, then V (±) = ∓B does NOT solve Problem (T±). Remark 4. We thank the referee for noting that Theorem 9 holds without assuming σ ± = 0.
Proof. The following bounds hold for the standard normal distribution function N (·), The first inequality follows from the identity ∞ r (1 + y −2 )e −y 2 /2 dy = r −1 e −r 2 /2 for all r > 0, and the second is given in (22).
Assume now that σ ± = 0. Let and note that for all large t > 0 we have Z(t) < Z(t) < 0 and the equality n (Z(t)) = x y holds. The representations in (20) and (21) imply The inequalities in (24) imply the limit In order to deal with the second factor on the right-hand side of (26), we note the following inequalities, n( Z(t))Z(t) which are a consequence of two applications of the second inequality in (24) and identity (25).
Definition (5) and assumption σ ± = 0 imply |σ + | > |σ − | > 0 and hence µ/(2σ 2 + ) < µ/(2σ 2 − ). The mirror coupling is therefore not efficient for Problem (T+) since it has a strictly thicker exponential tail than the synchronous coupling. Likewise, the synchronous coupling is not efficient for Problem (T−), which requires the thickest possible exponential tail among all couplings, since it has a thinner tail than the mirror coupling.
Remark 5. It is the presence of the positive drift µ > 0 that makes the mirror coupling suboptimal in Problem (T+) (see Theorem 8). The proof of Theorem 9 suggests that if the drift µ is strictly positive and the time horizon T is large, it is in fact better (according to the exponential tail criterion) to use synchronous coupling. This naturally leads to the following conjecture: If µ > 0, the synchronous (resp. mirror) coupling is efficient in Problem (T+) (resp. (T−)).

Appendix A. Family of Brownian motions on a filtered probability space
Recall that V is defined in (2). See e.g. [6, Lemma 2.1] for the proof of Lemma 10.
Lemma 10. For any Brownian motion V ∈ V, there exists an (F t )-Brownian motion W ∈ V and a process C = (C t ) t≥0 , such that B and W are independent, C is progressively measurable with −1 ≤ C t ≤ 1 for all t ≥ 0 P-a.s., and the following representation holds: Remark 6. The proof of this lemma requires the existence of a Brownian motion B ⊥ ∈ V that is independent of B. If our probability space did not support such a Brownian motion, we could enlarge it, which would only increase the set V. Since the optimal Brownian motions in Theorems 1 and 8(b) are constructed from B alone, they would also have to be optimal in the original problem.
Therefore we can assume that B ⊥ exists.