Three essays on stopping

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This modifies a formula by Perry et al (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown threshold, if and only if the diffusion characteristic mu/sigma^2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum at a fixed drawdown threshold is exponentially distributed for any spectrally negative L\'evy process, a result due to Mijatovic and Pistorius (2012).

This paper comprises three essays on stopping. In section 1, we compute the Laplace transform of the first hitting time of a fixed upper barrier for a reflected Brownian motion with drift. This expands on, and corrects a result by Perry et al. (2004).
In section 2 we show, by using an intrinsic delay differential equation, that for a diffusion process, the maximum before a fixed drawdown threshold is generically exponentially distributed, only if the diffusion characteristic µ/σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). We further construct diffusions, where the exponential law only holds for specific drawdown sizes.
Section 3 uses Lehoczky (1977)'s argument to show that the maximum before a fixed drawdown threshold is exponentially distributed for any spectrally negative Lévy process, the parameter being the right-sided logarithmic derivative of the scale function. This yields an alternative proof to the original one in Mijatović and Pistorius (2012).
1 The first hitting time for a reflected Brownian motion with drift Let X be a reflected Brownian motion on [0, ∞), with drift µ and volatility σ. By Graversen et al. (2000) the RBM(µ, σ 2 ) can be realized as |ξ x t |, where ξ x is the unique strong solution of dξ t = µ sign(ξ t )dt + σ dB t , ξ 0 = x, where B is a standard Brownian motion. 1 We therefore assume, in the following, a filtered probability space given that supports B, and identify X = (X t ) t≥0 with the X t = |ξ x t |, x ≥ 0. By Tanaka's formula, we have where L 0 (X) is the local time of X at 0. Since the latter is supported on {X = 0}, Itô's formula implies for any f ∈ C 2 b ((0, ∞)), for which f ′ (0+) = 0, the process is a martingale, where A is the differential operator, defined by Af = σ 2 2 f ′′ (x) + µf ′ (x). 2 Since, before reaching the boundary 0, the process cannot be distinguished from a Brownian motion with drift, for 0 < δ + x < x, the first hitting time τ δ := inf{t ≥ 0 | X t = δ + x} equals, in distribution, to the first hitting time τ δ of a Brownian motion with drift, starting at x. Therefore, we may confine ourselves to computing τ δ for barriers δ + x, where δ > 0. Our aim is to compute the Laplace transform Theorem 1.1. For δ ≥ 0, the Laplace transform of the first hitting time of a reflected Brownian motion with drift µ and volatility σ is given by Proof. Pick Φ ∈ C ∞ c (R) such that Φ(ξ) = 1 for |ξ| ≤ x + δ. Furthermore, let κ > 0, then for any θ ≥ 0 and t ≥ 0, the function F (t, x) := e −θt Φ(x) e −κx + κx satisfies f := F (t, ·) ∈ C 2 b and f ′ (0) = 0. According to the introductory notes of this section, the process F (t, X t ) − t 0 ∂ s F (s, X s )ds − t 0 AF (s, X s )ds is a uniformly bounded martingale, and therefore also the stopped process is a true martingale, which starts at zero, P x -almost surely. Using the fact that Φ(X t∧τ δ ) = 1, we find that the stopped process satisfies for any t ≥ 0, Letting t → ∞, we thus get by optional sampling, For the two choices κ ∈ {κ − , κ + }, where we thus obtain two equations, for two unknown moments, Solving this linear system for the involved moments yields the Laplace transform of τ δ , equation (1).

Sanity Check: driftless case
For a first "sanity check" of Theorem 3.2, we compute the LT (1) independently when µ = 0 and x = 0. In this case, the reflected Brownian motion is equal to |σB| in law, where B is a standard Brownian motion. But then τ δ equals, in distribution, to Now it is well known that the Laplace transform of τ δ is given by which indeed coincides with (1) for µ → 0 (that is, zero Sharpe Ratio, zero absorption and null killing).  (Perry et al., 2004, Formula (5.2)) state a different Laplace transforms than our Theorem 3.2. Letting µ → 0 in (Perry et al., 2004, Formula (5.2)) indeed yields (σ 2 = 1) which contradicts (2). The proof of (Perry et al., 2004, Lemma 5.1) can however not be rectified, by merely fixing the (obviously) missing factor of 1/2. Indeed, in the second line of their proof, they forget a factor e −κW (s) in the second integrand, and thus by inserting special values of κ into the process in line 2, one does not get rid of the local-time term, as claimed.
2 Diffusions with exponentially distributed gains before fixed drawdowns Let X be a diffusion process on the [−a, ∞), satisfying the SDE where µ(x) and σ(x) are locally Lipschitz continuous functions of linear growth on [−a, ∞), and σ(x) > 0 thereon. For a threshold 0 < δ ≤ a, we define M δ as the maximum of X, prior to a drawdown of size δ, that is . The following is due to Lehoczky (1977): Caution is needed when interpreting the original paper Lehoczky (1977): Lehoczky uses the letter "a" for three different objects: The drift µ(x) is denoted as a(x), while a is the left endpoint of the interval of the support of X; third, the threshold δ in his paper is also called a. An inspection of Lehozky's proof reveals that our more general version with δ ≤ a holds.
In terms of diffusion characteristics, Lehoczky's result holds in a more general context. First, the assumption of locally Lipschitz coefficients are too strong, and can be relaxed. For example, we can relax to Hölder regularity of σ(x) of order no worse than 1/2, due to Yamada et al. (1971). Also, we can allow reflecting or absorbing boundary conditions, thus include reflected diffusions. For instance, Proposition 2.1 holds for a Brownian motion with drift, starting at 0 and being reflected at −a, because, the process X cannot hit −a, before it reaches a strictly positive maximum, due to strict positive volatility σ(0) > 0.
From (4) it can be seen that when µ/σ 2 is constant, M δ is exponentially distributed (the special case for for a Brownian motion with drift is due to Taylor (1975), and independently discovered by Golub et al. (2016)). Mijatović and Pistorius (2012) extended this result to spectrally negative Lévy processes: For those, M δ is also exponentially distributed, with the parameter being the right-sided logarithmic derivative of the scale function, evaluated at the drawdown threshold.
Proof of the Theorem. Sufficiency of the first condition for the second one follows directly from Proposition 2.1. Suppose, therefore that for each 0 < δ ≤ a, there exists Λ(δ) > 0 such that M δ ∼ E(Λ(δ)). Then, due to (4), By this particular functional form, and, since µ/σ 2 is continuous, it follows that the functions Λ(δ) and Φ(x) are continuously differentiable. By differentiating (5) with respect to ξ, we have and differentiating with respect to δ yields, in conjunction with the previous identity, and dividing the last two equations yields Lobacevsky's functional equation 3 Note, Φ is continuously differentiable, and strictly positive. Hence, by taking derivatives with respect to δ, we get and by setting ξ = δ, we thus have where α = Φ ′ (0)/Φ(0) ∈ R. We conclude that for some β ≥ 0, Φ(ξ) = e βξ , 0 ≤ ξ ≤ 2a.
By (6) we can extend the exponential solution to −a ≤ ξ < 0: By setting ξ = 0, we indeed have Similarly, we can succesively extend the validity of (7) to the right, using the functional equation (6). Now that Φ(ξ) = e βξ for some β ≥ 0, we have, by taking the logarithmic derivative of Φ, that µ(x)/σ 2 (x) is indeed a constant.
Examples of processes for which the running maximum at drawdown is exponentially distributed, are the following: 1. (a = −∞): Brownian motion with drift σB t + µt.
These include reflected diffusions.
However, there are processes that do not satisfy Theorem 2.2, but exhibit exponentially distributed gains before δ drawdowns for specific choices of δ. One can, for instance, let µ/σ 2 be constant only on [−1, ∞), and modify µ, σ 2 on [−2, −1) in such a way, that the SDE (3) has unique global strong solution (or, alternatively, make −a an absorbing boundary). Then, by Proposition 2.1, for any δ < 1 the maximum at drawdown of size δ is exponentially distributed. It goes without saying, that there must exist δ > 1 for which this is not the case.

Lehoczky's proof for spectrally negative Lévy martingales
We study in this section the distribution of maximal gains 4 of processes, prior to the occurrence of a fixed loss δ > 0. Golub et al. (2016Golub et al. ( , 2018 claim that for a Brownian motion (the toy model of a fair game), this gain is exponentially distributed, with parameter δ; thus in average, one gains δ before experiencing a loss of size δ. This result is independent of the volatility of the Brownian motion. In private communication, Golub (2014) raised the question, of whether similar scaling laws hold for other processes, e.g., other diffusion models, or processes with jumps. Such models are useful as benchmark models in the context of certain event-based high-frequency trading algorithms, where the Brownian motion is used as a proxy for an asset, and the location of the maximum suggests the beginning of a trend reversal. 5 The conjecture that a fair game in average experiences the exact same gain, as is lost later on, may appear intuitive. And this is indeed the case for many continuous-time martingales, those who are time-changed Brownian motions, with a quadratic variation tending to infinity, along almost every path (because the timing is not relevant here). But it is not true for Lévy martingales, as can seen from Theorem 3.2. Nevertheless the (exponential) distribution of gains, not its parameter, is universal within the class of spectrally negative Lévy processes. Besides, the martingale property is not needed to arrive at this result.
After Theorem 3.2 was proved in summer 2019, F. Hubalek kindly pointed out that the result is, in identical form, preceded by Mijatović and Pistorius (2012). Our proof is, however, similar to the one of Lehoczky (1977), and is therefore an alternative, and simpler one. (Finally, we also found a replication of Lehoczky's proof in (Landriault et al., 2017, Lemma 3.1), however, also this proof is more difficult than ours, due the general discretization used therein).
We assume, that a Lévy process X is given with downward jumps only, but not equals the negative of a Lévy subordinator 6 . Such a process is defined by its Lévy exponent which is assumed to be of the form with Lévy-Khintchine triplet µ ∈ R, σ ∈ R and a sigma-finite measure ν(dξ) supported on (−∞, 0), integrating ξ 2 near 0. The scale function W , is the unique absolutely continuous function , θ > 0.
Since the processes lack positive jumps, they can only creep up. This assumption is essential to obtain exit probabilities from compact intervals, and also for the main Theorem 3.2.
Theorem 3.1. (Bertoin, 1996, Theorem VII.8) Let x, y > 0, the probability that X makes its first exit from [−x, y] at y is We are ready to state and proof the main theorem: Theorem 3.2. For a spectrally negative process, not equals to a negative subordinator, the maximal gain M δ before a δ-loss is exponentially distributed with parameter equals the logarithmic derivative of the scale function, that is, Proof of Theorem 3.2. The proof is inspired by Golub et al. (2016), however, the exact same idea can be traced back to Lehoczky (1977) in the general context of univariate diffusions processes. Let A k,n be the event that X reaches kξ/n before −δ+(k−1)/2 n ξ (k = 1, . . . , 2 n ). Then Then M δ ≥ ξ can be approximated by n k=1 A k,n , which are decreasing for increasing n. In other words, Due to state-independence of the process (translation invariance) and the Markov property where the last identity follows from Theorem 3.1. Since W is differentiable from the right at δ, applying L'Hospital's rule yields Remark 3.3. Theorem 3.2 implicitly requires right-differentiability of the scale functions, which is for free, because it can be rewritten as an integral of the tail of some finite measure, see (Bertoin, 1996, Chapter VII). However, in many models, full C 1 -regularity is guaranteed (cf. (Kuznetsov et al., 2012, Lemma 2.4)).

Examples
The scale functions for the below processes are taken from review article of Hubalek and Kyprianou (2011).
Example 3.4 (Compound Poisson Process). Assume we have a compound Poisson process with negative exponentially distributed jumps, We get Clearly W ∈ C 1 (0, ∞), Unlike the previous example, the following two examples exhibit the same qualitative dependence on the threshold δ, as the standard Brownian motion, where M δ ∼ E(1/δ): when δ → 0, the average maximum at drawdown of size δ tends to 0, and when δ → ∞, this average goes to infinity.
The process exhibits Infinite variation jumps, and drifts to −∞, because Ψ ′ (0) < 0. The Scale function is W (x) = (1 − e −x ) β−1 Using Theorem 3.2 we thus get The asymptotic behaviour of the logarithmic derivative of the scale function of a spectrally negative Lévy process can be characterized, using the asymptotic behaviour of W and W ′ , cf. (Kuznetsov et al., 2012, Chapter 3). For instance, W (0) = W (0+) = 0, if and only if the process is of infinite variation. In the case of finite variation, we can write the process as δt − J t , where J is a subordinator; and then W (0) = 1/δ > 0. Furthermore, W ′ (0+) = ∞, if a diffusion component is present, or if the Lévy measure is infinite. These general findings are consistent with the three examples.