Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.


Introduction
Let E be a locally compact separable metric space, m be a Radon measure on E with full support, and let {B (t) , t ≥ 0} be a family of regular semi-Dirichlet forms on L 2 (E; m) with common domain V satisfying some regularity conditions. By L t denote the operator corresponding to the form B (t) . In the present paper we study the obstacle problem with one and two irregular barriers. In the case of one barrier h : E → R it can be stated as follows: for given ϕ : E → R, f : [0, T ] × E × R → R and smooth (with respect to the parabolic capacity Cap associated with {B (t) , t ≥ 0}) measure µ on E 0,T ≡ (0, T ) × E find u :Ē 0,T ≡ (0, T ] × E → R such that      − ∂u ∂t − L t u = f (·, u) + µ on the set {u > h}, u(T, ·) = ϕ, − ∂u ∂t − L t u ≥ f (·, u) + µ on E 0,T , u ≥ h.
(1. 1) In the second part of the paper we show how the results on (1.1) can be used to solve some system of variational inequalities associated with so-called switching problem.
Problems of the form (1.1) are at present quite well investigated in the case where L t are local operators. Classical results for one or two regular barriers and L 2 -integrable data are to be found in the monograph [1]. Semilinear obstacle problem with uniformly elliptic divergent form operators L t and one or two irregular barriers was studied carefully in [13,14] in case of L 2 -data, and in [20] in case of measure data. In [2] it is considered evolutianry p-Laplacian type equation (with p ∈ (1, ∞)). In an important paper [30] linear problem (1.1) with one irregular barrier, L 2 -data and operators L t associated with Dirichlet forms is considered. The aim of the present paper is to generalize or strengthen the existing results in the sense that we consider semilinear equations involving measure data with two barriers and wide class of operators corresponding to semi-Dirichlet forms. As for the obstacles, we only assume that they are quasi-càdlàg functions satisfying some mild integrability conditions. The class of quasi-càdlàg functions naturally arises in the study of evolution equations. It includes quasi-continuous functions and parabolic potentials (which in general are not quasi-continuous).
When considering problem (1.1) with measure data, one of the first difficulties one encounters is the proper definition of a solution. Because of measure data, the usual variational approach is not applicable. Moreover, even in the case of L 2 -data, the variational inequalities approach does not give uniqueness of solutions (see [25]). Therefore, following [30] and [20], we consider so-called complementary system associated with (1.1). Roughly speaking, by a solution of this system we mean a pair (u, ν) consisting of a quasi-càdlag function u :Ē 0,T → R and a positive smooth measure ν on E 0,T such that − ∂u ∂t − L t u = f (·, u) + ν + µ on E 0,T , u(T, ·) = ϕ in E, (1.2) "ν is minimal", (1.3) u ≥ h q.e., (1.4) where q.e. means quasi-everywhere with respect to the capacity Cap. Of course, in the above formulation one has to give rigorous meaning to (1.2) and (1.3). As for (1.2), we develop some ideas from the paper [15] devoted to evolution equations involving measure data and operators associated with semi-Dirichlet forms.
In the paper we assume that µ belongs to the class M of smooth Borel measures with finite potential, which under additional assumption of duality for the family {B (t) , t ≥ 0}, takes the form where M ρ denotes the set of all smooth signed measures on E 0,T such that µ ρ = E ρ d|µ| < ∞, and the union is taken over all strictly positive excessive functions ρ. If ν ∈ M, then we define a solution of (1.2) as in [15]. To formulate this definition, denote by M a Hunt process {(X t ) t≥0 , (P z ) z∈E 0,T } with life time ζ associated with the operator ∂ ∂t +L t , and set ζ υ = ζ ∧(T −υ(0)), where υ is the uniform motion to the right, i.e. υ(t) = υ(0) + t and υ(0) = s under P z with z = (s, x). By A µ , A ν denote natural additive functionals of M in the Revuz correspondence with µ and ν, respectively. By a solution of (1.2) we mean a function u such that for q.e. z ∈ E 0,T , u(z) = E z ϕ(X ζυ ) + ζυ 0 f (X r , u(X r )) dr + Unfortunately, in general, the "reaction measure" ν need not belong to M (in fact, as shown in [14], in case of two barriers, it may be a nowhere Radon measure). In such case we say that u satisfies (1.2) if u is of class (D), i.e. there is a potential onĒ 0,T (see Section 3.2) dominating |u| on E 0,T , and there exists a local martingale M (with M 0 = 0) such that the following stochastic equation is satisfied under the measure P z for q.e. z ∈ E 0,T : u(X t ) = ϕ(X ζυ ) + ζυ t f (X r , u(X r )) dr (1. 6) In the above definition the requirement that u is of class (D) is very important. The reason is that (1.6) is also satisfied by functions solving equation (1.2) with additional nontrivial singular (with respect to Cap) measure on its right-hand side (see [17]). These functions are not of class (D). If ν ∈ M then (1.6) is equivalent to (1.5). Furthermore, if the time dependent Dirichlet form E 0,T on L 2 (E 0,T ; m 1 := dt ⊗ m) determined by the family {B (t) , t ≥ 0} has the dual Markov property and ϕ ∈ L 1 (E; m), f (·, u) ∈ L 1 (E 0,T ; dt ⊗ m), µ, ν ∈ M 1 (i.e. µ, ν are bounded), then the solution u in the sense of (1.5) is a renormalized solution of (1.2) in the sense defined in [21]. In particular, this means that u has some further regularity properties and may be defined in purely analytical terms. More precisely, u is a renormalized solution if the truncations T k u = −k ∨ (u ∧ k) of u belong to the space L 2 (0, T ; V ), and there exists a sequence {λ k } ⊂ M 1 such that λ k 1 → 0 as k → ∞, and for every bounded v ∈ L 2 (0, T ; V ) such that ∂v ∂t ∈ L 2 (0, T ; V ′ ) and v(0) = 0 we have for all k ≥ 0. In case of local operators of Leray-Lions type the above definition of renormalized solutions was proposed in [29] (for the case of elliptic equations see [4]).
We now turn to condition (1.3). Intuitively, it means that ν "acts only if necessary". If h is quasi-continuous, this statement means that ν acts only when u = h, because then the right formulation of the minimality condition takes the form [20,30]). If h is only quasi-càdlàg, the situation is more subtle. For such h the function u satisfying (1.5) is also quasi-càdlàg, so the left-hand side of (1.8) is well defined, because ν is a smooth measure. However, in general, the left-hand side is strictly positive. M. Pierre has shown (in the case of linear equations with L 2 -data and Dirichlet forms), that for general barrier h the condition Hereũ is a precise m 1 -version of u andh is an associated precise version of h. The notions of a precise m 1 -version of a potential and an associated precise version of a function h dominated by a potential (which is not necessarily m 1 -version of h) were introduced in [30,31]. In the paper, in case of semi-Dirichlet forms, we use probabilistic methods to introduce another notion of a precise m 1 -versionû of a quasicàdlàg function u (note that potentials are quasi-càdlàg). Since our barriers as well as solutions to (1.2)-(1.4) are quasi-càdlàg, we do not need the notion of an associated precise version. We show that if u is an L 2 potential, then u =ũ q.e, and for any quasi-càdlàg function u which is dominated by an L 2 potential, u ≤ũ q.e., (ũ −û) dν = 0.
It follows in particular that in case of L 2 data and Dirichlet form, (1.9) is equivalent to One reason why we introduce a new notion of a precise version is that it is applicable to wider classes of operators and functions then those considered in [30,31]. The second is that our definition is more direct that the construction of a precise version given in [30,31]. Namely, in our approach by a precise version of a quasi-càdlàg function u on E 0,T we mean a functionû onĒ 0,T such that for q.e. z ∈ E 0,T , As a consequence, our definition appears to be very convenient for studying obstacle problems and may be applied to quite general class of equations (possibly nonlinear with measure data and two obstacles).
In case of one obstacle, the main result of the paper says that if ϕ, f (·, 0) satisfy mild integrability conditions, and f is monotone and continuous with respect to u, then for every µ ∈ M there exists a unique solution (u, ν) of (1.1), i.e. a unique pair (u, ν) consisting of a quasi-càdlàg function u onĒ 0,T and positive smooth measure ν on E 0,T such that (1.6), (1.10) and (1.4) are satisfied. We also give conditions under which ν ∈ M or ν ∈ M 1 , i.e. when equivalent to (1.6) formulations (1.5) or (1.7) may be used. Moreover, we show that u n ր u q.e., where u n is a solution of the following Cauchy problem Our probabilistic approach allows us to prove similar results also for two quasicàdlàg barriers h 1 , h 2 satisfying some separation condition. In the case of two barriers, the measure ν appearing in the definition of a solution is a signed smooth measure. Moreover, we replace the minimality condition (1.9) by (1.12) and replace condition (1.4) by h 1 ≤ u ≤ h 2 q.e. We show that under the same conditions on ϕ, f, µ as in the case of one barrier, there exists a unique solution (u, ν) of the obstacle problem with two barriers. We also show thatū n ր u and λ n ր ν − , where (ū n , λ n ) is a solution of problem of the form (1.1), but with one upper barrier h 2 and f replaced by We prove these results under two different separation conditions. The first one, more general, may be viewed as some analytical version of the Mokobodzki condition considered in the literature devoted to reflected stochastic differential equations (see, e.g., [16]). The second one, which is simpler and usually easier to check, as yet, has not been considered in the literature on evolution equations. It says that If h 1 , h 2 are quasi-continuous, then (1.13) reduces to the condition h 1 < h 2 q.e., because quasi-continuous functions are precise (see [14] for this case).
Note also that at the end of the Section 6 we show that the study of the obstacle problem with one merely measurable barrier (or two measurable barriers satisfying the Mokobodzki condition) can be reduced to the study of the obstacle problem with quasi-càdlàg barriers. It should be stressed, however, that when dealing with merely measurable barriers, our definition of a solution is weaker. Namely, instead of (1.4) we only require that u ≥ h m 1 -a.e. (or h 1 ≤ u ≤ h 2 m 1 -a.e. in case of two barriers).
In the last part of the paper we use our results on (1.1) to study so-called switching problem (see Section 7). This problem is closely related to system of quasi-variational inequalities, which when written as a complementary system, has the form 14) where In (1.14)-(1.16), we are given f j : E 0,T × R N → R N , h j,i : E 0,T × R → R, µ j ∈ M and sets A j ⊂ {1, . . . , j − 1, j + 1, . . . , N }, and we are looking for a pair (u, ν) = ((u 1 , . . . , u N ), (ν 1 , . . . , ν N )) satisfying (1.14)-(1.16) for j = 1, . . . , N . Note that in (1.16) the barrier H j depends on u. Systems of the form (1.14)-(1.16) were subject to numerous investigations, but only in the framework of viscosity solutions and for local operators (see [6,7,9,10,12,11,24]) or for special class of nonlocal operators associated with a random Poisson measure (see [23]). In the paper we prove an existence result for (1.14)-(1.16). In the important special case where h j,i (z, y) = −c j,i (z) + y i , we show that there is a unique solution of (1.14)-(1.16), and moreover, that u is the value function for the optimal switching problem related to (1.14)-(1.16).

Preliminaries
In the paper E is a locally compact separable metric space and m a Radon measure on E such that supp[m] = E. For T > 0 we set E 0,T = (0, T ) × E,Ē 0,T = (0, T ] × E. Recall (see [26]) that a form (B, V ) is called semi-Dirichlet on L 2 (E; m) if V is a dense linear subspace of L 2 (E; m), B is a bilinear form on V × V , and the following conditions (B1)-(B4) are satisfied: (B2) B satisfies the sector condition, i.e. there exists K > 0 such that In the paper {B (t) , t ∈ R} is a family of regular semi-Dirichlet forms on L 2 (E; m) satisfying (B2), (B3) with some constants K, α 0 not depending on t. We also assume that for all u, v ∈ V the mapping R ∋ t → B (t) (u, v) is measurable, and for some λ ≥ 1, Let (E, D[E]) denote the time-dependent semi-Dirichlet form on L 2 (E 1 ; m 1 ) (E 1 := R × E) associated with the family {B (t) , t ∈ R} (see [26, Section 6.1]), and Cap denote the associated capacity. We say that some property holds quasi-everywhere (q.e. for short) if it holds outside some set B ⊂ E 1 such that Cap(B) = 0. Capacity Cap on E 0,T is equivalent to the capacity considered in [30,31] (see [32]) in the context of parabolic variational inequalities.
Let µ be a signed measure on E 1 . By |µ| we denote the variation of µ, i.e. |µ| = µ + + µ − , where µ + (resp. µ − ) denote the positive (resp. negative) part of µ. Recall that a Borel measure on E 1 is called smooth if µ charges no exceptional sets and there exists an increasing sequence {F n } of closed subsets of E 1 such that |µ|(F n ) < ∞ for n ≥ 1 and Cap(K \ F n ) → 0 for every compact K ⊂ E 1 .
It is known (see [26,Section 6.3]) that there exists a unique Hunt process M = where υ is the uniform motion to the right, i.e. υ(t) = υ(0) + t and υ(0) = s, P z -a.s. for z = (s, x) (see [26,Theorem 6.3.1]). In the sequel, for fixed T > 0 we set Let us recall that from [26, Lemma 6.3.2] it follows that a nearly Borel set B ⊂ E 1 is of capacity zero iff it is exceptional, i.e. P z (∃ t>0 ; X t ∈ B) = 0, q.e. Recall also (see [26,Section 6]) that there is one-to-one correspondence, called Revuz duality, between positive smooth measures and positive natural additive functionals (PNAFs) of M. For a positive smooth measure µ we denote by A µ the unique PNAF in the Revuz duality with µ. For a signed smooth measure µ we put A µ = A µ + − A µ − . For a fixed positive measurable function f and a positive Borel measure µ we denote by f · µ the measure defined as By S 0 we denote the set of all measures of finite energy integrals, i.e. the set of all smooth measures µ having the property that there is whereη is a quasi-continuous m 1 -version of η (for the existence of such version see [26,Theorem 6.2.11]). By M we denote the set of all smooth measures on E 0,T such that For a given positive smooth measure µ onĒ 0,T we set where ·, · is the duality pairing between V ′ and V (V ′ stands for the dual of V ). It is known (see [33,Example I.4.9(iii)]) that E 0,T is a generalized semi-Dirichlet form. The operator associated with E 0,T has the form we denote the (unique) strongly continuous contraction resolvent on L 2 (0, T ; L 2 (E; m)) associated with E 0,T (see Propositions I.3.4 and I.3.6 in [33]).

Precise versions of parabolic potentials in the sense of Pierre and its probabilistic interpretation
We first recall the notion of a precise version of a parabolic potential introduced in [31].
The set of all parabolic potentials will be denoted by P 2 .
Proposition 3.1. Let u ∈ P 2 . Then there exists a unique positive µ ∈ S 0 onĒ 0,T such that In the sequel, for u ∈ P 2 we set E(u) = µ, where µ ∈ S 0 is the measure from Proposition 3.1.

Definition.
A measurable function u on E 0,T is called precise (in the sense of M. Pierre) if there exists a sequence {u n } of quasi-continuous parabolic potentials such that u n ց u q.e. on E 0,T .
The following result has been proved in [31].
In what follows we denote byũ a precise version of u ∈ P 2 in the sense of Pierre. It is clear thatũ it is defined q.e. In [31] it is proved that the mapping t →ũ(t, ·) ∈ L 2 (E; m) is left continuous and has right limits, whereas in [15,Proposition 3.4] it is proved that u defined by (3.1) has the property that the mapping t → u(t, ·) ∈ L 2 (E; m) is right continuous and has left limits. Since for any B ∈ B(E) and t ∈ [0, T ], Cap({t} × B) = 0 if and only if m(B) = 0 (see [31,Proposition II.4]), it follows that in general Cap({u = u}) > 0. In the sequel, for given u ∈ P 2 we will always consider its version defined by (3.1).
Let us recall that function x : [a, b] → R is called càdlàg (resp. càglàd) iff x is right-continuous (resp. left-continuous) and has left (resp. right) limits. Proof. The proof is analogous to that of [28, Lemma 2.2], so we omit it. ✷ Lemma 3.4. Assume that for each n ∈ N, where A n is a predictable increasing process with A n 0 = 0, and M n is a local martingale with M n Since M is a Hunt process each F-martingale M has the property that M τ − = M τ for all predictable τ ∈ T (see [3,Proposition 2]). Therefore for any predictable τ ∈ T , . From the above convergence,Â σ ≤Â τ for all predictable σ, τ ∈ T such that σ ≤ τ . Therefore applying the predictable cross-section theorem (see [5, [Theorem 86, p. 138]) we conclude thatÂ is an increasing process. Consequently, by Lemma 3.3,Ŷ is càglàd. ✷ Now we are ready to prove the main result of this section. In the sequel, for a function u onĒ 0,T we set u( Theorem 3.5. Assume that u ∈ P 2 . Then for q.e. z ∈ E 0,T , Proof. By the definition of a precise version of a potential, there exists a sequence {u n } of quasi-continuous parabolic potentials such that u n ցũ q.e. on E 0,T . Since ϕ(u) ∈ P 2 for any bounded concave function ϕ on R + and any u ∈ P 2 we may assume that u n , u are bounded. Since u n , u ∈ P 2 , then by Proposition 3.1 and the strong Markov property there exist measures µ n , µ ∈ M and martingales M n , M such that Since u n ցũ q.e. on E 0,T , we have s. for q.e. z ∈ E 0,T , because u n is quasi-continuous. Therefore applying Lemma 3.4 shows thatũ(X t− ) is càglàd. We are going to show that

Probabilistic approach to precise versions of quasi-càdlàg functions
Definition. We say that u : Of course, any quasi-continuous function is quasi-càdlàg. In the sequel the class of smooth measures µ onĒ 0,T for which E z A |µ| ζυ < ∞ q.e. on E 0,T we denote by M T . We say that u is a potential onĒ 0,T iff for a positive µ ∈ M T .
for q.e. z ∈ E 0,T . By [15,Proposition 3.4], each potential onĒ 0,T is quasi-càdlàg. By Proposition 3.1, each u ∈ P 2 is a potential onĒ 0,T . Theorem 3.6. Let u be a quasi-càdlàg function onĒ 0,T . Then there exists a unique (q.e.) functionû such that for q.e. z ∈ E 0,T , Proof. By [26, Theorem 3.3.6] there exists a measurem 1 equivalent to m 1 such that (X, P z ) has the dual Hunt process (X,P z ) with respect tom 1 . Therefore by [8,Theorem 16.4] applied to the process u(X) − there exists a Borel measurable function u satisfying (3.9). Uniqueness follows from the very definition of exceptional sets and  In the sequel we will need the following result.
Proposition 3.9. Let µ be a positive smooth measure on E 0,T , and let u be a positive quasi-càdlàg function on E 0,T . Then for q.e. z ∈ E 0,T . Moreover, s. It is well known that X has only totally inaccessible jumps (as a Hunt process), which combined with the fact that A µ is predictable gives (3.10). The second part of the proposition follows directly from the Revuz duality. ✷

Associated precise versions in the sense of Pierre
Let u be a measurable function onĒ 0,T bounded by some element of W. In [30] M. Pierre introduced the so-called associated precise m 1 -versionũ of u. By the definition, u is the unique quasi-u.s.c. function such that {v ∈ W 0,T + P 2 :ṽ ≥ u q.e.} = {v ∈ W 0,T + P 2 :ṽ ≥ũ q.e.} In general, it is not true thatũ = u m 1 -a.e. However, if u is quasi-càdlàg, then u ≤ũ q.e. (3.11) Indeed, by [30, Proposition IV-I] there exists a sequence {u n } ⊂ W 0,T such that inf n≥1 u n =ũ q.e. Since u ≤ u n q.e. and u is quasi-càdlàg, we have Taking infimum, we getû By the above and [8, Proposition 11.1] we get (3.11). Set
By M (resp. M loc ) we denote the space of martingales (resp. local martingales) with respect to F. [M ] denotes the quadratic variation process of M ∈ M loc . By M 0 (resp. M p ) we denote the subspace of M consisting of all M such that M 0 = 0 (resp. By V (resp. V + ) we denote the space of all F-progressively measurable processes (resp. increasing processes) V of finite variation such that Let f : Ω × [0, T ] × R → R be a measurable function such that f (·, y) is Fprogressively measurable for every y ∈ R, ξ be a F T -measurable random variable and V be a càdlàg process of finite variation such that V 0 = 0.
Definition. We say that a pair of processes (Y, M ) is a solution of the backward stochastic differential equation with terminal condition ξ and right-hand side f + dV (BSDE(ξ, f + dV ) for short) if (a) Y is an F-adapted càdlàg process of Doob's class (D), M ∈ M 0,loc , Definition. We say that a triple of processes (Y, M, K) is a solution of the reflected backward stochastic differential equation with terminal condition ξ, right-hand side f + dV and lower barrier L (RBSDE(ξ, f + dV, L) for short) if (a) Y is an F-adapted càdlàg process of Doob's class (D), M ∈ M 0,loc , K ∈ p V + ,  In the sequel we will need the following lemma.

Lemma 4.2. Assume that
(i) there is µ ∈ R such that for a.e. t ∈ [0, T ] and all y, y ′ ∈ R, Let f n = f ∨ (−n), and let (Y n , M n , R n ) be a solution of RBSDE(ξ, f n + dV, L, U ). Then where (Y, M, R) is a solution of RBSDE(ξ, f + dV, L, U ).
Proof. By [16, Proposition 2.14, Proposition 3.1, Without loss of generality we may assume that µ ≤ 0 in (i). Then From this we conclude that the sequence {M n } is locally uniformly integrable. Hencē M t := lim n→∞ M n t , t ∈ [0, T ], is a local martingale. We shall show that the triple (Ȳ ,M ,R) is a solution of RBSDE(ξ, f + dV, L, U ). It is clear thatR is a càdlàg process of finite variation. Moreover, by (ii) and (iii), It is also clear thatȲ is of class (D) and L ≤Ȳ ≤ U . By the Hahn-Saks theorem, Assume that ∆K t > 0. Then there exists n 0 such that ∆R +,n Thus the triple (Ȳ ,M ,R) is a solution of RBSDE(ξ, f + dV, L, U ). From this and the minimality of the Jordan decomposition of the measure R it follows that R + = K, R − = A. ✷

PDEs with one reflecting barrier
In this section T > 0 is a real number, ϕ : E → R, f :Ē 0,T × R → R are measurable functions, and h :Ē 0,T → R is a quasi-càdlàg function such thatĥ(T, ·) ≤ ϕ.
Definition. We say that a quasi-càdlàg function u onĒ 0,T is a solution of the Cauchy problem Definition. We say that a pair (u, ν), where u :Ē 0,T → R is a quasi-càdlàg function and ν is a positive smooth measure on E 0,T , is a solution of problem (1.1), i.e. obstacle problem with data ϕ, f, µ and lower barrier h (we denote it by OP(ϕ, f + dµ, h)), if (a) (1.6) is satisfied for some local martingale M (with M 0 = 0), Definition. We say that a pair (u, ν) is a solution of the obstacle problem with data ϕ, f, µ and upper barrier h (we denote it by OP(ϕ, f + dµ, h)), if (−u, ν) is a solution of OP(−ϕ,f − dµ, −h), wheref (t, x, y) = −f (t, x, −y).
We will need the following duality condition considered in [15].
(∆) For some α ≥ 0 there exists a nest {F n } on E 0,T such that for every n ≥ 1 there is a non-negative η n ∈ L 2 (E 0,T ; m 1 ) such that η n > 0 m 1 -a.e. on F n andĜ 0,T α η n is bounded, whereĜ 0,T α is the adjoint operator to G 0,T α .
Note that (∆) is satisfied if for some γ ≥ 0 the form E γ has the dual Markov property (see [15,Remark 3.9]).

Remark 5.2.
It is an elementary check (see [15, (3.7)]) that E z |ϕ(X ζυ )| = E z A |β| ζυ (on E 0,T ), where β = δ {T } ⊗ (ϕ · m), so by [15,Proposition 3.8] under condition (∆) if ϕ ∈ L 1 (E; m) then E z |ϕ(X ζυ )| < ∞ for q.e. z ∈ E 0,T . Proof. Let (u 1 , ν 1 ), (u 2 , ν 2 ) be two solutions of OP(ϕ, f + dµ, h). Set u = u 1 − u 2 , ν = ν 1 − ν 2 By the Tanaka-Meyer formula, for any stopping time τ such that τ ≤ ζ υ we have for q.e. z ∈ E 0,T . Observe that by the minimality condition and Proposition 3.9, Let {τ k } be a chain (i.e. an increasing sequence of stopping times with the property P z (lim inf k→∞ {τ k = T }) = 1, q.e.) such that M τ k is a martingale for each k ≥ 1. Then by (H2) and (5.2), Letting k → ∞ in (5.3) and using the fact that u is of class (D) we get Applying Gronwall's lemma shows that E z |u|(X t )| = 0, t ∈ [0, ζ υ ] for q.e. z ∈ E 0,T . This implies that u = 0 q.e. From this, (1.6) and the uniqueness of the Doob-Meyer decomposition we conclude that A ν = 0, which forces ν = 0. Proof. By [15,Theorem 5.8] there exists a unique solution u n of (1.11). Set By the definition of a solution of (1.11) and [15,Proposition 3.4] there exists a martingale M n such that the pair (u n (X), M n ) is a solution of BSDE(ϕ(X ζυ ), f n (X, ·) + dA µ ) under the measure P z for q.e. z ∈ E 0,T . By [16,Theorem 4.1] (see also [34]) there exists a solution (Y z , M z , K z ) of RBSDE(ϕ(X), f (X, ·) + dA µ , h(X)) under the measure P z for q.e. z ∈ E 0,T , and From (5.4) it follows that u n ≤ u n+1 q.e. (since the exceptional sets coincide with the sets of zero capacity) for n ≥ 1 and for q.e. z ∈ E 0,T , where u := sup n≥1 u n . This implies that u is quasi-càdlàg. What is left to show that there exists a smooth measure ν such that K z = A ν for q.e. z ∈ E 0,T . Since the pointwise limit of additive functionals is an additive functional, we may assume by Lemma 4.2 and [16, Theorem 2.13] that EK z ζυ < ∞ for q.e. z ∈ E 0,T . By [16,Proposition 4.3] and Theorem 3.6, for every predictable stopping time τ , is a PNAF (without jump in ζ υ sinceĥ(T, ·) ≤ ϕ), which implies that there exists β ∈ M such that J = A β (by the Revuz duality). Set It is clear that the process C z is continuous. By Remark 4.1, the triple (Y z , M z , C z ) is a solution of RBSDE(ϕ(X), f (X, ·) + dA µ + dA β , h(X)). By [15,Theorem 5.8] there exists a solution v n of the equation Therefore, by the definition of solution and [15,Proposition 3.4] there exists a martingale N n such that the pair (v n (X), N n ) is a solution of BSDE(ϕ(X), f n (X, ·) + dA µ + dA β ) under the measure P z for q.e. z ∈ E 0,T . Let C n t = t 0 n(v n (X r ) − h(X r )) − dr, t ∈ [0, ζ υ ]. By [16,Theorem 2.13] the sequence {C n } converges uniformly on [0, ζ υ ] in probability P z for q.e. z ∈ E 0,T . Since C n is a PCAF for each n ≥ 1, the process C defined by is a PCAF. Therefore there exists a measure α ∈ M such that C = A α . It is clear that e., f 1 (·, y) ≤ f 2 (·, y) m 1 -a.e. for every y ∈ R, dµ 1 ≤ dµ 2 and h 1 ≤ h 2 q.e. Then u 1 ≤ u 2 q.e. on E 0,T , and if h 1 = h 2 q.e., then dν 1 ≥ dν 2 .
Proof. By the proof of Theorem 5.4, u i,n ր u i q.e., where u i,n is a solution to PDE(ϕ i , f n,i + dµ i ) with By [15,Corollary 5.9], u 1,n ≤ u 2,n q.e. Hence u 1 ≤ u 2 q.e. As for the second assertion of the theorem, by Lemma 4.2 we may assume that ν 1 , ν 2 ∈ M. By the proof of Theorem 5.4, , P z -a.s. for q.e. z ∈ E 0,T and some positive β i ∈ M, α i ∈ M c such that A β i is purely jumping and . We already know that u 1 ≤ u 2 q.e. Moreover, by the assumptions and Revuz duality, dA µ 1 ≤ dA µ 2 . Therefore dA β 1 ≥ dA β 2 . Furthermore, by the proof of Theorem 5.4, for q.e. z ∈ E 0,T , where α i,n = n(v i,n − h) − · m 1 and v i,n is a solution to PDE(ϕ i , f n + dµ i + dβ i ). By [15,Corollary 5.9], v 1,n ≤ v 2,n , which implies that α 1,n ≥ α 2,n , n ≥ 1.
We denote by M 1,T the set of all finite Borel measures onĒ 0,T .
Definition. We say that a pair (u, ν) consisting of a quasi-càdlàg function u :Ē 0,T → R of class (D) and a smooth measure ν on E 0,T is a solution of the obstacle problem with data ϕ, f, µ and barriers h 1 , h 2 (we denote it by OP(ϕ, f + dµ, h 1 , h 2 )) if Proof. The proof is analogous to the proof of Theorem 5.3. The only difference is the estimate of the integral involving dA ν . In the present situation, the estimate is as follows: ✷ Consider the following hypothesis: (H6) h 1 , h 2 are quasi-càdlàg functions of class (D) such thatĥ 1 (T, ·) ≤ ϕ ≤ĥ 2 (T, ·) and h 1 < h 2 ,ĥ 1 <ĥ 2 q.e. on E 0,T , or there exists β ∈ M T such that h 1 ≤ R 0,T β ≤ h 2 q.e. on E 0,T .
Remark 6.6. Let v be a difference of potentials onĒ 0,T , i.e. v = R 0,T β for some β ∈ M T . Observe that if a pair (ū,ν) is a solution of OP(ϕ, . It follows that the assumption f (·, 0) · m 1 ∈ M in Theorems 5.4 and 6.3 may be replaced by more general assumption saying that there exists a function v which is a difference of potentials onĒ 0,T such that f (·, v) · m 1 ∈ M. Similarly, counterparts of the results stated in Propositions 5.7, 6.4 and Remarks 5.8 and 6.5 hold true under the assumption f (·, v) · m 1 ∈ M. To get appropriate modifications of these results, we first apply them to the pair (ū,ν), and next we use the fact that (ū,ν) = (u − v, ν).
In condition (b) of the definition of the obstacle problem given at the beginning of this section we require that the solution lies q.e. between the barriers. Below we show that if we weaken (b) and require only that this property holds a.e., then our results also apply to measurable barriers h 1 , h 2 satisfying the following condition: there exits v of the form v = R 0,T β with β ∈ M T (i.e. v is a difference of potentials) such that h 1 ≤ v ≤ h 2 m 1 -a.e. (in case of one barrier h it is enough to assume that h + is of class (D)).
Before presenting our results for measurable barriers, we give a definition of a solution.
Definition. We say that a pair (u, ν) consisting of a quasi-càdlàg function u :Ē 0,T → R of class (D) and a smooth measure ν onĒ 0,T is a solution of the obstacle problem with data ϕ, f, µ and measurable barriers h 1 , h 2 :Ē 0,T → R if (a) (1.6) is satisfied for some local martingale M (with M 0 = 0), If the barriers are quasi-càdlàg, then the above definition agrees with the definition given at the beginning of Section 6, because for quasi-càdlàg functions condition (b*) is equivalent to (b) and in (c*) we may take η 1 = h 1 and η 2 = h 2 . Therefore the obstacle problem with data ϕ, f, µ and measurable barriers h 1 , h 2 we still denote by OP(ϕ, f + dµ, h 1 , h 2 ).
In case of measurable barriers the proof of uniqueness of solutions to the problem OP(ϕ, f + dµ, h 1 , h 2 ) goes as in the case of quasi-càdlàg barriers, the only difference being in the fact that in (6.1) we replace h 1 by u 1 ∧ u 2 and h 2 by u 1 ∨ u 2 , and then we apply (c*).
The problem of existence of a solution is more delicate. Observe that if (u, ν) is a solution to OP(ϕ, f + dµ, h 1 , h 2 ) with measurable barriers h 1 and h 2 , then it is a solution to OP(ϕ, f + dµ, η 1 , η 2 ) with η 1 , η 2 as in condition (c*). It appears that under the additional assumption on barriers (mentioned before the last definition) one can construct quasi-càdlàg η 1 , η 2 such that if (u, ν) is a solution to OP(ϕ, f +dµ, η 1 , η 2 ), then it is a solution to OP(ϕ, f + dµ, h 1 , h 2 ). This shows that as long as we only require (b * ), the study of the obstacle problem with measurable barriers can be reduced to the study of the obstacle problem with quasi-càdlàg barriers. Finally, note that, unfortunately, there is no construction of η 1 , η 2 depending only on h 1 and h 2 (the barriers η 1 , η 2 depend also on ϕ, f, µ). The reason is that the class of càdlàg functions is neither inf-stable nor sup-stable.
Proof. First observe that the data f (X, ·), H j (X, ·), ξ := ϕ(X ζυ ), Y := u(X), Y := u(X), V = A µ satisfy the assumptions of [18,Theorem 3.11] under the measure P z for q.e. z ∈ E 0,T . Set u 0 = u and Y 0 = Y . By Theorem 5.4 (see also Remark 6.6), for every n ≥ 1, u j n (X t ) = Y n,j for q.e. z ∈ E 0,T , where (Y, M, K) is the minimal solution of (7.2) such that Y ≤ Y ≤ Y . Hence, since the exceptional sets coincide with the sets of zero capacity, we have u j (X t ) = Y j t , t ∈ [0, ζ υ ], P z a.s.