{'fraction_non_alphanumeric': 0.08590480244023552, 'fraction_numerical': 0.024047669716961057, 'mean_word_length': 3.586206896551724, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 12, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the strong uniqueness of a solution to singular stochastic differential equations

We prove the existence and uniqueness of a strong solution for an SDE on a semi-axis with singularities at the point 0. The result obtained yields, for example, the strong uniqueness of non-negative solutions to SDEs governing Bessel processes.


Introduction
We consider a stochastic process the state space of which is a non-negative semi-axis. Assume that up to the first hitting time of zero the process (x(t)) t≥0 satisfies an SDE where x 0 ≥ 0, a, σ are supposed to be locally Lipshitz continuous on (0, ∞), (w(t)) t≥0 is a Wiener process. Possible singularities of the coefficients generate different types of behavior of the process in a neighborhood of zero. As a consequence, the integral representation of (x(t)) t≥0 may acquire various forms.
As an example let us consider the following SDE It is known that β-dimensional Bessel process with β > 1 is a unique non-negative strong solution to (1) (cf. [2]). Note that this equation possesses no additional terms. Otherwise, an additional summand can be represented by the local time (l(t)) t≥0 of unknown process (x(t)) t≥0 at the point 0 like in Skorokhod equation (2) x(t) = x 0 + t 0 a(x(s))ds + t 0 b(x(s))dw(s) + l(t), or by principal values of some functionals of the unknown process like in the following representation for a β-dimensional Bessel process with β ∈ (0, 1) (cf. [13], Ch. XI) Here k(t) = V.P. t 0 ρ −1 (s)ds which, by definition, is equal to ∞ 0 a β−2 (L ρ a (t) − L ρ 0 (t))da, L ρ a (t) ia a local time of the process ρ(t) at the point a.
It seems improbable to describe all possible forms of integral representations. Let f be a twice continuously differentiable function on [0, ∞) which is a constant in a neighborhood of zero. Applying Itô formula (additional tricks are needed in some cases) we see that the stochastic differential of the function f (x(t)) has identical form for solutions 1 of equations (1)- (3). Namely, (4) f (x(t)) = f (x 0 ) + t 0 a(x(s))f ′ (x(s)) + 1 2 σ 2 (x(s))f ′′ (x(s)) ds σ(x(s))f ′ (x(s))dw(s).
The differential has no singularities. Intuitively, the singularities at the point 0 are killed by zero derivatives of the function f . We use this fact to formulate the problem as an analogue of a martingale problem (see Section 1). The main result of this paper is as follows: the existence of a weak non-negative solution for equation (4) spending zero time at the point 0 implies the existence and uniqueness of a non-negative strong solution spending zero time at 0. The pathwise uniqueness is obtained by method of Le Gall [5] based on the fact that the maximum of two solution also solves the equation. The formulation of a martingale problem involving a class of functions that are constant in a neighborhood of possible singularities was used by many authors (see, for example, [12], [15]). The notations and definitions used are collected in Section 1. We prove the main Theorem in Section 2. In Section 3 some examples are represented.
This and Condition A ensure the existence of all the integrals on the right-hand side of (5).
Remark 2. It is not hard to verify that the existence of a weak solution (x(t), w(t)) t≥0 to equation (5) on a probability space (Ω, F, P ) with initial condition x 0 is equivalent to the existence of a probability measureP on some probability space (Ω,F,P ) satisfying conditions (i) and (ii) of Definition 1. The process (x(t)) t≥0 induces the measureP on (C + ([0, ∞)), G), namelyP = P x −1 .
For the proof see Appendix.
Definition 4. The weak uniqueness holds for equation (5) if, for any two weak solutions (x, w) and (x,w) (which may be defined on different probability spaces) with a common initial value, i.e. x(0) = x 0 P -a.s.,x(0) = x 0P -a.s. the laws of processes x andx coincide.
Denote by F w t the filtration of w completed with respect to P .
Definition 6. Given a process (w(t)) t≥0 , and x 0 ≥ 0, we say that the process (x(t)) t≥0 is a strong solution to equation (5) with initial condition x 0 if it is adapted to the filtration

The main result
Unfortunately, we are not able to write equation (5) for the process (x(t)) t≥0 itself because the function f (x) = x does not belong to C 2 c ([0, ∞)). Instead, in the next Lemma we obtain an SDE for the process ζ δ (x(t)) = x(t) ∨ δ, t ≥ 0, which will often be used in the sequel.
Suppose (x(t)) t≥0 is a weak solution to equation (5). Then the equality (6) ζ δ (x(t)) = ζ δ (x(0)) + t 0 a(x(s))1 (δ,+∞) (x(s))ds is valid for all t ≥ 0. Here (L x δ (t)) t≥0 is a local time of the process (x(t)) t≥0 at the point δ defined by the formula Proof. We make use of a standard approximation of non-smooth function by smooth ones like the construction in the proof of Tanaka's formula (see, for example, Theorem 4.1, Ch. 3 of [8]). For a > 0, let us approximate the function ξ a (x) = x ∨ a by twice continuously differentiable functions. Put Then the function u n is twice continuously differentiable and u n ( for any continuous and bounded function ϕ. ) be a non-decreasing and such that η δ (x) = x on [δ/2, ∞). The process (η δ (x(t))) t≥0 can be represented in the form (5), and thus it is a semimartingale. By Itô formula we have By occupation times formula (cf. [11], Corollary 1, p. 216), the last integral on the right-hand side of (9) is equal to as n → ∞.
Passing to the limit in (9) as n → ∞ and taking into account that η δ (x) = x, x > δ/2, we arrive at the equation (6).
The main result of the paper is the following theorem. We split the proof of Theorem into two steps. At the first one we show that the existence of a solution to the martingale problem provides well-posedness. At the second one the pathwise uniqueness is obtained from weak uniqueness. These two steps are formulated as Lemmas in the following way.
Lemma 2 (weak uniqueness). Suppose a, σ satisfy the conditions of Theorem 1. Let for each x 0 ≥ 0, there exists a solution to martingale problem M (a, σ, x 0 ). Then the weak uniqueness holds for equation (5).
Proof. We would like to get a law of the process (x(t)) t≥0 . But we don't know an integral representation for (x(t)) t≥0 itself. Instead, we consider the process (x(t) ∨ δ) t≥0 . Applying a space transformation and change of time to the process (x(t) ∨ δ) t≥0 we will see that the law of the process obtained coincides with that of the Wiener process with reflection at the point δ.
Similarly to Theorem 12.2.5 of [14] it can be shown that the existence of solution to the martingale problem for each x 0 ≥ 0 implies the existence of a strong Markov, time homogeneous measurable Markov family {P x0 : is a solution to the martingale problem starting from x 0 . And by Theorem 12.2.4 of [14] to prove the uniqueness of a solution to the martingale problem it is sufficient to prove the uniqueness only for the family of strong Markov, time homogeneous solutions. IfP x0 is such a solution starting from x 0 , then according to Remark 2 there exists a pair (x, w) on some probability space (Ω, F , P x0 ) which is a weak solution to equation (5) and P x0 x −1 =P x0 . This yields that (x(t)) t≥0 is a strong Markov and time homogeneous process.
Note that if the process (x(t)) t≥0 does not hit zero the assertion of Lemma is trivial. So from now on we suppose that starting from x 0 the process (x(t)) t≥0 hits zero P -a.s.
Proof. Let (x 1 (t)) t≥0 , (x 2 (t)) t≥0 be processes defined on the same probability space (Ω, F, (F t ) , P ) and let each of them is a weak solution to equation (5). The idea of the proof is as follows. We will see that the process ((x 1 ∨ x 2 )(t)) t≥0 is also a weak solution to equation (5).
Let q ∈ [0, ∞) ∩ Q be such that ζ δ (x 1 (q)) > δ, ζ δ (x 2 (q)) > δ.Define a q = sup{t < q : (ζ δ (x 1 ) ∧ ζ δ (x 2 ))(t) = δ}, b q = inf{t > q : (ζ δ (x 1 ) ∧ ζ δ (x 2 ))(t) = δ}, and I q = (a q , b q ). Suppose ζ δ (x 1 )(q) = ζ δ (x 2 )(q). Then by Theorem on homeomorphisms of flows (cf. Theorem V-46, [11]) applied to equation (6), we have ζ δ (x 1 (t)) = ζ δ (x 2 (t)), t ∈ [q, b q ). On the other hand, if there exists r < q such that ζ δ (x 1 (r)) = ζ δ (x 2 (r)), by the same theorem ζ δ (x 1 (q)) = ζ δ (x 2 (q)). Thus ζ δ (x 1 (q)) = ζ δ (x 2 (q)) implies ζ δ (x 1 (t)) = ζ δ (x 2 (t)), t ∈ I q , and (ζ δ (x 1 ) ∨ ζ δ (x 2 ))(t) = ζ δ (x 1 (t)), t ∈ I q . Comparison (14) with (6) permits the conclusion that L ζ δ (x1)−ζ δ (x2) 0 (t) = 0, t ∈ I q . In the case of ζ δ (x 1 (q)) = ζ δ (x 2 (q)) we get ζ δ (x 1 (t)) = ζ δ (x 2 (t)), t ∈ I q . So for every [α, β] ∈ I q there exists ε 0 > 0 such that From Corollary 3 of [11], p.225 we obtain Let f ∈ C 2 c ([0, ∞)). Then there exists δ > 0 such that f is constant on [0, 2δ], and we have We have seen that the local times L Taking into account that f ′ (x) = f ′′ (x) = 0 on [0, 2δ] and making use of Itô formula we obtain Therefore, the process ((x 1 ∨ x 2 )(t)) t≥0 satisfies equation (5). By the weak uniqueness for all t ≥ 0, Proof of Theorem. Let for each x 0 ≥ 0, there exists a solution to the martingale problem M (a, σ, x 0 ). Then by Lemma 2 the weak uniqueness holds for equation (5). Then the assertion of Theorem follows from Lemma 3 similarly to Yamada-Watanabe theorem (cf. Theorem IV-1.1 of [8]).  (5) stays at the point x 0 forever. Suppose a(x 0 ) = 0. If for some y ∈ B such that y < x 0 , a(y) = 0, a solution starting from x 0 never hits the point y due to homeomorphic property of solutions of SDE (see [9], Ch. 4.4). If there exists y ∈ B such that y ≤ x 0 and a(y) > 0, then a solution of (5) never attends the half-interval [0, y). In two last cases the assertion of the Theorem is fulfilled because a, σ are Lipshitz continuous on [y, ∞) (see, for example [7]). If for all y ∈ B the inequality y > x 0 holds, we need to prove the uniqueness only up to the time of hitting B by a solution. The case when for all y ∈ B such that y ≤ x 0 , we have a(y) < 0 is reduced to the previous one. Thus, if the set B does not have limit points in some neighborhood of 0, then a solution to equation (5) either attends a point of B just once or does not attends it or lives in it forever. The assertion of Theorem holds true in this case. Now, suppose that 0 is a limit point of the set B. Suppose there exists a subsequence {y n : n ≥ 1} ⊂ B such that y n → 0 as n → ∞ and a(y n ) ≥ 0. Then starting away from 0, a solution never hits some neighborhood of 0 and the assertion of Theorem holds true. If such a subsequence does not exist, then there is a neighborhood of 0, say U , such that for all y ∈ B ∩ U , a(y) < 0. In this case, starting from 0 a solution does not attend the interval (y, +∞) for all y ∈ B ∩ U . Thus, if 0 is a limit point of B, a solution either hits 0 in finite time with positive probability or does not hit 0 a.s. In the former case a solution stays at the point 0 forever. But this contradicts with item (iii) of Definition 1. In the latter case the assertion of Theorem is obvious.
Let (x, l) be a pair of continuous processes adapted to the filtration (F w t ) and such that (i) x is non-negative, (ii) l(0) = 0, l(·) is nondecreasing, (iii) l(·) increases only at those moments of time when x(t) = 0, i.e. for each t ≥ 0, holds and all the integrals in the right-hand side of (16) are well-defined. Then the pair (x, l) is called a strong solution to equation (16).
If (x, l) is such a solution, then for each f ∈ C 2 c ([0, ∞)), by Itô formula for semimartingales, we have According to (15), the last member in the right-hand side of (17) is equal to 0. Thus, if the pair of the processes (x, l) is a strong solution to equation (16), then the process x is a strong solution to equation (5) in the sense of Definition 6.
Cherny [2] has shown that there exists a unique non-negative strong solution to equation (18). Let ρ be a non-negative solution to (18). Applying Itô formula we get the equation Thus, (ρ, w) is a weak solution to equation (5) in the sense of Definition 3. Then according to Theorem there exists a strong solution to (5) and the strong uniqueness holds. Therefore we obtain the result of Cherny from ours. 2) Let 0 < β < 1. Let (ρ(t)) t≥0 be a Bessel process on some probability space (Ω, F, (F t ), P ).
Then the process ρ is not a semimartingale. Nonetheless it has the family of local times defined by the formula valid for all t > 0 and for every positive measurable function φ on [0, ∞) a.s. The Bessel process of dimension β is a weak solution to the following equation (cf. [13], Ch.XI, Ex. 1.26) da. Let us check that the pair (ρ, w) is a weak solution to equation (5) in the sense of Definition 3. Then the Theorem yields that there exists a unique strong solution to equation (20). To prove this we need the following Lemma.
Example 3. Let the process (x(t)) t≥0 be a weak solution to an SDE of the form where the coefficients a and b are locally Lipshitz continuous on (0, ∞). Then for each even function being constant in a neighborhood of zero according to Itô formula, we get Note that if the process (x(t)) t≥0 is a weak solution to equation (25) spending zero time at the origin then the process y(t) = |x(t)|, t ≥ 0, satisfies equality (26) for each f ∈ C 2 c ([0, +∞)). By Theorem 1 the process y(t), t ≥ 0, is a unique non-negative strong solution to (26) spending zero time at the origin.
Consider an SDE which can be regarded as an example of equation of the form (25) It is known that there exists a weak solution to (27) spending zero time at the point 0 (cf. [10], 3.10b).
Remark 4. Girsanov [6] has shown that without additional assumption this equation has infinitely many weak solutions.
As a consequence of the following simple Lemma we have that for each m ≥ 1 the process (w m (t)) t≥0 is adapted w.r.t. the filtration generated by the process (x(t)) t≥0 and for all k ≥ m, The proof is trivial. The sequence {w m : m ≥ 1} defined in (30) is fundamental in mean square on compact intervals. Indeed, for k ≥ m, T > 0, using martingale inequality (cf. [8], Theorem I-6.10) and (31), we get The process (w(t)) t≥0 ∈ M c,loc 2 (P ) and w(t) (t) = t 0 1 (0,∞) (x(s))ds = t.