Application of Stochastic Flows to the Sticky Brownian Motion Equation

We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.


Introduction
A θ-sticky Brownian on the half line [0, ∞) is a diffusion with generator where θ > 0 is the stickiness parameter. This is a special case of Feller one dimensional diffusions introduced by Feller by means of their infinitesimal generators [4]. For comparison of the boundary condition f ′′ (0 + ) = 2θf ′ (0 + ) with other examples, see [3]. Sticky Brownian motion has an intermediate behavior, depending on θ, between Brownian motion absorbed at 0 and reflected Brownian motion. One possible path construction of a θ-sticky Brownian motion X started from 0 consists in slowing down a reflected Brownian motion R started from 0 whenever it is at 0 in the following way X t = R inf{u:u+ 1 θ Lu>t} (1) Laboratoire de l'Intégration du Matériau au Système, Bordeaux. Email: Hatem.Hajri@imsbordeaux.fr (2) Koc University, Istanbul. Email: mcaglar@ku.edu.tr The research of M. Caglar is supported by TUBITAK Project No. 115F086 (3) Institut de Mathématiques de Bordeaux, Bordeaux. Email: marc.arnaudon@math.ubordeaux.fr 1 where L t = lim ǫ→0 1 2ǫ t 0 1 {0≤Rs≤ǫ} ds is the local time of R [7,8]. As a consequence of this construction, the amount of time spent at 0 by X up to t, t 0 1 {Xs=0} ds, has positive probability of being greater than 0. More precisely, the following equality holds in law Proposition 5 in [7]).
In this paper, we are interested in sticky Brownian motion as solution of the following stochastic differential equation driven by a standard Brownian motion W and where θ > 0 is a given constant and x ∈ [0, ∞) is a given initial condition. It has been proved by Chitashvili [2] that (1) has a weak solution X which is a θ-sticky Brownian motion started from x, the law of (X, W ) is unique but X is not a strong solution to (1). Later on, Warren [15] derived the following remarkable result describing the law of X t given W (the form given here follows Theorem 2 [16]).
with T an exponential variable with mean 1 2θ . This theorem shows, in particular, that X cannot be a strong solution to (1). Subsequently, Warren [16] described all couplings of solutions to (1) which leave the diagonal. Before going on, we mention the work of Engelbert and Peskir [3] where a third proof of the non strong solvability of (1) and a two sided version of it can be found (see also [1]).
A remarkable and attractive fact in Warren's conditional law identity is that it involves the well known and habitual process W + strong solution to This raises the question whether there is a link between (1) and (2) explaining Theorem 1.1.
In this paper, it is shown that stochastic flows of kernels [12] provide an answer to the previous question. More precisely, define Then ϕ is a stochastic flow of maps which solves the flow version of (2) where L s,t (x) = lim ǫ→0 Then K is a stochastic flow of kernels which is a strong solution to the flow of kernels version of (1): for all t ≥ s, f ∈ D(A) and x ≥ 0 a.s.
K, called the Wiener solution of (5) in [12], is characterized by being the unique (up to modification), strong solution of (5). This leads to Theorem (1.1) as the conditional law of X t given W should coincide with K 0,t (0, dy). Note that Equation (5) encapsulates the flow property (iv) of Definition 2.1 for K. Therefore, identifying the complete flow K s,t (x, dy) for every s, t and x is crucial in proving this result, not only for s = 0 and x = 0. The semigroup and Feller properties also play an important role in this fact. See [12] for further discussion on E[f (X t )|F W ] satisfying Equation (5). As a complete proof, we argue that (4) being the Wiener flow satisfies this equation and the theorem follows for G f (W + t ) in the special case X 0 = 0. Section 2 gives details and proofs of the previously claimed facts. It can be remarked that the proofs only rely on the definition of stochastic flows with no additional results of the theory. The present paper provides, in particular, a direct application of stochastic flows to the study of weak solutions (see [6] for another recent application). In Section 3, we conclude the paper with the Wiener chaos expansion of the conditional law.

The generalized sticky Brownian motion equation.
Let us now recall the definition of stochastic flows from [12]. In this definition P(R + ) denotes the space of all probability measures on R + and B(E) indicates the Borel σ-field of E.
We say that ϕ is a stochastic flow of mappings on R + if K s,t (x) = δ ϕs,t(x) is a stochastic flow of kernels on R + .
For K, a stochastic flow of kernels on R + , defines a Feller semigroup on R n + . Moreover (P n ) n≥1 is a compatible family (in a sense explained in [12]) of Feller semigroups acting respectively on C 0 (R n + ) that uniquely characterize the law of K. Conversely, it has been proved in [12] that to each family of compatible Feller semigroups (P n ) n≥1 is associated a (unique in law) stochastic flow of kernels such that (6) holds for every n ≥ 1.

For a family of random variables
Definition 2.3. Let K be a stochastic flow of kernels and W be a real white noise defined on the same probability space. We say that (K, W ) is a (generalized) solution of the sticky equation if for all f ∈ D(A), t ≥ s and x ∈ R + a.s.
Let us explain the link between this equation and the original sticky equation (1). We start with the following Lemma 2.4. If (K, W ) is a solution of the generalized sticky equation, then Then along the same lines of the proof of Lemma 3.1 in [13], one can show that K s,t (x) = δ x+Ws,t for all s ≤ t ≤ τ ǫ s (x). As ǫ > 0 is arbitrarily small, K s,t (x) = δ x+Ws,t also for t ≤ τ s (x) := inf{u ≥ s : x + W s,u = 0}. Since this holds for x arbitrarily distant from 0, the lemma follows.
In view of Lemma 2.4, we may sometimes say K is a solution of the generalized sticky equation without specifying the white noise since it is determined by K.
Assume now (K, W ) satisfies Definition 2.3 and set By the previous lemma, (Q t ) t defines a Feller semigroup. Denote by L its generator and D(L) its domain. A simple application of Itô's formula shows that D 1 ⊗C 2 K (R) ⊂ D(L) where D 1 = {f ∈ D(A) : f ′ (0+) = 0} and C 2 K (R) denotes the space of C 2 functions on R with compact supports. Moreover for all f ∈ D 1 and g ∈ C 2 K (R), Let (X, B) be the Markov process associated to (Q t ) t and started from (x, 0). Then X is a θ-sticky Brownian motion started from x and B is a standard Brownian motion started from 0. Now for f ∈ D 1 and g ∈ C 2 K (R), As X is a θ-sticky Brownian motion, it satisfies X t = x + M t + θ t 0 1 {Xs=0} ds with M a martingale with quadratic variation M t = t 0 1 {Xs>0} ds. Writing Itô's formulas for f (X)g(B) and using (7) shows that On the other hand, there exists a sequence (g n ) ⊂ C 2 K (R) such that the support of g n is [−n, n] with sup x g ′ n (x) ≤ 1 and g ′ n (x) → 1 as n → ∞ for each x ∈ R. In view of (8), it follows from bounded convergence theorem that More generally, considering the semigroups Q n t (f ⊗ g)(x, w) = E[K ⊗n 0,t f (x)g(w + W t )] for n ≥ 1, one can prove that there exists a one to one correspondence between the laws of stochastic flows of kernels satisfying Definition 2.3 and compatible weak solutions to the sticky equation (see Proposition 2.1 in [5] for more details in a similar context).
To close this subsection, we mention that if ϕ is a flow of mappings such that K = δ ϕ satisfies Definition 2.3, then necessarily ϕ is a flow of mappings solution of and vice versa. Warren [17] proved that such a flow ϕ exists and its law is uniquely determined. This flow can also be constructed by applying the general results of [12]. Proof. We follow the proof of Proposition 4.2 [5]. Note that K s,t (x, y) = K 1 s,t (x) ⊗ K 2 s,t (y) is a stochastic flow of kernels on R 2 + and Q t (f ⊗ g ⊗ h)(x, y, w) := E[K 1 0,t f (x)K 2 0,t g(y)h(w + W t )] is a Feller semigroup on (R + ) 2 × R. Fix x ∈ R + and let (X 1 , X 2 , B) be the Markov process associated to Q started from (x, x, 0), then B is a standard Brownian motion and X 1 , X 2 are two θ-sticky Brownian motions. Moreover X 1 , X 2 are solutions of the sticky equation driven by B and in particular (X 1 , B) and (X 2 , B) have the same law. Since K 1 and K 2 are two Wiener solutions, there exist two measurable func- We will prove that for all measurable bounded f : R → R a.s.
To prove (9), we will check by induction on n that for all t 1 ≤ · · · ≤ t n−1 ≤ t n = t and all bounded functions f, g 1 , · · · , g n : R → R, we have Let us prove this for i = 1 and set Q 1 t (f ⊗ g) = Q t (f ⊗ Id ⊗ g). For n = 1, (10) is immediate from the definition of Q. Let us prove (10) for n = 2. We have ]. Now (10) holds using a uniform approximation of Q 1 t 2 −t 1 (f ⊗ g) by a linear combination of functions of the form h ⊗ k, h, k ∈ C 0 (R + ). It is clear now from (9), that N 1 0,t (x) = N 2 0,t (x) since (X 1 , B) and (X 2 , B) have the same law.
Now in the rest of the paper, we take W a real white noise and will check that K defined in (4)  Proof. To check that K is a stochastic flow of kernels, we will only check the flow property for all f ∈ C 0 (R + ), s ≤ t ≤ u and x ∈ R + , with probability 1, The other claims in Definition 2.1 are easy to verify. Let us now check (11). For this, we will use the fact that ϕ defined in (3) is a stochastic flow of mappings (for non specialists of stochastic flows, this is a rather simple exercise). Note that G f writes as (with λ = 2θ) We first check (11) for x = 0. By the flow property of ϕ, K s,u f (0) = G f (ϕ s,u (0)) = G f (ϕ t,u • ϕ s,t (0)) and K s,t K t,u f (0) = G Kt,uf (ϕ s,t (0)). Using the independence of increments of ϕ, it suffices to prove that for all y ≥ 0, a.s G f (ϕ t,u (y)) = G Kt,uf (y) which is equivalent to To prove this identity, note that for all y > 0, z → ϕ t,u (z) is differentiable at y with derivative given by 1 {u<τt(y)} . Thus by a simple calculation, the derivative of z → e λz ϕ t,u (z) at y coincides with λe λy K t,u f (y). This proves (11) for x = 0. Now take x > 0 and let y = ϕ s,t (x).
On the event {t ≤ τ s (x) < u}, we still have τ s (x) = τ t (y) and K s,u f (x) = G f (ϕ s,u (x)) = G f (ϕ t,u (y)). Moreover K s,t (K t,u f )(x) = G Kt,uf (y) and so the flow property holds by the calculations above.