The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative.


Introduction
Let a probability measure on a subset of the real line have moments of all orders. Under which conditions do these moments pin down the probability measure uniquely? This is the well-studied moment problem. When the subset is compact, the answer is always affirmative. In the noncompact case uniqueness is more delicate (see [Sch17]).
In stochastic analysis one is usually concerned with measures on some space of paths, the prime example being Wiener measure on the space of continuous functions. It turns out that for many purposes a good replacement for "monomials" in this setting are the iterated integrals of paths. The collection of all of these integrals is called the iteratedintegrals signature.
For smooth paths X : [0, T ] → R d (and with respect to a time horizon T > 0), it is defined, using Riemann-Stieltjes integration, as S(X) 0,T := 1 + (1) It is well-known (see [Boe+16] and references therein) that • S(X) 0,T ∈ G, where G ⊂ T ((R d )) is the group of grouplike elements, • S(X) 0,T completely characterizes the path X up to reparametrization and up to tree-likeness.
Let X : Ω × [0, T ] → R d now be a stochastic process. For fixed ω ∈ Ω, t → X t (ω) is usually not smooth, so that we have to assume that the stochastic process posesses a "reasonable" integration theory. In particular assume that integrals of the form g(X s )dX s , exist for a large class of functions g ∈ C(R d , L(R d , R n )) and that the fundamental theorem of calculus holds, An important example is Brownian motion with Stratonovich integration. Other examples include: the Young integration against a fractional Brownian motion with Hurst parameter strictly larger than 1/2, . . . (see [FV10]). The iterated-integrals signature S(X) 0,T defined by the expression (1) -now, using the given integration theory -is then a random variable. Let us assume that we can take its expectation level-by-level, i.e. for all n ≥ 1 (we postpone the discussion of the choice of norm on (R d ) ⊗n to later) E T 0 rn 0 · · · r 2 0 dX r 1 ⊗ · · · ⊗ dX rn < +∞.
We can then define expected signature level-by-level (2) where E p denotes the expectation level-by-level (p for "projective") of a G-valued random variable. The question arises: Does ExpSig(X) completely characterize the law of X?
As we have seen above, the computation of S(X) 0,T already incurs a loss of information: the parametrization of X and any tree-like parts are lost. The relevant question is hence Does ExpSig(X) T completely characterize the law of X, up to parametrization and tree-likeness?
Since this formulation is a bit awkard, and since the (deterministic) step S(X) → X is completely understood, we can instead focus on Does ExpSig(X) T completely characterize the law of S(X) 0,T on G?
A sufficient condition for this to be the case is given in [CL16]: if ExpSig(X) T has infinite radius of convergence, that is for all λ > 0 then the law of S(X) 0,T on G is the unique law with this (projective) expected value. Here proj n : T ((R d )) → (R d ) ⊗n denotes projection onto tensors of length n. Let us give two examples. Let µ be a probability measure on R having all moments and define a n := x n µ(dx).
Consider the stochastic process X t := tZ, where Z is distributed according to µ. Since X is smooth, its signature is well-defined and actually has the simple form

Then
ExpSig(X) T = 1 + T a 1 + T 2 2! a 2 + T 3 3! a 3 + . . . , and a sufficient condition for n a n T n λ n /n! to have infinite radius of convergence is |a n | ≤ C n , for some C > 0. 1 Then [CL16, Proposition 6.1] applies, and the law of S(X) 0,T on G is uniquely determined by these moments.
Consider now the expected signature of a standard Brownian motion B calculated up to some fixed time T > 0. It is known (see for example [LV04,Proposition 4.10]) that It follows that and hence for any λ > 0. Again, by [CL16, Proposition 6.1], the law of S(B) 0,T is uniquely determined by ExpSig(B) T . In this paper we study properties of the expected signature, not up to deterministic time T , but up to a stopping time τ . Concretely, we consider the Brownian motion B z in R 2 started at some point z in the unit circle D := {z ∈ R 2 : |z| ≤ 1}, and stopped at hitting the boundary, that is (4) In the notation introduced above, we are interested in where X z t := B z t∧τ . In [LN15] it was shown that for every n ∈ N and n ≥ 2, the nth term of Φ satisfies the following PDE: with the boundary condition that for each |z| = 1, Additionally, one has proj 0 (Φ(z)) = 1 and proj 1 (Φ(z)) = 0 for all z ∈ D. Using this, they were able to obtain the bound proj n (Φ(x)) ≤ C n for some C > 0 ([LN15, 1 The condition |an| ≤ C n is of course more than enough in the classical moment problem to have uniqueness for the law µ on R ([RS75, Example X.6.4]).
Theorem 3.6]). This bound is not enough to decide whether the radius of convergence for ExpSig(X z ) ∞ is infinite or not, but it is enough to deduce that ExpSig(X z ) ∞ has radius of converge strictly larger than 0. In this work we show that the radius of convergence is indeed finite. Recall from [CL16, Proposition 6.1] that if A, B are G-valued random variables such has an infinite radius of convergence, then A D = B. Our main theorem, proven in Section 6, is the following.
The condition of E p [A] having an infinite radius of convergence is equivalent to E p [A] lying in E. Here E is defined as the closure of T ((R 2 )) under the coarsest topology such that for all normed algebras A and all M ∈ L(R 2 , A), the extension M : T ((R 2 )) → A is continuous. Recall that for M ∈ L R 2 , A , we may define M firstly as a map on the k-times algebraic tensor product (R 2 ) ⊗ak , by the relation and then extended it to T ((R 2 )) by linearity.
We want to show that Φ(z) does not lie in the space E. It is sufficient to show that there exists λ ∈ R, and M ∈ L R 2 , M 3×3 (R) , such that (λM ) (Φ(z)) diverges as λ tends to a finite number λ * . In fact, we will choose M to be Such a map M first appeared in [HL98] to study the signature of bounded variation paths and is also subsequently used in [LX17]. We proceed as follows. In Section 2, for λ > 0 we let λM act on Φ(z). For λ small enough, we show that resulting linear map in L(R 3 , R 3 ), evaluated at (0, 0, 1) ∈ R 3 is smooth in z and solves a certain PDE. Using rotational invariance of Brownian motion, in Section 3 we rewrite said PDE solution in polar coordinates. In Sections 4 and 5, we obtain an explicit solution for the PDE (still, for λ small enough) in terms of Bessel functions. Finally, in Section 6 we show that the solution blows up as λ →λ for somẽ λ < +∞, proving our main theorem. The Appendix, Section 7, contains some auxiliary results on PDEs.

Differentiability of the development of expected signature
We first need two technical lemmas which assert that the development of the expected signature is twice differentiable, and satisfies the PDE we expect it to. In the lemma below we will adopt the multi-index notation Observe that Let · be the projective norm.
Lemma 1. The function z → proj n (Φ(z)) is twice continuously differentiable. There exists a constant C > 0 such that for all n ∈ N, all z ∈ D and all α ∈ N 2 satisfying |α| ≤ 2, one has the bound D α proj n (Φ(z)) ≤ C n .
Proof. Let m ∈ N. By Theorem 16 in the appendix, the function z → proj n (Φ(z)) is twice continuously differentiable (it is in fact infinitely differentiable on D). By Lemma 15 in the Appendix, there exists C > 0 such that for all n ∈ N where the norm · W m,2 (D) is the Sobolev norm on the unit disc D with respect to the variable z, By Theorem 2.2 in [LN15], which bounds the values of a function u in terms of the Sobolev norm of u, there is some constantC(2) such that for all z ∈ D and |α| ≤ 2, Since M proj n (Φ(z)) is a linear image of proj n (Φ(z)), the function M proj n (Φ(z)) is twice continuously differentiable in z, and moreover, there exists c > 0 such that for all z ∈ D, This bound also allows us to deduce that for |α| = 2, the series converges uniformly and hence the series is twice continuously differentiable and the derivatives can be taken inside the infinite summation.
Lemma 2. There exists λ * > 0 such that if λ < λ * , the function F λ defined by is twice continuously differentiable on D, and satisfies Here (e 1 , e 2 ) denotes the canonical basis of R 2 .
Proof. If we apply the linear map M to the PDE (5), then we have a matrix-valued PDE together with the boundary condition We may multiply both sides with λ n , sum to infinity and apply to the vector (0, 0, 1) to get By Lemma 1, each of the infinite sums converges and we may take the derivatives outside the infinite sum.

A polar decomposition for the development
We may consider M (x) as a linear endomorphism of Lemma 3. For any linear map R : where R * is the transpose of R.
Proof. Brownian motion B R(z) starting at R(z) has the same distribution as the rotated Brownian motion R (B z ), where B z starts from z. Let •d denote the Stratonovich differential. Then By Lemma 3 As R is orthogonal, we have Therefore, where F λ is the function defined by (9). In polar coordinates, the expression of F λ reads Additionally, there exists λ * > 0 such that if λ < λ * , then A λ , B λ , C λ are twice continuously differentiable functions in the variable r for all r ∈ [0, 1].
Then the real-valued functions defined for all r > 0 by 2 Im(ᾱJ 1 (λζ)J 0 (λζr)) (19) are the unique solution of the differential system (12) satisfying the boundary conditions stated in Lemma 6.

Concluding
Lemma 8. In the notation of Lemma 7, there existsλ > 0 such that C(0), viewed as a function of λ, has a pole atλ.
Proof. Let us first show that d(λ) has a zero lying in the interval (2.5, 3). Consider the series expansions [Olv+10,(10 For x ∈ C and n ∈ N such that |x| < 2(n + 1), the remainders starting at index n of both series are bounded by In particular, for x = λζ or x = λζ with 0 < λ 3, we have |x/2| < 1.784. For n = 5, the quantity (27) is bounded by 0.025. By replacing J 0 and J 1 by the first five terms of the series (26) in the expression of d(λ) and propagating this bound by the triangle inequality, one can check that d(2.5) < −0.06. A similar calculation shows that d(3) > 0.03. Since d(λ) is a continuous function of λ, it follows that d(λ) vanishes for someλ ∈ (2.5, 3).
The claim follows.
Remark 9. Instead of doing the calculation sketched in the proof manually, one can easily prove the result using a computer implementation of Bessel functions that provides rigorous error bounds. For example, using the interval arithmetic library Arb [Joh17] via SageMath, the check that d(λ) has a zero goes as follows. Remark 11. Since the condition of [CL16] of uniqueness of laws is only sufficient, the questions remains on whether there exists another law on G having the same moments as S(X 0 ) 0,T .
Proof. Assume for contradiction that Φ(0) has an infinite radius of convergence. Then F λ (0) is an entire function in λ. We also know from Corollary 5 that there exists λ * > 0 such that for real λ < λ * where A λ , B λ , C λ are defined by Lemma 7. By the Identity theorem, A λ , B λ , C λ are entire functions and (28) holds for all λ. This contradicts Lemma 8, and therefore Φ(0) has a finite radius of convergence.

Appendix
Let Γ be a domain in R d .
Definition 12. Let u be a locally integrable function in Γ and α be a multi-index. Then a locally integrable function r α u such that for every g ∈ C ∞ c (Γ), will be called weak derivative of u and r α is denoted by D α u. By convention, D α u = u if |α| = 0.
Definition 13. The Sobolev space W k,p (Γ) for p, k ∈ N is defined to be the set of all Rd-valued functions u ∈ L p (Γ) such that for every multi-index α with |α| ≤ k, the weak partial derivative D α u belongs to L p (Γ), i.e.
It is endowed with the Sobolev norm defined as follows: When k = 0, this norm coincides with the L p (Γ)-norm, i.e.
Theorem 14. Let M be a second order differential operator with coefficients {a i,j }. Let u be a weak solution of M u = f (x), u − g ∈ H 1,2 0 (Γ).
Proof. It is proved by using Theorem 8.13 in [GT15] and setting the boundary condition ϕ = 0.
In the following we prove Lemma 15 for m ≥ 2, which is a generalization of Lemma 3.11 for the case m = ⌊ d 2 ⌋ in [LN15].
Proof. The proof of Lemma 3.11 in [LN15] can be applied here directly, except for that we need to check that proj n (Φ) ∈ W m,2 , which is proved in the following theorem 16.
Theorem 16. Suppose that Γ is a non-empty bounded domain in E. It follows Φ is infinitely differentiable in componentwise sense, i.e. for all index I, proj n •Φ is infinitely differentiable for all n.
Since Ψ is smooth (in polynomial form) and K ε is a smooth function with compact support, G ε is a smooth function with compact support. It is easy to show that for any partial derivative D α G ε is L 1 integrable.