Strong solutions of a stochastic differential equation with irregular random drift

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$\mathrm{d} X= u(\omega,t,X)\, \mathrm{d} t + \frac12 \sigma(\omega,t,X)\sigma'(\omega,t,X)\,\mathrm{d} t + \sigma(\omega,t,X) \, \mathrm{d}W(t), $$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $\sigma$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(\Omega; L^\infty([0,T];\dot{H}^1(\mathbb{R})))$ for any finite $p\ge 1$, $\mathbb{E}\left|u(t)-u(0)\right|_{\dot{H}^1(\mathbb{R})}^2 \to 0$ as $t\downarrow 0$, and $u$ satisfies the one-sided gradient bound $\partial_x u(\omega,t,x) \le K(\omega, t)$, where the process $K(\omega,t )>0$ exhibits an exponential moment bound of the form $\mathbb{E} \exp\Big(p\int_t^T K(s)\,\mathrm{d} s\Big) \lesssim {t^{-2p}}$ for small times $t$, for some $p\ge1$. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter--Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation (SPDE).

1. Introduction 1.1. Main result. In this paper, we prove strong existence and pathwise uniqueness for a class of one-dimensional SDEs with rough random drift u = u(ω, t, x) and a noise coefficient σ = σ(ω, t, x) that is random and possibly degenerate. We fix a stochastic basis S = (Ω, F , {F t } t≥0 , P) consisting of a complete probability space (Ω, F , P) and a complete right-continuous filtration {F t } t≥0 . Moreover, we fix a standard Brownian motion W on S adapted to the filtration {F t } t≥0 .
Here we use the notation h 1 α h 2 if h 1 ≤ C(α)h 2 for some constant C that may depend on α, and non-negative functions h 1 , h 2 . Finally, we require a strong temporal continuity condition at t = 0: The conditions imposed on the drift u are motivated by the work [9], in which u solves a nonlinear stochastic transport equation, and the one-sided gradient bound (1.3) acts as a selection principle for dissipative weak solutions of this SPDE. We will return to the motivation behind the key condition (1.3) later.
The exponential moments (1.8) are used to prove the existence of a solution X that belongs (locally) to L p (Ω; C([0, T ])) for any finite p ≥ 1. Dropping the requirement of arbitrary p-moments, one can relax (1.8) somewhat. By Jensen's inequality, the condition (1.8) implies which will be used on certain occasions. Finally, we will also need the following technical conditions: Remark 1.1. It is possible to consider σ = σ(ω, t, x) such that σ 2 ′′ satisfies the same conditions (1.3) and (1.5) as q. In this case, for our existence result, we must additionally assume (1.9). However, these conditions will fail to include the linear case σ(x) = a + bx, which originally motivated this study (see Section 1.3).
Our main result is the following theorem. The central part of Theorem 1.1 is the uniqueness assertion (cf. Theorem 2.2). We prove pathwise uniqueness by a careful estimation of the difference between two solutions, making essential use of the Tanaka formula, the exponential moment bound (1.3), and a recent stochastic Gronwall inequality [18,21] (see Lemma 2.1 below). The exponential bound (1.3), along with (1.5), allows us to control the difference between the two solutions for short times t ≤ ε (ε ≪ 1), which is the main challenge in demonstrating pathwise uniqueness. When σ ≡ 0, our uniqueness result recovers [21,Prop. A]. The detailed proof reported in Section 2 can be viewed as a surprisingly non-trivial stochastic extension of the ODE proof in [21].
In Section 3, we demonstrate existence of strong solutions to the SDE (1.1) (cf. Theorem 3.7). We approximate (1.1) using "one-sided truncations" {u R } of the drift u, and then make use of Krylov's theorem [10] for SDEs with random coefficients to solve (1.1) with u = u R . This produces a family of solutions {X R }, indexed by the truncation level R with R → ∞. We show that {X R } constitutes a Cauchy sequence in the space L 1/2 (Ω; C([0, T ])), with metric d(X 1 , X 2 ) := E sup t∈[0,T ] |X 1 (t) − X 2 (t)| 1/2 (cf. Proposition 3.5) [1, 4.7.62]. The proof of this result proceeds along the lines of the uniqueness argument. The Cauchy property, along with (1.8) and R-independent p-moments of X R (cf. Lemma 3.3), implies the existence of a limit X ∈ L 2 (Ω; C([0, T ]) such that X R → X in L 2 (Ω; C([0, T ]). It is straightforward to deduce that X is a solution of (1.1) (cf. Theorem 3.7).
Before discussing the literature on SDEs with irregular drift and the motivation behind our particular class of drift coefficients u, let us supply a relevant example of the process K arising in (1.3).

1.2.
Background. Let us contextualise our result by discussing some previous studies on the well-posedness of SDEs. There is a very rich literature studying the existence and uniqueness of solutions, which begins with Itô's work on SDEs with globally Lipschitz coefficients (see [15,Chap. IX]). Often the Lipschitz condition is too strong. While weak existence is relatively easy to obtain for non-smooth coefficients (via, say, Girsanov's theorem), the construction of strong solutions is a more delicate matter. Strong solutions of SDEs with rough deterministic coefficients have been studied by many authors, beginning with [23,17], and later [7,8,11,6], to mention just a few examples. Most of these works use the Fokker-Planck PDE associated with the SDE, the Krylov estimate, and the Zvonkin transformation, which require the noise coefficient to be non-degenerate (uniformly elliptic). As a consequence, the results hold under very weak conditions on the drift, much weaker than in deterministic ODEs. For recent work on the well-posedness of SDEs with (Sobolev) rough coefficients and degenerate noise, see [2]. A probabilistic approach based on Malliavin calculus (nondegenerate noise) is developed in [12,13]. Most of the cited articles assume additive noise. The works [19,20] consider multiplicative noise under non-degeneracy and Sobolev regularity conditions on the noise coefficient. For a detailed study of one-dimensional SDEs, see the book [3]. The influential paper [5] studied stochastic regularisation in linear transport SPDEs with non-smooth velocity b, for which the characteristic equation is dX = b(t, X) dt + dW.
does not satisfy a one-sided bound of the form (1.3). Motivated by [5], there were many additional works studying strong solutions of SDEs like (1.12) with non-smooth drift b, but almost all of them assume that b is deterministic. Let us turn our attention to SDEs with random coefficients. In [10], Krylov established the existence and uniqueness of strong solutions to under some boundedness, monotonicity, and coercivity conditions on the random coefficients b and σ. His proof is based on a detailed convergence analysis of the Euler discretization scheme. We state Krylov's result as Theorem 3.1 below, and use it in Section 3 as a part of the existence proof. Because of an indispensable "logarithmic divergence" at t = 0, Krylov's theorem does not apply to the SDE (1.1) with u satisfying the one-sided gradient bound (1.3).
With a random drift b and σ ≡ 1 in (1.13), the work [4] partially recovered the results of [5] under an additional condition of Malliavin differentiability of b. The proof employed a Girsanov transformation idea [23], which extends the Itô-Tanaka trick in [5], by considering a backward parabolic SPDE instead of the Fokker-Planck PDE associated with X for a deterministic b. We also refer to [14] for a related result, which allows for the drift b(ω, t, where the deterministic part b 1 is measurable and of linear growth. In contrast, the random part b 2 is sufficiently smooth in t, x and Malliavin differentiable in ω. These results were extended and sharpened in [22] to the SDE (1.13) with non-degenerate noise and random coefficients b and σ satisfying similar (t, x)-regularity and Malliavin differentiability conditions. An illustrative example of random drift b covered by these recent works is b(ω, t, x) = f (t, x, W (t)) for a function f that is Lipschitz continuous in the last variable. The works [4,14,22] cannot handle the SDE (1.13) with random drift u ∈ L p (Ω; L ∞ ([0, T ];Ḣ 1 (R))) satisfying (1.3) and (1.5), even if we were to assume that σ(·) > 0. The proof of our Theorem 1.1 will not use ideas based on the associated backward SPDE, nor will we impose non-degeneracy or Malliavin differentiability conditions on our coefficients.

Motivation.
We conclude this introduction with a brief motivation of the current study, which stems from our ongoing investigation into the uniqueness and dissipation properties of solutions to the stochastic Hunter-Saxton equation [9] dq Existence results, along with a specific distribution for wave-breaking (finite-time blowup and continuation), were derived for the nonlinear transport-type SPDE (1.14) in [9]. These results were derived under the condition that σ is linear. Solutions to (1.14) were constructed from its characteristic equation, namely the SDE (1.1).
Using the Itô-Wentzell theorem and the characteristic equation (1.2), the following Lagrangian formulation of (1.14) can be postulated: This SDE can be solved exactly as a stochastic Verhulst equation. The solution is In [9], we constructed the drift u directly in such a way that it was obvious that (1.1) was well-posed, and Q(t, x) = ∂ x u(t, X(t, x)) solved (1.15), providing us with a way to construct solutions to the stochastic Hunter-Saxton equation (1.14) along characteristics. The solution to the SDE (1.15) identifies the dissipative solution of the SPDE (1.14) with an Oleȋnik-type (one-sided gradient) bound. This motivates our study of the SDE (1.1) with random drift u satisfying (1.11), and thus (1.3).
In an ongoing work, we study the uniqueness question for the stochastic Hunter-Saxton equation (1.14). In that work, starting from a solution to the SPDE (1.14), we must derive properties of the solution to the characteristic equation (1.2). The well-posedness theorem in the present paper, which we believe is of independent interest, is needed as a part of that endeavour. Remark 1.3. Finally, we present an example of a random drift u motivated by (1.14), cf. [9]. Fixing a number c ∈ R, let Z 1 (t) be the unique solution to Fixing a number v 0 > 0, we introduce Finally, we set Now we define the adapted and continuous drift coefficient u by Clearly, the gradient blows up (∂ x u → −∞ while |u| remains bounded) as t ↑ T ⋆ but evidently (1.11), and thus (1.3), holds. Besides, ∂ x u ∈ L p (Ω; L ∞ ([0, T ];Ḣ 1 (R))) for all p ≥ 1, and one can easily check that u obeys (1.5). Note that u(t) ≡ 0 for all t > T ⋆ , which corresponds to a dissipative solution of the stochastic Hunter-Saxton equation (1.14).

Pathwise uniqueness
In this section, we prove the uniqueness part of Theorem 1.1. We make essential use of the stochastic Gronwall inequality established recently by Scheutzow [16]. The proof in [16] relies on a martingale inequality of Burkholder that holds for continuous martingales. Below we recall a mild refinement due to Xie and Zhang [18,Lemma 3.8] which holds for general discontinuous martingales. The stochastic Gronwall lemma provides an upper bound for the pth moment of a process ξ that does not depend on the martingale part M of the inequality. It is this convenient "martingale uniformity" that forces p ∈ (0, 1).
. Fix a stochastic basis S. Let ξ(t) and η(t) be non-negative adapted processes, A(t) be a non-decreasing adapted process starting at A(0) = 0, and M be a local martingale with M (0) = 0. Suppose ξ is càdlàg in time and satisfies the following pathwise differential inequality: For any 0 < p < r < 1 and t ∈ [0, T ], We are now in a position to prove the following result.
Proof. Let X 1 , X 2 , and T be as in the statement of the theorem. Without loss of generality, we assume throughout the proof that for some N > 0. Indeed, introducing the stopping time We can therefore apply the upcoming argument toX 2 −X 1 on [0, τ N ] instead of to for any finite N . By the continuity of X 1 and X 2 , we have that τ N → T a.s. as N → ∞. Therefore, sending N → ∞, we arrive at (2.1).
In what follows, we consider X 2 − X 1 and assume (2.2). We have by linearity Set Y := |X 2 − X 1 |. By the Tanaka formula, Since the local time L 0 Y at 0 of Y is supported on the zero set of X 2 − X 1 , which is a subset of the zero set of σ(t, X 2 ) − σ(t, X 1 ), the local time correction term is zero. Set φ σ (t) := (σ(t, X 2 ) − σ(t, X 1 )) /Y (t), which is a process uniformly bounded in absolute value by Λ(t) of (1.9). Integrating in time yields where, in view of (1.9) and (2.2), the last term is a square-integrable martingale starting from zero; see (1.11) for the definition q. Making use of (1.10b) and taking the expectation, we obtain q(s, y) dy ds by the Cauchy-Schwarz inequality. Here, ∆ s denotes the (random) interval Taking the supremum over t ∈ [0, ε] on both sides gives The first term on the right-hand side can be absorbed by the term on the left-hand side. We then divide through by sup 1/2 and square both sides, eventually arriving at The estimate (2.4) allows us to control E Y (t) near t = 0. Using the one-sided bound (1.11), which deteriorates near t = 0 for every ω ∈ Ω, in combination with the quadratic short-time estimate (2.4), we will next deduce a global estimate on the entire time interval [0, T ].
Remark 2.1. We point out that whilst the result above holds for q(0) ∈ L 2 (R), that is, q 2 (0) ∈ L 1 (R), it fails for general q(t) for which the right-continuity limit lim t↓0 q 2 (t) exists only in the sense of measures-but not in L 1 as required by (1.5). An example comes from the deterministic Hunter-Saxton equation with an initial condition of the form q 2 (0) = δ 0 . Although it is possible to define characteristics for this case, the characteristics emanating from x = 0 are not unique. The temporal continuity condition (1.5) is essential.

Existence of solution
In this section, we establish the existence of strong solutions for the SDE (1.1) by approximating (1.1) using a truncated coefficient in a way that allows us to apply a well-posedness theorem of Krylov, reproduced below. We then show that the solutions to the approximating SDEs form a Cauchy sequence in an appropriate space, from which we recover a solution to our SDE.
We begin by recalling Krylov's theorem for the well-posedness of SDEs with random coefficients [10, Thm. 1.2].
[10] Let S be a stochastic basis. Assume that for any ω ∈ Ω, t ≥ 0, and x ∈ R d , we have V (ω, t, x) ∈ R d×d and b(ω, t, x) ∈ R d , and that V and b are continuous in x for any (ω, t), and measurable in (ω, t). Moreover, assume (i) boundedness: for any T, ℓ ∈ [0, ∞), ω ∈ Ω, and any matrix norm V , (ii) monotonicity: for all t, ℓ ∈ [0, ∞), x, y ∈ B ℓ (0), the ball with radius ℓ and centred at the origin, and ω ∈ Ω, (iii) coercivity: for all t, ℓ ∈ [0, ∞), x ∈ B ℓ (0), and ω ∈ Ω, whereK(t, ℓ) is an adapted non-negative processes satisfyinĝ has a solution which is unique up to indistinguishability. Moreover, Unfortunately, the factor multiplying |x − y| 2 is not sufficiently well controlled at t = 0 to ensure (3.1). There is the possibility of a logarithmic divergence in the temporal integral. As a result, Theorem 3.1 does not apply to our problem.
Next we introduce an approximate SDE by truncating the gradient q = ∂ x u. The reason for doing so is explained in Remark 3.1. The strong well-posedness of these approximate SDEs then follows from Theorem 3.1.
has a unique strong solution.
By assumption, E u p p 1 for all p ∈ [1, ∞). Of course, the same bound holds for u R :
The next lemma supplies R-independent estimates for X R in L p (Ω; C([0, T ])) for any finite p. Note carefully that the L 2 -estimate on X R (t) coming from Theorem 3.1, cf. (3.2), is useless because our K depends on R. Lemma 3.3. Let X R be the solution constructed in Lemma 3.2. Assume in addition that (1.8) and (1.10d) hold. We have the uniform-in-R bound Proof. We make frequent use of the following elementary inequalities, which hold for all r ≥ 2 and a, b, ǫ > 0: By Itô's formula, |X R (t)| 2p = |x| 2p + I 1 (t) + I 2 (t) + I 3 (t) + M (t), where Given (1.7), we readily derive the bounds for a constantC p depending only p. From this we obtain the inequality for another constant C p depending only p.
For any N > 0, introduce the stopping time By the continuity of X R we have that τ N → T , P-almost surely, as N → ∞. Clearly, for t ∈ [0, T ], where t → M (t ∧ τ N ) is a (square-integrable) martingale starting from zero. Using the stochastic Gronwall inequality (Lemma 2.1 with exponents 1 2 and 2 3 ), Given (1.10d) and (3.5), we conclude that Finally, sending N → ∞, we arrive at (3.6).
To show that {X R } is a Cauchy sequence, we will require some compactness properties of u R as R → ∞. Since u R is constructed from u in an explicit manner, this is not difficult to establish: Lemma 3.4. Suppose u ∈ L p (Ω; L ∞ ([0, T ];Ḣ 1 (R))), for p ∈ [1, ∞). Let u R be defined by the construction (3.3). We have the convergence Moreover, for any finite p ≥ 1, 1 by assumption, we find that E I R (t) tends to zero as R → 0, uniformly in t ∈ [0, T ]; hence (3.7) holds. We also havẽ By assumption, for all p ≥ 1 we have q ∈ L p (Ω; L ∞ ([0, T ]; L 2 (R))) and therefore This proves the claim (3.8).
The next result, which is the main contribution of this section, reveals that {X R } is a Cauchy sequence in L 1/2 (Ω; C([0, T ])). Proof. For N, R, R ′ > 0, define Applying the upcoming argument toX R −X R ′ on the time interval 0, τ R,R ′ N , wherẽ , we deduce that for any δ > 0 there exists R 0 = R 0 (δ) such that, for all t ∈ [0, T ], (3.15). To conclude from this, one notices that τ R,R ′ N → T as N → ∞, uniformly in R, R ′ . Indeed, the R-independent bound (Lemma 3.3) E sup as N → ∞, uniformly in R. Hence, τ R,R ′ N → T as N → ∞, uniformly in R, R ′ . Given the preceding discussion, in what follows, there is no loss of generality in assuming that for some given N > 0, when seeking to establish that satisfies the Cauchy property (3.15). The Tanaka formula gives (3.10) This is very similar to (2.3), except for the difference u R (s, First we seek to estimate Y (t) over a short time period t ∈ [0, ε]. In (3.10), as in the previous section, we writê ϑ R (q(s, y)) dy ds Estimating by the Cauchy-Schwarz inequality, where φ σ (t) := (σ(t, X R ) − σ(t, X R ′ )) /Y (t) is a process bounded in absolute value by Λ(t) of (1.9). Here, ∆ s denotes the (random) interval Given (3.9), the last term in (3.11) is a square-integrable martingale starting from zero. Taking the expectation, and estimating as in the proof of Theorem 2.2, Taking the supremum over t ∈ [0, ε], and applying Young's inequality (in the In what follows, we fix ε so small that 1 4 + 1 16 + ε 4 σ 2 ′′ (0) L ∞ (Ω×R) ≤ 1 2 . Since ε and R, R ′ are independent parameters, given (3.8) of Lemma 3.4, we can take R 0 = R 0 (ε) so large that (1), as ε → 0, (3.12) for all R, R ′ ≥ R 0 (ε). This gives us . Importantly, from (1.10c) and (2.7) we conclude that (3.13) sup As in the proof of Theorem 2.2, we estimate Y again (this time on the entire time interval [0, T ]). From (3.11), we arrive at the integral inequality where, for t ∈ [0, T ], and, as in (2.5), Since we have not assumed an exponential moment bound for the difference u R − u R ′ L ∞ ([0,T ]×R) , it becomes imperative to include this term as a part of η and not A. The process η is non-negative and, by (1.10c), (3.13) and (3.12), is controlled thus: Now we apply Lemma 2.1, the stochastic Gronwall inequality with p = 1 2 and a suitable r ∈ 1 2 , 1 . In view of (1.3) and (3.14), Therefore, given any δ > 0, we can find ε = ε(δ) and such that, for all t ∈ [0, T ], This concludes the proof of the proposition. Proof. By Chebyshev's inequality and Proposition 3.5, we obtain P sup so that {X R } is a Cauchy sequence in the space of continuous processes with respect to locally (in t) uniform convergence in probability. Since this space is complete, the lemma follows.
It remains to identify the limit X as a solution to the original SDE (1.1). Proof. Fix a finite number T > 0. By Lemma 3.2, there exists a unique strong solution X R to the SDE (3.4), such that X R (t) = x +ˆt 0 u R (s, X R ) ds + 1 4ˆt 0 σ 2 ′ (s, X R ) ds +ˆt 0 σ(s, X R ) dW (s).
This completes the proof that X is a solution of the SDE (1.1).