Young differential equations with power type nonlinearities

Transcription

1 Avilble online t ScienceDirect Stochstic Processes nd their Applictions 127 (217) Young differentil equtions with power type nonlinerities Jorge A. León, Dvid Nulrt b, Smy Tindel c, Depto. de Control Automático, CINVESTAV-IPN, Aprtdo Postl 14-74, 7 México, D.F, Mexico b Deprtment of Mthemtics, University of Knss, 45 Snow Hll, Lwrence, KS, USA c Deprtment of Mthemtics, Purdue University, 15 N. University Street, W. Lfyette, IN 4797, USA Received 6 June 216; received in revised form 2 November 216; ccepted 24 Jnury 217 Avilble online 7 Februry 217 Abstrct In this note we give severl methods to construct nontrivil solutions to the eqution dy t = σ (y t ) dx t, where x is γ -Hölder R d -vlued signl with γ (1/2, 1) nd σ is function behving like power function ξ κ, with κ (, 1). In this sitution, clssicl Young integrtion techniques llow to get existence nd uniqueness results whenever γ (κ + 1) > 1, while we focus on cses where γ (κ + 1) 1. Our nlysis then relies on Zähle s extension (Zähle, 1998) of Young s integrl llowing to cover the sitution t hnd. c 217 Elsevier B.V. All rights reserved. Keywords: Frctionl Brownin motion; Frctionl clculus; Young integrtion; Integrl equtions 1. Introduction Let T > be fixed rbitrry horizon, nd consider noisy function x : [, T ] R d in the Hölder spce C γ ([, T ]; R d ), with γ > 1/2. Let σ 1,..., σ d be some vector fields on R m, be n initil dt in R m nd consider the following integrl eqution y t = + d j=1 t σ j (y u ) dx j u, t [, T ]. (1) Corresponding uthor. E-mil ddresses: jleon@ctrl.cinvestv.mx (J.A. León), nulrt@ku.edu (D. Nulrt), stindel@purdue.edu (S. Tindel) / c 217 Elsevier B.V. All rights reserved.

2 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) When σ 1,..., σ d re smooth enough, Eq. (1) cn be solved thnks to frctionl clculus [12,16] or Young integrtion techniques [7,8]. Extensions of these methods, thnks to the rough pths theory (see e.g. [3,4,9]), lso llow to hndle cses of signls with regulrity lower thn 1/2. In the current pper, we re concerned with different, though very nturl problem: cn we define nd solve Eq. (1) for coefficients which re only Hölder continuous? Stted in such generlity the question is still open, but we consider here the specil cse of coefficient σ behving like power function. This problem hs quite long story, nd full nswer in the cse of 1-dimensionl eqution driven by stndrd Brownin motion is given in [6,15]. The bsic ide on which Wtnbe Ymd s contribution relies, is the following priori estimte. Consider Eq. (1) driven by Brownin motion B, with non-linerity σ (ξ) = ξ κ where κ > 1/2. Nmely, let y be solution to y t = + t y u κ d B u, t [, T ], (2) where the differentil with respect to B is understood in the Itô sense. Then obviously the min problem in order to estimte y is its behvior close to, since elsewhere ξ ξ κ is Lipschitz function. For n 1 we thus consider n pproximtion ϕ n of the function ξ ξ such tht ϕ n Cb 2(R), ϕ n nd ϕ n (2) n. Then pplying Itô s formul to Eq. (2) we get E [ϕ n (y t )] = ϕ n () t E ϕ n (2) (y u) y u 2κ du. (3) The right hnd side of Eq. (3) is then controlled by noticing tht, whenever y u 1/n, we hve ϕ n (2) (y u ) y u 2κ n (2κ 1). This quntity converges to s n, which is the key step in order to control E[ϕ n (y t )] in [15]. The method described bove in order to hndle the Brownin cse is short nd elegnt, but fils to give true intuition of the phenomenon llowing to solve Eq. (1) with power type coefficient. This intuition hs been highlighted in [1,11], though in the much more technicl context of the stochstic het eqution. In order to understnd the min ide, let us go bck to Eq. (1) understood in the Young sense. Then two cses cn be thought of (we restrict our considertions to 1-dimensionl pths in the reminder of the introduction for nottionl ske): (i) One expects y to be n element of C γ, since the eqution is driven by x C γ. This mens tht σ (y) should lie in C κγ. When κ stisfies κ γ + γ > 1, ech integrl t σ (y u) dx u cn thus be defined s usul Young integrl, nd Eq. (1) is solved thnks to clssicl methods s in [4,8,12,16]. (ii) Let us now consider the cse κ γ + γ 1. If one wishes to define the integrl t σ (y u) dx u properly when y u is close to, the heuristic rgument is s follows: when y u is smll the eqution is bsiclly noiseless, so tht σ (y) should be considered s C κ -Hölder function insted of C κγ -Hölder function. This mens tht the expected condition on κ in order to solve Eq. (1) is just κ + γ > 1. As mentioned bove, this strtegy hs been successfully implemented in [1,11] in Brownin SPDE context. It hevily relies on the regulrity gin when y hits. In our cse, we will follow two directions which re somehow different in their nture: (i) We will see tht if y does not hit too shrply, this condition being quntified in n integrl wy, then the integrls t σ (y u) dx u

3 344 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) still hve good chnce to be defined even if κ γ + γ < 1. One cn then construct solution of (1) in this lndmrk. (ii) Another pproch consists in quntifying the regulrity gin enforced by Eq. (1) when the solution y pproches. In this wy, one cn get some uniform priori Hölder bounds on y nd invoke some compctness rguments. To be more specific, we shll proceed s follows: (1) We strt with generl lemm on Young integrtion. Nmely (see Proposition 2.4 for precise sttement), we consider η such tht (κ + η)γ > 1 γ. We lso consider pth y C γ nd function σ behving like power function ξ κ. By dding the ssumption y 1 L q ([, τ]) with q = η γ (κ+η), we prove tht t σ (y u) dx u is well defined s Youngtype integrl nd gives rise to γ -Hölder function. Notice tht we hve crried out this prt of our progrm with frctionl integrtion techniques becuse the clcultions re esily expressed in this setting. We cn however link the integrl we obtin with Riemnn sums, s will be shown in Theorem 2.6. (2) With this integrtion result in hnd, we consider the 1-dimensionl version of Eq. (1) nd perform Lmperti-type trnsformtion y t = φ 1 (x t ), where φ(ξ) = ξ [σ (s)] 1 ds. Then we prove tht y is solution to our eqution of interest by identifying the Young integrl t σ (y u) dx u for y t = φ 1 (x t ). Our result is vlid for ny κ such tht γ (1 + κ) < 1, nd we refer to Theorem 3.7 for precise sttement. (3) In cse of multidimensionl setting, our globl strtegy is different. Nmely, we will bse our considertion on the fct tht when y u is close to, its regulrity is higher thn expected (s mentioned bove). Specificlly, our bsic priori estimte for (1) sttes tht whenever solution y stisfies y u 2 k for u lying in n intervl I, then we lso hve y t y s of order 2 κk t s γ for s, t I. Our regulrity gin is thus expressed by the coefficient 2 κk bove. This gin is sufficient to get to the existence of γ -Hölder continuous solution to Eq. (1) in the d-dimensionl cse. We will then construct solution which vnishes s soon s it hits the origin (see Theorem 4.15). Summrizing the considertions bove, we re ble to get existence theorems for Eq. (1) with power type nonlinerities in wide rnge of cses. The sitution would obviously be clerer if we could get the corresponding pthwise uniqueness results, like in the forementioned Refs. [6,1,11,15]. However, these rticles hndle the cse of Itô type equtions, for which uniqueness is expected. In our Strtonovich Young cse uniqueness of the solution is ruled out, since both the nontrivil solution we shll construct nd the solution y solve Eq. (1) when =. We shll go bck to this issue below. Our pper is structured s follows: the Young s integrl relted to our power type coefficient is studied in Section 2. Section 3 dels with its ppliction to the existence of solutions to Eq. (1) in dimension 1. The other pproch, bsed on the priori regulrity gin of the solution when it hits, is developed in Section 4. Finlly, in Section 5 we discuss the ppliction of these results to the cse of stochstic differentil equtions driven by frctionl Brownin motion. Nottions. Throughout the rticle, we use the following conventions: for 2 quntities nd b, we write b if there exists universl constnt c (which might depend on the prmeters of the model, such s, γ, κ, η, α, T,...) such tht c b. If f is vector-vlued function defined on n intervl [, T ] nd s, t [, T ], δ f st denotes the increment f t f s.

4 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) An extension of Young s integrl This section is devoted to n extension of Young s integrl using frctionl clculus techniques, which will be suitble to hndle Eq. (1) with Hölder-type nd singulr nonlinerities. We shll first recll some generl elements of frctionl clculus Elements of frctionl clculus We restrict this introduction to rel-vlued functions for nottionl ske. Consider < b T nd n L 1 ([, T ])-function f. For t [, b] nd α (, 1) the frctionl integrls of f re defined s I α + f t = 1 Γ (α) t (t r) α 1 f r dr, nd I α b f t = 1 Γ (α) b t (r t) α 1 f r dr. For ny p 1, we denote by I+ α (L p ) the imge of L p ([, b]) by I+ α, nd similrly for I b α (L p ). The inverse of the opertors I+ α nd I b α re clled frctionl derivtives, nd re defined s follows. For f I+ α (L p ) nd t [, b] we set D+ α f t = L p 1 f t ε t lim ε Γ (1 α) (t ) α + α f t f r dr, (4) (t r) 1+α where we use the convention f r = on [, b] c. In the sme wy, for f Ib α (L p ) nd t [, b], we set Db α f t = L p 1 f b t lim ε Γ (1 α) (b t) α + α f t f r dr. (5) (r t) 1+α By [14, Remrk 13.2] we hve tht, for p > 1, f I+ α (L p ) (resp. f Ib α (L p )) if nd only if f L p ([, b]) nd the limit in the right-hnd side of (4) (resp. (5)) exists. In this cse f = I+ α (Dα + f ) (resp. f = I b α (Dα b f )). It is not difficult to see tht, s consequence of the proof of [14, Theorem 13.2], the fct tht f L p f ( ) ([, b]), ( ) α nd f ( ) f r f ( ) dr (resp. ( r) 1+α (b ) α nd b f ( ) f r dr) belong to L p ([, b]) implies tht f I α (r ) 1+α + (L p ) (resp. f Ib α (L p )) nd D+ α f 1 f t t t = Γ (1 α) (t ) α + α f t f r dr (6) (t r) 1+α resp. D α b f t = 1 f t Γ (1 α) (b t) α + α b t t+ε f t f r dr. (r t) 1+α Notice tht C α+ε ([, b]) I+ α (L p ), with ε >. In the sme mnner, we hve C α+ε ([, b]) Ib α (L p ). Let g, f L 1 ([, T ]) be two functions such tht, for some α (, 1), f I+ α (L1 ) nd g b Ib 1 α (L1 ), where gr b g if nd only if (D+ α f )D1 α following wy = g r g b. In this cse we sy tht f is integrble with respect to L 1 ([, b]). In this cse we define the integrl b f dg in the b gb r b f r dg r := b (D α + f r )D 1 α b gb r dr. (7)

5 346 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) If f C α ([, b]) nd g C λ ([, b]) (i.e., f nd g re α-hölder nd λ-hölder continuous, respectively) with α + λ > 1, then it cn be checked tht b f r dg r is well-defined, nd tht it coincides with Young s integrl defined s limit of Riemnn sums (see [16, Theorem 4.2.1]). We shll nlyze below this integrl under hypotheses suited to our purposes (see Proposition 2.4), tht is, to solve Eq. (1) The frctionl integrl We ssume in this section tht x is γ -Hölder continuous nd rel vlued signl. In this section, s in the remining of this pper, we ssume tht γ (1/2, 1). Consider the following dditionl ssumption on the coefficient σ : R m R m. Hypothesis 2.1. The function σ : R m R m stisfies σ () = nd σ (ξ 2 ) σ (ξ 1 ) ξ 2 κ ξ 1 κ, ξ 1, ξ 2 R m, (8) for some κ (, 1) such tht γ (κ + 1) < 1. Remrk 2.2. In order to understnd the implictions of Hypothesis 2.1, note tht if σ fulfills condition (8) nd if we consider ξ 1, ξ 2 R m such tht ξ 1 = ξ 2, then we obviously hve σ (ξ 2 ) = σ (ξ 1 ). Thus (8) implies tht σ is rdil function, tht is, σ (ξ) = ρ( ξ ), where ρ : [, ) R m. On the other hnd, it is not difficult to see tht rdil function σ (ξ) = ρ( ξ ) such tht ρ C 1 ((, )), ρ() = nd ρ (1) (y) y κ 1, y >, stisfies inequlity (8). For function σ stisfying Hypothesis 2.1, we define σ (ξ2 ) σ (ξ 1 ) N κ,σ := sup ξ 2 κ ξ 1 κ : ξ 2, ξ 1 R m, ξ 1 ξ 2. (9) We now lbel the following uxiliry result for further use. Lemm 2.3. Assume σ stisfies Hypothesis 2.1. Then we hve σ (ξ 2 ) σ (ξ 1 ) κ κ + η N κ,σ ξ2 η + ξ 1 η ξ 2 ξ 1 κ+η, for ny η 1 κ nd ξ 1, ξ 2 R m \ {}. Proof. The cse η = or η = 1 κ is obvious, so we ssume < η < 1 κ. Without loss of generlity, we cn ssume tht ξ 1 ξ 2. According to (8), we cn write σ (ξ 2 ) σ (ξ 1 ) N κ,σ ξ2 κ ξ 1 κ which yields our clim. ξ2 ξ2 = κn κ,σ z κ 1 dz κn κ,σ ξ 1 η z κ+η 1 dz ξ 1 ξ2 κn κ,σ ξ 1 η (z ξ 1 ) κ+η 1 dz, ξ 1 We re now redy to provide result on the integrl defined in (7). To do this, for ny λ (, 1) nd η >, we introduce the spce C λ η ([, T ]; Rm ) = {y C λ ([, T ]; R m ) : y 1 L η ([, T ]; R)} (1) ξ 1

6 nd use the convention f λ := J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) f (t) f (s) sup s<t T (t s) λ, for ny f C λ ([, T ]; R m ). Proposition 2.4. Assume tht σ stisfies Hypothesis 2.1 nd following results hold true: (i) If y C γ η ([, T ]; R m ), then, for ny t [, T ], the integrl [Λ(y)] t := t σ (y s ) dx s, 1 γ (1+κ) γ < η < 1 κ. Then the is well defined in the sense of reltion (7). (ii) Consider y C γ η ([, T ]; R m ). Then Λ(y) belongs to the spce C γ ([, T ]; R m ), nd the γ (κ+η) following bound holds true: Λ(y) γ x γ σ (y) + N κ,σ y κ+η γ where N κ,σ hs been introduced in (9). Remrk 2.5. Tking into ccount tht the function η γ (κ+η) T η η+κ γ (κ+η) y s η γ (κ+η) ds, (11) is strictly incresing we deduce tht η > γ 1 η 1 κ if nd only if. Therefore, the integrbility condition for y 1 in sttement (ii) is stronger thn tht in sttement (i). > 1 γ κγ γ (1 γ ) Proof of Proposition 2.4. Let α be such tht 1 γ < α < γ (κ +η), which implies αγ 1 κ < η < 1 κ. Let t 1 < t 2 T. Recll tht the integrl t 2 t 1 [σ (y)] s dx s is defined by formul (7). To show tht this integrl exists nd to estblish suitble estimtes, we first nlyze the frctionl derivtive of x Dt 1 α 2 xt 2 s = 1 x s x t2 t2 x s x r Γ (α) + (1 α) dr (t 2 s) 1 α (r s) 2 α t2 x γ (t 2 s) α+γ 1 + x γ (r s) α+γ 2 dr s s x γ (t 2 s) α+γ 1, (12) where we hve used the fct tht α + γ > 1 for the lst step. Hence, we cn write t2 [Dt α 1 + σ (y)] s Dt 1 α 2 xt 2 s ds x γ Jt 1 1 t 2 + Jt 2 1 t 2, with nd t 1 J 1 t2 t 1 t 2 = σ (y) (s t 1 ) α (t 2 s) α+γ 1 ds t 1 t2 s Jt 2 σ (y s ) σ (y u ) 1 t 2 = t 1 t 1 (s u) α+1 du (t 2 s) α+γ 1 ds.

7 348 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) It is now redily checked tht J 1 t 1 t 2 σ (y) (t 2 t 1 ) γ. (13) For the term Jt 2 1 t 2, invoking Lemm 2.3 nd some elementry lgebric mnipultions, we get J 2 t 1 t 2 N κ,σ t2 t 1 N κ,σ y κ+η γ N κ,σ y κ+η γ + t2 t 1 α+γ 1 (t 2 s) y u η t2 t 1 s t 1 α+γ 1 (t 2 s) t2 (t 2 s) α+γ 1 y s η t 1 t2 u ys η + y u η y s y u κ+η duds (s u) α+1 s ys η + y u η (s u) γ (κ+η) α 1 duds t 1 s t 1 (s u) γ (κ+η) α 1 duds (t 2 s) α+γ 1 (s u) γ (κ+η) α 1 dsdu. (14) Notice tht η > αγ 1 κ implies tht γ (κ + η) α >. This implies tht the integrl t2 t 1 [σ (y)] s dx s is well defined, provided y 1 L η ([, T ]; R). Applying Hölder s inequlity with p 1 = γ (κ + η) nd q 1 = 1 p 1, nd ssuming y 1 L η/(γ (κ+η)) ([, T ]; R), yields t2 1/p Jt 2 1 t 2 N κ,σ y κ+η γ y u pη du t 1 t2 (t 2 s) q(α+γ 1) (s t 1 ) q(γ (κ+η) α) ds + t 1 t2 t 1 1/q (t 2 u) qγ (κ+η+1) q du. 1/q Now simple nlysis of the exponents in the bove reltion implies T γ (κ+η) Jt 2 1 t 2 N κ,σ y κ+η γ y s η γ (κ+η) ds (t 2 t 1 ) γ. (15) Finlly, the estimte (11) follows from (13) nd (15). The proof is now complete The integrl vi Riemnn sums The next gol is to see tht the integrl Λ(y) given in Proposition 2.4 cn be pproximted by Riemnn sums. Towrds this end, for ny n 2, we consider uniform prtition Π n = { = t 1 < t 2 < < t n = b} of the intervl [, b] [, T ], such tht Π n := b n 1 = t j+1 t j for ll j {1, 2,..., n 1}. For y s in Proposition 2.4(i), we define the following pproximtion bsed on Π n n zs n = 1 ti σ (y r )dr 1 (ti 1,t Π n i ](s), s [, b]. (16) i=2

8 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) We observe tht, owing to [16, Corollry 2.3], we hve b n zs n dx 1 ti s = σ (y s )ds δx ti 1 t Π n i, i=2 where the left hnd side is understood s in reltion (7) nd where we recll tht δx uv := x v x u. The convergence of b zn s dx s is given in the following theorem, which is the min result of this subsection. Here we use the Definitions (1) nd (16). Theorem 2.6. Suppose tht σ stisfies Hypothesis 2.1. Let η be such tht Consider y C γ η ([, T ]; R m ). Then for ll < b T we hve b lim n z n s dx s = b where z n is defined in (16). σ (y s )dx s, 1 γ (1+κ) γ In order to prove this theorem, we first go through series of uxiliry results. < η < 1 κ. Lemm 2.7. Let σ stisfy Hypothesis 2.1, y C γ ([, T ]; R m ) nd consider [, b] [, T ]. Then for ll s [, b] we hve σ (y s ) zs n N κ,σ y κ γ Π n κγ. Proof. For s (, b], the definition of z n gives σ (ys ) zs n n = σ (y s) 1 ti σ (y r )dr Π i=2 n 1 (,t i ](s) n 1 ti σ (y s ) σ (y r ) dr Π i=2 n n 1 ti N κ,σ ys κ y r κ dr Π n i=2 Since y is γ -Hölder continuous, we thus hve σ (ys ) zs n Nκ,σ y κ γ N κ,σ y κ γ which completes the proof. n i=2 1 Π n ti s r κγ dr n Π n κγ 1 (ti 1,t i ](s), i=2 1 (ti 1,t i ](s) We now estimte the Hölder regulrity of our pproximtion z n. 1 (ti 1,t i ](s). 1 (ti 1,t i ](s) Lemm 2.8. Let σ nd y be functions verifying the ssumptions of Theorem 2.6. Then, for < u < s b, we hve z n s zu n y κ+η γ Φ n u,s + Ψu,s n,

9 35 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) where nd Φ n u,s = Π n γ (κ+η) 1 Ψ n u,s = (s u)γ (κ+η) Π n 2 j<i n 2 j<i n ti y r η dr + ti y r η dr + t j t j 1 y r η dr t j t j 1 y r η dr 1 (t j 1,t j ](u)1 (ti 1,t i ](s) 1 (t j 1,t j ](u)1 (ti 1,t i ](s). Proof. Assume s (, t i ]. If u lies into (, t i ] too, then zs n zn u = by definition of zn. We now ssume tht u (t j 1, t j ] with j {2,..., i 1}. Then it is redily checked tht zs n zn u = 1 ti t j σ (y r ) dr σ (y r ) dr Π n t j 1 = 1 Π n t j t j 1 σ (yr+ti 1 t j 1 ) σ (y r ) dr. Therefore, thnks to Lemm 2.3 we obtin z n s zu n 1 t j Nκ,σ yr+ti 1 t Π n j 1 η + y r η yr+ti 1 t j 1 y r κ+η dr t j 1 from which we derive y κ+η γ t j 1 γ (κ+η) N κ,σ Π n zs n zn u y κ+η γ (s u + Π n ) γ (κ+η) Π n ti y r η dr + 2 j<i n Our clim is now esily deduced. t j t j t j 1 yr+ti 1 t j 1 η + y r η dr, t j 1 y r η dr 1 (t j 1,t j ](u)1 (ti 1,t i ](s). The next result will help to hndle some of the terms ppering in Lemm 2.8. Lemm 2.9. Let the ssumptions of Theorem 2.6 previl, nd consider the pth Φ n : [, b] 2 R + introduced in Lemm 2.8. We lso introduce the following mesure on [, b] 2 µ(du, ds) = (s u) α 1 (b s) α+γ 1 1 {u<s} duds, (17) where α is such tht 1 γ < α < γ (κ + η). Then Φ n converges to zero in L 1 ([, b] 2, µ), s n. Proof. We cn write Φ n L 1 ([,b] 2,µ) Π 1+γ (κ+η) n ti t j 2 j<i n ti y r η dr + t j t j 1 y r η dr t j 1 (s u) 1 α duds I n 1 + I n 2, (18)

10 where nd J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) n I1 n = Π 1+γ (κ+η) n n I2 n = Π 1+γ (κ+η) n ti t j i=3 ti i=4 j=2 y r η dr + i 2 ti t j 1 (s u) 1 α duds. ti 1 t i 2 y r η dr + We now bound the terms I1 n nd I 2 n seprtely. It is esily seen from the expression of I1 n tht I n 1 Π n γ (κ+η) α n ti i=2 ti y r η dr t j y r η dr = Π n γ (κ+η) α t j 1 y r η dr b y r η dr. Hence, due to the fct tht γ (κ + η) α >, we obtin lim n I1 n =. As fr s I2 n is concerned, simple scling rgument entils I2 n Π n γ (κ+η) α n i 2 ti t y r η j dr + y r η dr t j 1 i i 1 j j 1 i=4 j=2 (s u) 1 α duds, ti 1 nd roughly bounding the term s u by i j 1 in the integrl bove, we get I2 n Π n γ (κ+η) α n i 2 ti t y r η j dr + y r η dr (i j 1) 1 α t j 1 Π n γ (κ+η) α i=4 j=2 n ti i=2 t i 2 y r η n 1 b dr k 1 α Π n γ (κ+η) α y r η dr. k=1 (s u) 1 α duds We thus get lim n I2 n =, gin ccording to the fct tht γ (κ + η) α >. Finlly, tking into ccount lim n I1 n =, lim n I2 n = nd reltion (18), our clim is now proved. Still hving in mind bound on the terms of Lemm 2.8, we stte the following intermedite result. Lemm 2.1. Assume the hypotheses of Lemm 2.9 hold true nd let Ψ n be s in Lemm 2.8. Then s n, Ψ n converges in L 1 ([, b] 2, µ) to the function Ψ defined s follows Ψ u,s = y s η + y u η (s u) γ (κ+η) 1 {u<s}. Proof. The result is n immedite consequence of the fct tht y η L 1 ([, b]), together with the conditions α + γ 1 > nd γ (κ + η) > α. We re now redy to give the proof of Theorem 2.6.

11 352 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Proof of Theorem 2.6. Let α be such tht 1 γ < α < γ (κ + η). Owing to (12) we cn write b z n s σ (y s ) b dx s = D α + σ (y) z n s D1 α b xb s ds x γ L n 1 + L n 2, where b σ L n (ys 1 = ) zs n (s ) α (b s) α+γ 1 ds nd b L n 2 = s σ (y s ) zs n σ (y u ) zu n (s u) α+1 du (b s) α+γ 1 ds. Moreover, notice tht invoking Lemm 2.7 we cn deduce tht L n 1 N κ,σ y κ γ Π n κγ. Therefore L n 1 goes to zero s n. Thus, in order to finish the proof we only need to see tht Ln 2 converges to zero s n. In order to study the limit of L n 2, first notice tht thnks to Lemm 2.7 we cn write σ (ys ) zs n σ (y u ) zu n σ (ys ) zs n + σ (yu ) zu n Nκ,σ y κ γ Π n κγ, (19) which implies tht the integrnd in L n 2 converges to zero s n tends to infinity, for ech u nd s such tht u < s b. On the other hnd, we cn lso bound the rectngulr increment σ (y s ) zs n (σ (z u) zu n ) s follows σ (ys ) zs n σ (y u ) zu n σ (ys ) σ (y u ) + z n s zu n. (2) Lemm 2.3 plus the fct tht y C γ η imply tht σ (y s ) σ (y u ) y s η + y u η y s y u κ+η y s η + y u η (s u) (κ+η)γ. Since (κ + η)γ > α, we get tht the term σ (y s ) σ (y u ) is integrble in [, b] 2 with respect to the mesure µ(du, ds) = (s u) α 1 (b s) α+γ 1 1 {u<s} duds introduced in Eq. (17). Moreover, the term zs n zn u is bounded, up to constnt, by Φu,s n + Ψ u,s n (see Lemm 2.8). Applying the dominted convergence theorem s stted in [13, Theorem ], together with Lemms 2.9 nd 2.1, we deduce tht L n 2 tends to s n tends to infinity, which finishes the proof. 3. One-dimensionl differentil equtions The purpose of this section is to obtin existence results for the system (1) in dimension 1, tht is for the following eqution: y t = t σ (y s )dx s, t. (21) Here, recll tht we ssume x C γ ([, T ]; R), with γ (1/2, 1). We now give generl condition on the coefficient σ in (21), which will previl for the reminder of this section. Hypothesis 3.1. We suppose tht σ : R R + stisfies Hypothesis 2.1, nd moreover (i) σ is continuous function such tht σ (ξ) > for ξ. (ii) 1/σ is integrble on compct neighborhoods of zero.

12 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Remrk 3.2. Notice tht bsic exmple of function σ stisfying Hypothesis 3.1 is ny power coefficient of the form σ (ξ) = C ξ κ, where κ < 1 is such tht γ < 1+κ 1. Another exmple is given by σ (ξ) = ξ κ + sin( ξ κ ). With Hypothesis 3.1 in mind, we shll solve Eq. (21) thnks to n pproximtion procedure. We first stte the following lemm, whose elementry proof is left to the reder. Lemm 3.3. Let σ be function stisfying Hypothesis 3.1 nd n N. We consider the sequence { ξ n, n 1}, where ξ 1 [, 1] is the first time such tht σ ( ξ 1 ) = mx ξ 1 σ (ξ) nd ξ n+1 is the first time such tht σ ( ξ n+1 ) = mx ξ ξn 2 σ (ξ). Let us lso define the following function on R: σ n (ξ) = σ (ξ), ξ > ξ n, σ ( ξ n ), ξ ξ n. Then σ n stisfies (8), with N κ,σn N κ,σ, where N κ,σ is given in (9). We shll construct solution to Eq. (21) by mens of Lmperti type trnsformtion for σ, which hs been used, mong nother pplictions, to study the existence of unique solution for ordinry differentil equtions (see, for instnce, the proof of Theorem 5.1 in Hrtmn [5]). This trnsform is clssiclly defined in the following wy. Lemm 3.4. Let σ be function fulfilling Hypothesis 3.1 nd σ n be defined s in Lemm 3.3. For those two functions nd ξ R, we set φ(ξ) = ξ ds σ (s) nd φ n (ξ) = ξ ds σ n (s). (22) Then φ nd φ n re both invertible nd, for ny ξ R, we hve φ 1 (ξ) φn 1(ξ), where φ 1, φn 1 stnd for the respective inverse of φ nd φ n. Proof. The result is n immedite consequence of the inequlities φ n φ on R + nd φ φ n on R, which follow from our definition (22). The next result sttes the uniform (in n) Lipschitz regulrity of φ 1 n. Lemm 3.5. Let M >. Then, there is constnt c M > such tht φn 1 (ξ 1) φn 1 (ξ 2) c M ξ 1 ξ 2, for ll ξ 1 nd ξ 2 such tht ξ 1, ξ 2 M nd for ll n N. Proof. Suppose ξ M. By (22) nd Lemm 3.4, we get dφn 1(ξ) dξ = σn (φn 1 φ (ξ)) 1 φ Nκ,σ n (ξ) 1 Nκ,σ n (M). In ddition, observe tht lim n φn 1(M) = φ 1 (M), which mens in prticulr tht the sequence {φn 1 (M), n 1} is bounded. Thus direct ppliction of the men vlue theorem finishes the proof. We now proceed to the pproximtion of Eq. (21).

13 354 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Proposition 3.6. Suppose tht γ (1/2, 1), x C γ ([, T ]) nd x =. Also suppose tht Hypothesis 3.1 holds. Let n N nd yt n = φn 1(x t). Then y n solves the following eqution y n t = t σ n (y n s ) dx s, for ll t, where the integrl with respect to x is understood in Young s sense. Proof. We first observe tht the function φn 1 is loclly Lipschitz due to Lemm 3.5. The function σ n is lso loclly Lipschitz ccording to Lemm 2.3. Therefore, σ n (φn 1(x s)) is loclly γ -Hölder continuous. Thus, invoking the usul chnge of vrible in Young s integrl (see e.g. [16, Theorem 4.3.1]) nd reclling tht γ > 1/2, we obtin y n t = t σ n (φ 1 n (x s))dx s = nd the proof is complete. t σ n (y n s )dx s, t, We now turn to the min result of this section which sttes the convergence of y n to solution to Eq. (21). We recll tht γ > 1/2 gin. 1 γ (1+κ) Theorem 3.7. Assume tht σ stisfies Hypothesis 3.1. Consider η such tht γ < η < 1 κ. Let φ be the function given by (22), nd suppose tht x C γ ([, T ]) is such tht φ 1 (x) η L 1 ([, T ]) nd x =. Then the function y = φ 1 (x) is solution of the eqution y t = t σ (y s )dx s, t, where the integrl t σ (y s)dx s is understood s in Proposition 2.4. Remrk 3.8. Note tht y is non-trivil solution (i.e., it is not identiclly zero) nd tht z is lso solution of Eq. (21). So, in generl, this eqution my hve severl solutions. Proof of Theorem 3.7. Let y n be s in Proposition 3.6. For ech ξ R we hve φn 1(ξ) φ 1 (ξ) s n tends to infinity. Hence, y n converges point-wise to y s n tends to infinity. Therefore, thnks to Proposition 3.6, we re reduced to show tht for ll t I (t) := lim n t σn (y n s ) σ (y s) dx s =. Otherwise stted, ccording to Proposition 2.4, we hve to check tht, for t, t lim n D α + σ (y) σn (y n ) s D1 α t x t s ds =, (23) where α is such tht 1 γ < α < γ (κ + η). In order to prove reltion (23), we first invoke definition (6) nd reltion (12). For s [, T ], this gives D α σ (y) + σn (y n ) s s D1 α x γ I 1,n (s) + I 2,n (s, r)dr, (24) t x t s where σ (φ 1 (x s )) σ n (φn 1 I 1,n (s) = (x s)) s α

14 nd J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) σ (φ 1 (x s )) σ n (φn 1 I 2,n (s, r) = (x s)) σ (φ 1 (x r )) σ n (φn 1(x r )) (s r) 1+α. Going bck to our im (23), we re reduced to prove tht t lim n I 1,n (s) ds =, nd lim n t s I 2,n (s, r) drds =. (25) Moreover, thnks to the very definition of σ n, we hve tht for ll r < s t, I 1,n (s) nd I 2,n (s, r), s n. Our clim (25) is thus ensured if we cn bound I 1,n (s) nd I 2,n (s, r) properly. Let us strt with bound on the term I 1,n (s). As in the proof of Lemm 3.5 we cn show tht I 1,n (s) is bounded by constnt times s α for ll n N. This is enough to pply the dominted convergence theorem. In order to bound the term I 2,n, we pply Lemms 2.3, 3.4 nd 3.5, nd the fct tht σ n stisfies (8) with N κ,σn N κ,σ (see Lemm 3.3) to estblish I 2,n (s, r) (s r) α 1 σ (φ 1 (x s )) σ (φ 1 (x r )) + σ n (φn 1 (x s)) σ n (φn 1 (x r )) (s r) γ (κ+η) α 1 φ 1 (x s ) η + φ 1 (x r ) η + φ 1 n (x s) η + φ 1 (x r ) η (s r) γ (κ+η) α 1 φ 1 (x s ) η + φ 1 (x r ) η. We cn thus conclude by the dominted convergence theorem, thnks to the fct tht γ (κ + η) α >. We get the second clim in (25), which completes the proof of our theorem. Remrk 3.9. A smll vrint of our clcultions lso llows to construct solution to the initil vlue problem y t = + t n σ (y s )dx s, t, (26) for generl R. Indeed, long the sme lines s for Theorem 3.7, one cn prove tht y t = φ 1 (x t + φ()) is solution of (26) if φ 1 (x t + φ()) η L 1 ([, T ]). 4. Multidimensionl differentil equtions We now turn to the multidimensionl setting of Eq. (1). As mentioned in the introduction, our considertions will rely on regulrity gin estimtes for the solution when it pproches, similrly to [1,11]. Before we del with these regulrity estimtes, we will first introduce some new nottion Setting In the reminder of the rticle, we ssume tht x C γ ([, T ]; R d ) nd tht ech component σ j, j = 1,..., d in the coefficients of Eq. (1), stisfies Hypothesis 2.1. As in the previous section, we need n dditionl hypothesis tht sys tht σ j behves s power function.

15 356 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Hypothesis 4.1. We suppose tht for ech j = 1,..., d, σ j : R m R m stisfies Hypothesis 2.1 (with the sme κ s in the previous sections), nd moreover: (i) For ny ξ R m we hve σ j (ξ) ξ κ. (ii) σ j is differentible with σ j loclly Hölder continuous of order lrger thn γ 1 1 in the set { ξ }. Fix R m,, nd we consider eqution y t = + d j=1 t σ j (y u ) dx j u, t [, T ]. (27) Using n pproximtion of σ j similr to Lemm 3.3 nd pplying known results on existence nd uniqueness of solutions to equtions driven by Hölder continuous functions (see e.g. [4]), it is esy to show the following result. Proposition 4.2. Suppose tht Hypothesis 4.1(ii) holds, nd let T be given strictly positive time horizon. Then, there exist continuous function y defined on [, T ] nd n instnt τ T, such tht one of the following two possibilities holds: (A) τ = T, y is nonzero on [, T ], y C γ ([, T ]; R m ) nd y solves Eq. (27) on [, T ], where the integrls σ j (y u ) dx j u re understood in the usul Young sense. (B) We hve τ < T. Then for ny t < τ, the pth y sits in C γ ([, t]; R m ) nd y solves Eq. (27) on [, t]. Furthermore, y s on [, τ), lim t τ y t = nd y t = on the intervl [τ, T ]. Notice tht our option (A) bove leds to clssicl solutions of Eq. (27). In the rest of this section, we will ssume (B), tht is the function y given by Proposition 4.2 vnishes in the intervl [τ, T ]. We remrk tht the integrl in cse (B) is not the one defined in Proposition 2.4, which requires suitble integrbility conditions on σ j. Our im is thus to prove the following two fcts: The pth y is globlly γ -Hölder continuous on [, T ]. The integrls σ j (y u ) dxu j cn be understood s limits of Riemnn sums, nd y solves Eq. (27) on [, T ]. Observe tht in order to chieve this im, we will need some dditionl hypotheses on x. We shll lso ssume γ + κ > 1, which is nturl condition in our context (s explined in the introduction). Remrk 4.3. As mentioned in the introduction, we implement here the regulrity gin strtegy inspired by the Brownin SPDE cse (cf [1,11]). An outline of this strtegy is the following: (i) Our bsic regulrity gin result is Proposition 4.8. It sttes tht if solution y stisfies y u 2 k for u lying in n intervl I, then we lso hve y t y s of order 2 κk t s γ for s, t I. (ii) Proposition 4.8 enbles to get lower bound on the mount of time tht y spends in intervls of the form [2 k, b2 k ]. We cn get mtching upper bound by dding roughness ssumption on x. This roughness ssumption mounts to ssert tht the min contributions in the increments of solution y re lwys of the form y t y s σ (y s )[x t x s ]. Our considertions in this direction re summrized in Section 4.3.

16 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Fig. 1. An exmple of pth with stopping times. (iii) With the shrp bound of step (ii) in hnd, creful nlysis of the increments of y enbles to obtin the desired globl bound in γ -Hölder spces. This is the contents of Proposition Remrk 4.4. As the reder might see in the sequel, the mount of efforts devoted to prove tht y is globlly γ -Hölder continuous is rgubly very lrge. However, the stbility of C γ by the mp x y is of fundmentl importnce in the nlysis of differentil systems like (27). In ddition, we believe tht some of the techniques developed here might lso be useful to nlyze rough PDEs with power type coefficients. Now we split the intervl [, τ) s follows. We first define q = 2 q nd we introduce decomposition of the spce R +, which is the stte spce for y, into the following sets: I 1 = [1, ), nd I q = [ q+1, q ), q. We lso need to define the intervls: q+2 + q+1 J 1 = [3/4, ), nd J q =, q+1 + q =: â q+1, â q, q. 2 2 Notice tht â q = 3 2 q+2. We now construct prtition of [, τ) s follows. Assume tht I q, nd set λ = nd τ = inf t : y t I q. In this cse y τ Jˆq with ˆq {q, q 1}. We then set: λ 1 = inf t τ : y t Jˆq. In this wy we recursively construct sequence of stopping times λ < τ < < λ k < τ k such tht b1 y t 2 q, b 2 k 2 q, for t [λ k, τ k ] [τ k, λ k+1 ], (28) k where b 1 = 3 8, b 2 = 3 4 nd q k+1 = q k + l, with l { 1,, 1}, ssuming tht q k 1. Notice tht if q k = or q k = 1, then the upper bound b 2 my be infinity. This construction is depicted in Fig. 1. Finlly, let us justify simplifiction in nottions which will previl until the end of this Section.

17 358 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Remrk 4.5. Notice tht, owing to our Hypothesis 2.1, our problem relies hevily on rdil vribles in R m. Therefore, in order to llevite vectoril nottions, we will crry out the computtions below for m = d = 1. This llows us in prticulr to drop the exponents j in our formule. The reder will esily generlize our considertions to higher dimensions Regulrity estimtes Let us strt with decomposition lemm for the solution to the regulrized Eq. (27). We recll convention which will previl until the end of the pper: for function f defined on [, T ], we set δ f st = f t f s. Lemm 4.6. Let s < t < τ. For l we consider the dydic prtition Πst l of [s, t] defined by ti l = s + 2 l i(t s) for l nd i =,..., 2 l. Then one cn write: where δy st = σ (y s ) δx st + Kst l 2 l 1 = i= Kst l, (29) l=1 [δσ (y)] t l+1 2i t2i+1 l+1 δx t l+1. 2i+1 tl+1 2i+2 Proof. Since s, t [, τ), the integrl t s σ (y u) dx u is usul Young integrl, which is thus limit of Riemnn sums long dydic prtitions. Let us write Jst l for those Riemnn sums, nd notice tht Jst l 2 l 1 = = i= 2 l 1 i= σ y t l i δxt l i t l i+1 σ y t l+1 δx 2i t l+1 2i t2i+1 l+1 + δx t l+1 2i+1 tl+1 2i+2 (3). (31) Then, we know from usul Young integrtion tht Jst l converges, s l, to t s σ (y u) dx u. Therefore, we cn write t σ (y u ) dx u = σ (y s ) δx st + Jst l+1 Jst l. s l= Resorting to expression (3) for Jst l+1 nd to expression (31) for Jst l bove, some elementry lgebric mnipultions revel tht Jst l+1 Jst l = K st l, which ends the proof. Let us stte n dditionl (hrmless) hypothesis on our noise x, which will be crucil in order to get shrp regulrity estimtes. Hypothesis 4.7. There exists ε 1 > such tht for γ 1 = γ + ε 1, we hve x γ1 < nd γ 1 + γ κ < 1. We re now redy to give the bsis of the strtegy lluded to bove, bsed on regulrity gin when y is close to.

18 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Proposition 4.8. Assume σ stisfies Hypothesis 4.1 nd x is such tht Hypothesis 4.7 is fulfilled. Then the following bounds hold true: (i) There exist constnts c,x nd c 1,x such tht for s, t [λ k, λ k+1 ) stisfying t s c,x 2 αq k, with α := 1 κ, (32) γ we hve the following bound: δy st c 1,x 2 q kκ t s γ. (33) (ii) With Hypothesis 4.7 in mind, we get refined decomposition for δy st. Nmely, if s, t re two instnts in [λ k, λ k+1 ) such tht (32) holds true, we hve the following reltion for δy st : δy st = σ (y s ) δx st + r st, with r st c 2,x 2 κ ε 1 q k t s γ, (34) where we hve set κ ε1 = κ + ε 1 α. Proof. For k 1 nd ν > we set y γ,k,ν = sup δyuv v u γ : u, v [λ k, λ k+1 ), v u c 2 ν where the constnts c nd ν will be tuned on lter. Step 1: Proof of (33). Pick s, t [λ k, λ k+1 ) such tht s t c 2 ν. Recll tht we consider the dydic prtitions of [s, t], with ti l = s + 2 l i(t s) for l 1 nd i =,..., 2 l. Strt from decomposition (29). Then, since both y s nd y t lie into [b 1 2 q k, b 2 2 q k ] nd σ verifies Hypothesis 2.1, we obviously hve σ (y s ) δx st c 1 x γ t s γ 2 q kκ, (35) where c 1 = N κ,σ b2 κ. In the reminder of this proof, we denote t l+1 2i, t l+1 2i+1 by t 2i, t 2i+1, respectively, to simplify the nottion. We now bound the quntity [δσ (y)] t2i t 2i+1 δx t2i+1 t 2i+2 popping up in (29). Thnks to Lemm 2.3, for ny η 1 κ we hve [δσ (y)] t2i t 2i+1 N κ,σ yt2i η + y t2i+1 η δy t2i t κ+η 2i+1. Thus, since y t2i, y t2i+1 [b 1 2 q k, b 2 2 q k ] we get [δσ (y)]t2i t 2i+1 δxt2i+1 t 2i+2 Nκ,σ 2b η 1 x γ y κ+η γ,k,ν 2q kη, t s 2 l (1+κ+η)γ. (36) We choose η bove such tht γ (1 + κ + η) = 2γ. It is redily checked tht such η verifies η = 1 κ. Furthermore, with this vlue of η in hnd, reltion (36) becomes [δσ (y)]t2i t δxt2i+1 2i+1 t 2i+2 Nκ,σ 2b κ 1 1 x γ y γ,k,ν 2 q k(1 κ) Plugging this inequlity into the terms Kst l of (29) we end up with t s 2 l 2γ. (37) Kst l c 3,x y γ,k,ν 2 qk(1 κ) t s 2γ, (38) l=1

19 36 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) where we hve set c 3,x = N κ,σ 2b1 κ 1 x 2 2γ 1 γ. Reporting (35) nd (38) into (29), this yields δy st c 1 x γ t s γ 2 q kκ + A 2 st, with A2 st = c 3,x y γ,k,ν 2 q k(1 κ) t s 2γ. (39) We should now bound the term A 2 st s γ -Hölder increment. Indeed, reclling tht we ssume t s c 2 ν, we get A 2 st c 3,xc γ 2q k(1 κ) νγ y γ,k,ν t s γ. (4) We now choose c nd ν so tht c 3,x c γ 2qk(1 κ) νγ 2 1. It is redily checked tht this is chieved for c smll enough nd ν = αq k := γ 1 (1 κ)q k given by (32). With those vlues of c nd ν in hnd, reltion (39) becomes y γ,k,ν c 1 x γ 2 q kκ y γ,k,ν, from which (33) is esily deduced, with c 1,x = 2c 1 x γ. Step 2: Proof of (34). Go bck to reltion (37) nd invoke Hypothesis 4.7 in order to get [δσ (y)]t2i t δxt2i+1 2i+1 t 2i+2 Nκ,σ 2b κ 1 1 x γ1 y γ,k,ν 2 q k(1 κ) t s +ε 1 2 2γ l. Moreover, ccording to (29), the term r st in (34) is given by l=1 Kst l. Proceeding s for reltions (38) nd (39), we obtin tht r st Kst l l=1 A 2 st = c 3,x y γ,k,ν 2 qk(1 κ) t s 2γ +ε 1, (41) where c 3,x = N κ,σ 2b1 κ 1 2 2γ +ε 1 1 x γ 1. We now plug the priori bound (33) on y γ,k,ν we hve just obtined, nd red the regulrity of A 2 in γ -Hölder norm. Similrly to (4), we cn recst (41) s: A 2 st c 3,xc γ +ε 1 2 q k(1 κ) ν(γ +ε 1 ) c 1,x 2 q kκ t s γ. Let us recll tht ν = αq k. Therefore we obtin: A 2 st c 3,xc γ +ε 1 c 1,x 2 q k(κ+αε 1 )γ t s γ. Tking into ccount the fct tht κ ε1 = κ + αε 1, this finishes the proof of (34). In the sequel we shll need some regulrity estimtes for y on time scles slightly lrger thn 2 αq k with α = γ 1 (1 κ). This is the contents of the following property. Corollry 4.9. Under the sme hypotheses s in Proposition 4.8, consider ε 2 > such tht ε 2 < min γ 1 (1 κ), κ(1 γ ) 1, κ + γ 1 (1 κ)ε ε 1 Then there exists constnt c 4,x = 2 1 γ c,x such tht for s, t [λ k, λ k+1 ) stisfying t s c 4,x 2 (α ε 2)q k with α = γ 1 (1 κ) we hve δy st c 5,x 2 q kκ ε 2 t s γ, with κ ε 2 = κ (1 γ )ε 2. (42)

20 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Moreover, under the sme conditions on s, t, decomposition (34) still holds true, with r st c 6,x 2 q kκ ε1,ε 2 t s γ, where κ ε1,ε 2 = κ + αε 1 ε 2 ε 1 ε 2. (43) Proof. We tke up the nottion introduced for the proof of Proposition 4.8, nd we split gin our computtions in 2 steps. Step 1: Proof of (42). Strt from inequlity (33), which is vlid for t s c,x 2 αq k. Now let m N nd consider s, t [λ k, λ k+1 ) such tht c,x (m 1)2 αq k < t s c,x m2 αq k. We prtition the intervl [s, t] by setting t j = s + c,x j2 αq k for j =,..., m 1 nd t m = t. Then we simply write m 1 δy st δy t j t j+1 c 1,x 2 q kκ m 1 γ t j+1 t j c1,x 2 qkκ m 1 γ t s γ, j= j= where the lst inequlity stems from the fct tht t j+1 t j (t s)/m. Now the upper bound (42) is esily deduced by pplying the bove inequlity to m = [2 ε 2q k ] + 1. Step 2: Proof of (43). Once (42) is proven, we go gin through the estimtion of K l st. Replcing y γ,k,ν by c 5,x 2 q kκ ε 2 in (41), we end up with r st c 6,x 2 q kκ ε 2 2 q k(1 κ) 2 q k(α ε 2 )(γ +ε 1 ) t s γ = c 6,x 2 q kκ ε1,ε 2 t s γ, which is our clim (43) Estimtes for stopping times Thnks to the regulrity estimtes of the previous section, we get bound on the difference λ k+1 λ k which roughly sttes tht solution to Eq. (27), cnnot go too shrply to. Proposition 4.1. The sequence of stopping times {λ k, k 1} defined by (28) stisfies λ k+1 λ k c 7,x 2 αq k, where α = (1 κ)/γ. Proof. We shll prove tht τ k λ k stisfies lower bound of the form τ k λ k c 7,x 2 αq k. Along the sme lines we cn prove similr bound for λ k+1 τ k, nd this will prove our clim (44). Inequlity (45) is obtined in the following wy. We observe tht in order to get out of the intervl [λ k, τ k ), n increment of size 2 (q k+1) must occur. Indeed, t λ k the solution is t the middle point of I qk nd the length of this intervl is of order 2 q k. However, reltion (33) sserts tht if δy st 2 (q k+1) nd t s c,x 2 αq k, then we must hve c 1,x t s γ 2 κq k which implies 1 2 q k+1, t s 2c 1,x 1 γ 2 (1 κ)q k γ = 2c 1,x 1 γ 2 αq k. This finishes our proof. (44) (45) (46)

21 362 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) In order to shrpen Proposition 4.1, we introduce roughness hypothesis on x, borrowed from [1]. As we shll see, this ssumption is stisfied when x is frctionl Brownin motion. Hypothesis We ssume tht for ˆε rbitrrily smll there exists constnt c > such tht for every s in [, T ], every ϵ in (, T/2], nd every u in R d with u = 1, there exists t in [, T ] such tht ϵ/2 < t s < ϵ nd u, δx st > c ϵ γ +ˆε. The lrgest such constnt c is clled the modulus of (γ +ˆε)-Hölder roughness of x, nd is denoted by L γ,ˆε (x). Under this hypothesis, we re lso ble to upper bound the difference λ k+1 λ k in useful wy. To this im, recll tht option (B) in Proposition 4.2 is ssumed below. It yields the reltion lim k q k =. Also remember tht {λ k, k 1} is given in (28), nd tht α = (1 κ)/γ. Proposition For ll ε 2 < αε 1 times {λ k, k 1} stisfies 1+γ +ε 1 κ 1 γ nd q k lrge enough, the sequence of stopping λ k+1 λ k c x,ε2 2 q k(α ε 2 ). (47) Furthermore, inequlity (42) cn be extended s follows: there exists constnt c x such tht for s, t [λ k, λ k+1 ) we hve δy st c x 2 κ ε 2 q k t s γ. (48) Proof. If (47) does not hold, this implies tht there exists ε 2 < αε 1 condition of Corollry 4.9 so tht for ny constnt C the inequlity 1+γ +ε 1 1 γ κ stisfying the λ k+1 λ k C2 q k(α ε 2 ) (49) holds for infinitely mny vlues of k. This implies tht λ k+1 λ k C 2 q k(1 κ)/(γ +ˆε), (5) if we choose ˆε smll enough so tht (1 κ)/(γ + ˆε) α ε 2. We wish to exhibit contrdiction, nmely tht one cn find s, t [λ k, λ k+1 ] such tht δy st > J qk, where J qk denotes the size of J qk. In order to lower bound δy st, let us first invoke Hypothesis Since our computtions re performed in the one-dimensionl cse for nottionl ske, we cn in fct recst Hypothesis 4.11 s follows. Choose ε := c 1 2 q k (1 κ) γ +ˆε Lγ,ˆε (x) 1 γ +ˆε C 2 q k (1 κ) γ +ˆε, which cn be chieved by tking the constnt C lrge enough, for given constnt c 1. Then there exist s, t [λ k, λ k+1 ] stisfying ε 2 t s ε, nd δx st c γ +ˆε 1 2 q k(1 κ). (51)

22 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Notice tht c 1 cn be mde rbitrrily lrge, by plying with k nd ˆε. In ddition, we cn use the fct tht σ (y s ) c2 q kκ whenever s [λ k, λ k+1 ]. Indeed, this follows from Hypothesis 4.1 nd the fct tht y s b 1 2 q k 2 q k 2. This entils, for s, t s in (51) σ (y s )δx st cc γ +ˆε 1 2 q k. If (49) holds true, we cn now choose c 1 so tht cc γ +ˆε 1 6. This yields σ (y s )δx st 6 2 q k = 2 J qk. In prticulr the size of this increment is lrger thn twice the size of J qk. We now ssume gin tht we hve chosen ˆε smll enough so tht (1 κ)/(γ + ˆε) α ε 2. Then the upper bound on t s in (51) lso implies t s c 8,x 2 q k(α ε 2 ). For the two instnts s, t exhibited in reltion (51), we resort to decomposition (29) together with the bound (43). This yields δy st A 1 st A2 st, with A1 st = 6 2 q k, A 2 st c 6,x2 q kκ ε1,ε 2 t s γ c 9,x 2 q kµ ε2, where we recll tht κ ε1,ε 2 = κ + αε 1 ε 2 ε 1 ε 2 nd where we obtin µ ε2 = κ ε1,ε 2 + (α ε 2 )γ = 1 + αε 1 (1 + γ + ε 1 )ε 2. Our im is now to prove tht A 2 st cn be mde negligible with respect to 2 q k when q k is lrge enough. This is chieved whenever µ ε2 > 1, nd this condition cn be met by picking ε 1 lrge enough nd ε 2 smll enough. Summrizing our considertions, we hve thus shown tht A 1 st is lrger thn twice J qk = 3 2 q k nd tht A 2 st is negligible with respect to A1 st s q k gets lrge. This proves our clim (47) Hölder continuity We shll use the following nottion, vlid for λ (, 1), time horizon t [, T ] nd function from [, t] to R m : f λ,t := sup s<u t δ f st u s λ, where δ f st = f t f s. (52) Then, we hve the following result, which is our first min objective in this section. Proposition Suppose tht σ stisfies Hypothesis 4.1 nd tht our noise x stisfies Hypotheses 4.7 nd We lso ssume tht γ + κ > 1. Then, the function y given in Proposition 4.2 belongs to C γ ([, T ]; R m ). Proof. Remember tht we re ssuming tht y stisfies condition (B) in Proposition 4.2. Consider first s = λ k nd t = λ l with k < l. We strt by decomposing the increments δy st s follows l 1 δy st δyλ j λ j+1. j=k Then owing to Proposition 4.12 we hve λ k+1 λ k c x,ε2 2 q k(α ε 2 ) for k lrge enough. We cn thus pply Corollry 4.9, which yields l 1 l 1 δy st δyλ j λ j+1 c5,x 2 q j κε 2 λ j+1 λ j γ. (53) j=k j=k

23 364 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) Furthermore, inequlity (44) entils: 2 q j (1 κ) γ c7,x 1 λ j+1 λ j 2 q j κ ε 2 (c 7,x ) γ κ ε 2 1 κ λ j+1 λ j γ κ ε 2 1 κ. Plugging this informtion into (53) nd setting c 1,x = c 5,x (c 7,x ) γ κ ε 2 1 κ, we end up with: l 1 δy st c 1,x λ j+1 λ j µ ε 2, with µ ε2 = γ j=k 1 + κ ε 2 1 κ We now wish the exponent µ ε2 to be of the form µ ε2 = 1+ε 3 with ε 3 >. Since κ ε 2 is rbitrrily close to κ, it is redily checked tht this cn be chieved s long s γ + κ > 1. Reclling tht s = λ k nd t = λ l, one cn thus recst the previous inequlity s l 1 δy st c 1,x λ j+1 λ j 1+ε 3 c 1,x λ l λ k 1+ε 3 c 1,x τ 1+ε3 γ t s γ, j=k which is consistent with our clim. The generl cse s < λ k λ l < t is treted by decomposing δy st s δy st = δy sλk + δy λk λ l + δy λl t. Then resort to (48) in order to bound δy sλk nd δy λl t. The next proposition sys tht if (B) holds, the function y cn be obtined s the limit of suitble sequence of Riemnn sums.. Proposition Let y be the function given in Proposition 4.2. For ll s < t T, let Π st be the set of prtitions of [s, t], denoted genericlly by π = {s = t < < t m = t}. For ε > rbitrrily smll, define Π ε st = π Π st ; there exists j such tht t j < τ t j +1 nd η τ t j 2η, where η = c x ε 1/γ for strictly positive constnt c x nd τ is introduced in Proposition 4.2. Then under the conditions of Proposition 4.13, one cn find π Πst ε such tht: t σ (y u ) dx u σ (y t j ) δx t j t j+1 ε. (54) t j π s Proof. Consider prtition π lying in Π ε st, nd set S π = t i π σ (y t i ) δx ti t i+1. Since y u = for u τ, it is worth noting tht we lso hve S π = S π + σ (y t j ) δx t j t j +1, where S π j< j σ (y t j ) δx t j t j+1. Then we cn write δy st S π δy st j S π + δy t j τ + σ (y t j ) δx t j t j +1 := I1 + I 2 + I 3. We now bound seprtely the 3 terms on the right hnd side bove. For the term I 2 we hve I 2 y γ τ t j γ c x (2η) γ.

24 J.A. León et l. / Stochstic Processes nd their Applictions 127 (217) We cn obviously choose constnt c x such tht if η = c x ε 1/γ, then I 2 3 ε. Thnks to the sme kind of elementry considertions, we cn lso mke the term I 3 smller thn 3 ε. In order to bound I 1, we invoke the fct tht τ t j η nd we set Q η = inf { y s : s < τ η}. Observe tht Q η >. In ddition, by Hypothesis 4.1(ii), σ is differentible nd loclly Hölder continuous of order 1 γ 1 on [Q η, ). By usul convergence of Riemnn sums for Young integrls, we thus hve lim I 1 = lim δy st S π =. π Π st j, π π Π st j, π Therefore we cn choose π so tht I 1 3 ε. Putting together our upper bounds on I 1, I 2 nd I 3, the proof of (54) is now finished. Finlly we cn summrize the considertions of this section into the following theorem. Theorem Consider Eq. (27), nd let T be given strictly positive time horizon. We suppose tht Hypothesis 4.1 holds for the coefficient σ, nd tht Hypotheses 4.7 nd 4.11 re stisfied for our noise x. Then, there exist continuous function y defined on [, T ] nd n instnt τ T, such tht one of the following two possibilities holds: (A) τ = T, y is nonzero on [, T ], y C γ ([, T ]; R m ) nd y solves Eq. (27) on [, T ], where the integrls σ j (y u ) dx j u re understood in the usul Young sense. (B) We hve τ < T. Then for ny t < τ, the pth y sits in C γ ([, T ]; R m ) nd y solves Eq. (27) on [, T ], where the integrls σ j (y u ) dx j u re understood s in Proposition Furthermore, y s on [, τ), lim t τ y t = nd y t = on the intervl [τ, T ]. 5. Appliction to frctionl Brownin motion Let B H = {Bt H, t [, T ]} be stndrd d-dimensionl frctionl Brownin motion with the Hurst prmeter H ( 2 1, 1) defined on complete probbility spce (Ω, F, P), tht is, the components of B H re independent centered Gussin processes with covrince E(Bt H,i Bs H,i ) = 1 2 t 2H + s 2H t s 2H, for ny s, t [, T ]. It is cler tht E B H t B H s 2 = d t s 2H, nd, s consequence, the trjectories of B H re γ -Hölder continuous for ny γ < H. Consider the m-dimensionl stochstic differentil eqution X t = x + d j=1 t σ j (X s )d B H, j s, t T, (55) where x R m. If σ is Hölder continuous of order κ > H 1 1, then, there exists solution X which hs Hölder continuous trjectories of order γ, for ny γ < H. This ws proved by Lyons in [8] using the Young s integrl nd p-vrition estimtes. An extension of this result where there is mesurble drift with liner growth ws given by Duncn nd Nulrt in [2]. Under this wek ssumption of σ we cnnot expect the uniqueness of solution, which requires σ to be differentible with prtil derivtives Hölder continuous of order lrger thn H 1 1 (see [8,12]).