Boundary crossing identities for Brownian motion and some nonlinear ode's

We start by introducing a nonlinear involution operator which maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated equations which turn out to be nonlinear ordinary differential equations. We study some algebraic and analytical properties of this involution operator as well as some properties of a two-parameter family of operators describing the set of solutions of Sturm-Liouville equations. Next, we show how a specific composition of these mappings allows to connect, by means of a simple analytical expression, the law of the first passage time of a Brownian motion over a curve to a two-parameter family of curves. We offer three different proofs of this fact which may be of independent interests. In particular, one is based on the construction of parametric time-space harmonic transforms of the law of some Gauss-Markov processes. Another one, which is of algebraic nature, relies on the Lie group symmetry methods applied to the heat equation and reveals that our two-parameter transformation is the unique non-trivial one.


Introduction and main results
Let B = (B t ) t≥0 be a standard Brownian motion, starting at 0, defined on a filtered probability space (Ω, (F) t≥0 , F, P). We are concerned with the distribution of the stopping time where f ∈ C([0, ∞), R) satisfies f (0) = 0 and the usual convention inf{∅} = ∞ applies. C(I, J), for some subintervals I and J ⊆ R, stands for the space of continuous functions from I into J. This boundary crossing problem has been intensively studied for over a century and traces back to Bachelier's thesis [2]. Although different methodologies have been suggested for solving it in some special instances, the finding of an explicit expression for the law of T f for a general curve remains an open problem. We refer to [1] for a survey of these techniques. The purpose of this paper is to describe a simple and explicit analytical expression relating the distributions of the first passage time of the Brownian motion to some two-parameter family of curves, extending the result obtained in [1]. Besides, we shall provide three completely different proofs among which two may be of independent interests. One of the proofs relies on the study of some Gauss-Markov processes and provides a procedure to produce Doob's h-transform of the law of these processes whenever the covariance functions are obtained as the image of a two-parameter transformation relating solutions of a Sturm-Liouville equation. This approach motivated us to introduce and study two families of nonlinear transformations which turn out to be useful for describing the set of solutions of some strongly nonlinear second order differential equations in terms of solutions of the associated Sturm-Liouville equations. Another proof, which is of algebraic nature, is based on the study of the Lie group symmetry of the heat equation, showing in particular that our main identity is the only non trivial one which is attainable. Before stating our main results, we introduce some notations which will be used throughout the paper. First, let the nonlinear operator τ be defined on the space of functions whose reciprocals are square integrable in some (possibly infinite) interval of R + by where a and b are positive reals. Observe that if we denote by A ∞ the set of continuous functions which are of constant sign on some non-empty interval with 0 as left endpoint, then we have the following decomposition We also introduce the family (Π α,β ) (α,β)∈R * ×R of nonlinear operators acting on A ∞ which are defined, for each fixed couple of reals α and β, by Next, let ̺ be the inversion operator acting on the space of continuous monotone functions, i.e. ̺f • f (t) = t where • denotes the composition of functions. We use the same symbol to define the composition of operators and, for example, by ̺ • τ f we mean the image by ̺ of τ f . Now, using the nonlinear operator we construct the family (S α,β ) α∈R * ,β∈R of operators as follows Finally, for all a > 0, we set and, for f ∈ A ∞ , we write We note that, as a → ∞, we have if αβ < 0, +∞ otherwise, and the inequality a α,β ≤ ζ α,β holds. Now, we are ready to state the first main result of this paper whose proof is postponed to Section 2.
2 Theorem 1.1. 1. For α = 0 and β reals, the mapping S α,β : A ∞ → A ∞ is a linear operator admitting the simple representation Let µ be a positive Radon measure on R + . Then, there exists a unique positive, increasing, concave and differentiable function f with f(0) = 1, which satisfies the following nonlinear differential equation on R + , where f ′′ is the second derivative of f considered in the sense of distributions. Furthermore, {S α,β f; α > 0, β ≥ 0} is the set of positive solutions of (4).
We carry on by providing an example illustrating Theorem 1.1. To this end, let us consider the positive measure µ a,b , for some fixed a > 0 and b > 0, which is specified on R + by We easily check that the function f γ given by satisfies the requirements of item 2. of Theorem 1.1. That is f γ is positive, concave and increasing and it solves the nonlinear differential equation (4) with µ = µ a,b . Moreover, for any α, β > 0, the function is also a positive solution of (4). We proceed by stating our second main result which relates the distributions of the family of stopping times (T S α,β f ) α∈R * ,β∈R .
be such that f (0) = 0 and α = 0, β two fixed reals. Then, for any t < ζ α,β , we have the relationship In what follows, we describe some interesting properties satisfied by the family of linear operators (Π α,β ) α∈R * ,β∈R defined in (1). Before doing that, we recall how these operators are related to a class of ordinary second order differential equations. More precisely, recalling that µ denotes a positive Radon measure on R + , we consider the following Sturm-Liouville equation where φ ′′ is defined in the sense of distributions. Clearly, if φ is a solution to (7) then so is Π 0,1 φ. Actually, the set of solutions to equation (7) is the vectorial space {Π α,β φ = αφ + βΠ 0,1 φ; α, β ∈ R}. Observe, that all positive solutions are convex and described by the set where ϕ is the unique positive decreasing solution satisfying ϕ(0) = 1. Moreover, ϕ satisfies lim t→∞ ϕ(t) ∈ [0, 1] and the strict inequality ϕ(∞) < 1 except in the trivial case µ ≡ 0 which we exclude. We point out that ϕ is also differentiable on the support of µ. Moreover, under the condition (1 + s)µ(ds) < ∞ we know that lim t→∞ ϕ(t) > 0. We refer to [11, Appendix §8] for a detailed account on these facts. We are now ready to state the following result where the study is restricted to α ≥ 0 since the other case can be recovered by using the identity Π −α,β = −Π α,−β . ) for some positive reals a and b. Then, we have the following assertions.
which completes the proof of the Proposition. Now, we are ready to study some properties of the family of linear operators (S α,β ) α∈R * ,β∈R which is defined in (2). In particular, the next result contains the claims of item 1. of Theorem 1.1.
Then, the following assertions hold true. 1 In particular, (S 1,β ) β≥0 is a semigroup of linear operators. 3. Assuming that f ∈ A(a, b) is concave and differentiable then we have the following statements.
a. S α,β f is also concave and differentiable.
Proof. Item 1. and the first part of item 2. follow readily from the definition of S α,β and propositions 2.1 and 2.2. Then, from Proposition 2.1, we get that Moreover, we see that Inverting yields Finally, combining (9) and (10), we can write which is easily simplified to get (3). It is clear from (3) that S α,β is a linear operator. Next, since Σ is an involution, item 2. follows from Proposition 2.2 and Item 3.a. follows readily from the definition of (S α,β ) α∈R + ,β∈R combined with the propositions 2.1 and 2.2. Finally, we note that for β α ≥ β 0 , S α,β f is positive since t → α 2 t/(1 + αβt) is increasing on R + , which completes the proof by means of the concavity property.

Proof of Theorem 1.2
We actually derive three proofs of Theorem 1.2. The first one is almost straightforward and hinges on a previous result obtained by the authors in [1] whereas the second one reveals some interesting results concerning time-space harmonic transforms of the law of Gauss-Markov processes and explains the connections with the analytical result stated in Theorem 1.1. The last one relies on the Lie group techniques applied to the heat equation. Before developing the proofs, we mention that the symmetry of the Brownian motion implies the following identity in distribution (11) T S α,β f d = T S |α|,sgn(α)β f for any (α, β) ∈ R * × R. Hence, it is enough to consider the case α > 0. For convenience, we set f α,β = S α,β f .
3.1. The direct approach. We get from item 2. of Proposition 2.3 that S α,β = S 1,αβ • S α,0 which when combined with [1, Theorem 1] gives our result. To be more precise, recall that, from the aforementioned reference, we have for all t < ζ 1,β . Thus, by using f α,β = S 1,αβ • S α,0 f , we can write for t < ζ α,β . Next, using the scaling property of B, we obtain the equality in distribution Using the linearity and again the composition properties of S α,β , we get which combined with formula (11) completes the proof of Theorem 1.2.

The proof via Gauss-Markov processes.
For the second approach, we take φ ∈ A a b ∩ AC([0, b)), where AC([0, b)) is the space of absolutely continuous functions on [0, b), and consider the associated Gauss-Markov process of Ornstein-Uhlenbeck type with parameter φ. More specifically, we denote by P φ = (P φ x ) x∈R the family of probability measures of the process X = (X t ) 0≤t<b which is defined to be the unique strong solution to the stochastic differential equation Clearly, X is the Gaussian process given, for each fixed 0 ≤ t < b, by which has mean and covariance function respectively. To simplify, we assume throughout that φ(0) = 1. Note that if we take φ(t) = e −λt , for some λ > 0, then X is the classical Ornstein-Uhlenbeck process. Moreover, if X 0 is a centered and normally distributed random variable, with variance 1/2λ, which is independent of B, then X is the unique Gauss-Markov process which is stationary see e.g [11, Excercise (1.13), p.86].
Our motivation for introducing this process stems from the following simple connection between two types of boundary crossing problems. 7 Lemma 3.1. Let, for any y ∈ R, T y = inf{0 < t < b; φ(t) t 0 φ −1 (s) dB s = y}. Then, for any f ∈ A(a, b), writing φ = Σf and T = T 1 , the identity holds almost surely. In particular, P φ 0 (T < b) = P T f < a . Proof. By means of Dumbis, Dubins-Schwarz theorem, see e.g. [11,Theorem V.1.6], there exists a standard Brownian motion (W t ) 0≤t<b defined on (Ω, F, P) such that we have a.s.
where we used item 1. of Proposition 2.1. The proof is now easy to complete.
We mention that relation (12) was used by Breiman [3] for relating the first crossing time of a Brownian motion over the square root boundary to the first passage time to a fixed level by the classical stationary Ornstein-Uhlenbeck process. Next, we need to introduce the notation Our aim now is to show that the parametric families of distributions (P Π α,β φ ) (α,β)∈R * ×R of Gauss-Markov processes are related by some simple space-time harmonic transforms.
Lemma 3.2. For (α, β) ∈ R * × R and φ as above, the process (H t (X t )) 0≤t<a φ α,−β is a P φmartingale. Furthermore, the absolute-continuity relationship holds for all x ∈ R and t < a φ α,−β . Consequently, for any reals x and y, we have Proof. First, the Itô formula yields β 2 Thus, in the special case φ ≡ 1, the process (H t (X t )) 0≤t<a φ α,−β is a P-local martingale. Moreover, from the well-known identity E e − λ 2 B 2 t = (1+λt) −1/2 , λ > −1/t, see e.g. [11, p.441], we deduce that, for all t < a φ α,−β , we have E[H t (B t )] = 1. Hence it is a true martingale. The P φ -martingale property of (H t (X t )) 0≤t<a φ α,−β follows from the fact it has the same distribution as the process we deduce the absolute continuity relationship by an application of Girsanov's theorem. Next, on the event {T y ≤ t} ∈ F t∧Ty , we have H t∧Ty (X t∧Ty ) = H Ty (y). Now, Doob's optional stopping theorem implies that Our claim follows then by differentiation. Now, we are ready to complete a version of the second proof of Theorem 1.2. For the sake of clarity, we assume that f is continuously differentiable and thus according to Strassen [12], the law of T f is absolutely continuous with a continuous density which we denote by p f . Next, let φ = Σf and thus, by definition, f α,β = Σ • Π α,−β φ. Since Σ is an involution, we have from Lemma 3.1 that T f α,β = τ • Π α,−β φ (T ) a.s. Using the fact that we get, for any t < ζ α,β , where we used the identity τ f α,β (t) = τ f α 2 t 1+αβt which follows readily by a change of variable. Then, Proposition 3.2 combined with the identities yields, for any t < ζ α,β , and thus p f α,β (t) dt = f −2 α 2 t 1 + αβt (1 + αβt) 1/2 e − βαf α,β (t) 2 2(1+αβt) P φ 0 T ∈ τ f α 2 t 1 + αβt .
3.3. The approach via the Lie group symmetries of the heat equation. The purpose of this proof is to show how the symmetry groups method may be used to derive our main identity (6). We recall that the application of Lie group theory to solve differential equations dates back to the original work of S. Lie. It provides an effective mechanism for computing a wide variety of new solutions of a specific differential equation from known ones. An excellent account of this technique can be found in the monograph of Olver [9]. We mention that recently Lescot and Zambrini [8] resort to Lie group techniques for the study of some diffusions with a view towards stochastic symplectic geometry. Before describing the symmetries of the heat equation, that is the one-parameter group of transformations leaving invariant the space of solutions of this equation, denoted throughout by H, we state the following result which relates the boundary crossing problem to the study of the heat equation. These claims can be found in Theorem 1.1 and Lemma 1.4 of Lerche [7].