Stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions

In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the $G$-framework and extend $G$-It\^o's formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions (RGSDEs).


Introduction
In classical framework, Skorokhod [17,18] firstly introduced the diffusion processes with reflecting boundary in 1960s.From then on, reflected solutions to SDEs and BSDEs were investigated by many authors.For one-dimensional case, El Karoui [3], El Karoui and Chaleyat-Maurel [4] and Yamada [24] studied reflected SDEs on a half-line, and El Karoui et al. [5] obtained the solvability of reflected BSDEs.For multidimensional case, the existence of weak solutions to reflected SDEs on a smooth domain was proved by Stroock and Varadhan [22].After that, Tanaka [23] solved the similar problem on a convex domain by a direct approach based on the solution to Skorokhod problem.Furthermore, Lions and Sznitman [11] extended these results to a non convex domain.On the other hand, the corresponding results for reflected BSDEs can be found in Gegout-Petit and Pardoux [7], Ramasubramanian [15] and Hu and Tang [9], etc.. Motivated by uncertainty problems, risk measures and the superhedging in finance, Peng [13,14] introduced a framework of time consistent nonlinear expectation E[•], in which a new type of Brownian motion is constructed and corresponding stochastic calculus was established.Moreover, Denis et al. [2] derived that G-expectation E[•] can be viewed as an upper expectation with respect to a weakly compact family P of probability measures, and they naturally defined a Choquet capacity C(•) on Ω: Thus, many concepts from capacity theory can be introduced to G-framework, especially, the notion of "quasi-surely" which means that a property holds true outside a polar set, i.e., outside a set A ⊂ Ω satisfies C(A) = 0. Using the notation of "quasi-surely", Gao [6] and Lin and Bai [1] have already worked on the solvability to SDEs driven by G-Brownian motion.
Meanwhile, Soner et al. [19,20,21] have established a complete theory for 2BSDEs under a uniform Lipschitz conditions which is closely related to G-expectation.In their framework, they also issued a similar notion, but their definition of "quasi-surely" means that a property holds P-a.s. for every probability measure P in a non-dominated class of mutually singular measures.We notice that this definition is a little different compared to that made by Gcapacity C(•) in Denis et al. [2], since by their definition the null set A P under each P ∈ P could be different.More recently, Matoussi et al. [12] have studied the problem of reflected 2BSDEs with a lower obstacle .
The aim of this paper is to study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with reflecting boundary conditions (RGSDEs) in the sense of "quasi-surely" defined by Denis et al. [2].The scalar valued RGSDE we consider is defined as following: g s (X s )dB s + K t , q.s., 0 ≤ t ≤ T ; where B is the quadratic variation process of G-Brownian motion B, and K is an increasing process which pushes the solution X upwards to be remaining above the obstacle S in a minimal way.Similarly to classical reflected SDEs, the uniqueness result is deduced from a priori estimates and a solution in M p G ([0, T ]) to (1.1) can be constructed by fixed-point iteration.To establish the comparison theorem, we need to develop an extension of G-Itô formula to deal with such process X involves both the stochastic integrals and an increasing process.This extended G-Itô's formula can have its own interest and may be used in other situation.
This paper is organized as follows: Section 2 introduces some notations and results in Gframework which is necessary for what follows.Section 3 introduces the stochastic calculus with respect to an increasing process in G-framework.Section 4 studies the reflected G-Brownian motion, while Section 5 is our main results.

G-Brownian motion, G-capacity and G-stochastic calculus
The main purpose of this section is to recall some preliminary results in G-framework which are needed in the sequel.The reader interested in a more detailed description of these notions is referred to Denis et al. [2], Gao [6] and Peng [14].
2.1.G-Brownian motion.Adapting the approach in Peng [14], let Ω be the space of all R-valued continuous paths with ω 0 = 0 equipped with the distance B the canonical process and C l,Lip (R n ) the collection of all local Lipschitz functions on R n .For a fixed T ≥ 0, the space of finite dimensional cylinder random variables is defined by Remark 2.2.Without loss of generality, we always assume that σ 2 = 1 in what follows.
Definition 2.3.We call a sublinear expectation E : where ψ(x 1 , . . ., For p ≥ 1, we denote by In this sense, the domain of G-expectation can be extended from Naturally, we can define a corresponding regular Choquet capacity on Ω: C(A) := sup with respect to which, we have the following notions: A property is said to hold quasi-surely (q.s.) if it holds outside a polar set.
Definition 2.5.A random variable X is said to be quasi-continuous (q.c.) if for arbitrarily small ε > 0, there exists an open set Definition 2.6.We say that a random variable X has a q.c.version if there exists a q.c.random variable Y such that X = Y , q.s..
In a language of G-capacity, Denis et al. [2] proved that for p ≥ 1, the function space L p G (Ω T ) has a dual representation which is much more explicit to verify: Theorem 2.7.
Unlike in classical framework, the downwards monotone convergence theorem only holds true for a sequence of random variables from a subset of L 0 (Ω T ) (cf.Theorem 31 in Denis et al. [2]).
. Remark 2.9.We note that dominated convergence theorem does not exist in G-framework, even though we assume that {X n } n∈N is a sequence in L 1 G (Ω T ).The lack of this theorem is one of the main difficulties we shall overcome in the following sections.

G-stochastic calculus.
In Peng [14], generalized Itô integrals with respect to G-Brownian motion are established: Definition 2.10.A partition of [0, T ] is a finite ordered subset π N [0,T ] = {t 0 , t 1 , . . ., t N } such that 0 = t 0 < t 1 < . . .< t N = T .We set For p ≥ 1, define and denote by G ([0, T ]), we define The mapping ) is continuous and linear and thus can be uniquely extended to ), the stochastic integral with respect to G-Brownian motion B is defined by T 0 η s dB s := I [0,T ] (η).Unlike the classical theory, the quadratic variation process of G-Brownian motion B is not always a deterministic process (unless σ = σ) and it can be formulated in L 2 G (Ω t ) by Definition 2.13.For each η ∈ M 1,0 G ([0, T ]), we define The mapping ) is continuous and linear and thus can be uniquely extended to ), the stochastic integral with respect to the quadratic variation process B is defined by In view of the dual formulation of G-expectation as well as the properties of the quadratic variation process B in G-framework, the following BDG type inequalities are obvious.
where C p is a positive constant independent of η.

Stochastic calculus with respect to an increasing process
In this section, we define the stochastic integrals with respect to an increasing process with continuous paths and then we extend G-Itô's formula to the case where an increasing process appears in the dynamics.In the sequel, C and M denote two positive constants whose values may vary from line to line.
3.1.Stochastic integrals with respect to an increasing process.
Definition 3.1.We denote by M c ([0, T ]) the collection of all q.s.continuous processes X whose paths X • (ω) : t → X t (ω) are continuous in t on [0, T ] outside a polar set A.
Remark 3.2.For example, from the proofs to Theorem 2.1 and Theorem 2.2 in Gao [6], ( Remark 3.4.Obviously, an increasing process K in M I ([0, T ]) has q.s.finite total variation on [0, T ], and thus its quadratic variation is q.s.0. Definition 3.5.We define, for a fixed X ∈ M c ([0, T ]), the stochastic integral with respect to a given K ∈ M I ([0, T ]) by where A is a polar set, and on the complementary of which, X • (ω) is continuous and K • (ω) is increasing in t.
Remark 3.6.Since for a fixed ω ∈ A c , the function X • (ω) is continuous and the function K • (ω) is of bounded variation on [0, T ], the Riemann-Stieltjes integral on the right side always exists (cf.Hildebrandt [8]).Thus, (3.1) is well defined.Similar definition can be made for those X whose paths are q.s.piecewisely continuous and without discontinuity of the second kind , i.e., for each ω ∈ A c , the function X • (ω) is discontinuous at a finite number of points, and these discontinuous points are removable or of the first kind.

Remark 3.7. Given a sequence of refining partitions {π
) → 0, we set a sequence of binary functions: from which we deduce that The construction of the sequence (3.2) provides a q.s.approximation to the stochastic integral T 0 X t dK t .We note that the convergence (3.3) depends only on the sequence of refined partitions (π N [0,T ] ) N ∈N but is independent of the selection of the points of division and the representatives The following propositions can be verified directly by Definition 3.5 and the Heine-Cantor theorem.
Proposition 3.8.Let X, X 1 and u , q.s.. Remark 3.9.By a classical argument, a q.s.continuous and bounded variation process can be viewed as the difference of two increasing processes K 1 − K 2 , where K 1 and K 2 ∈ M I ([0, T ]).By Proposition 3.8 (3), the stochastic integral with respect to K 1 − K 2 can be defined in the same way as Definition 3.5.
As showed above, (3.1) defines a random variable T 0 X t dK t in L 0 (Ω T ).A nature question comes out: if we assume that for some appropriate p and q, X ∈ M p G ([0, T ]) and K ∈ M q G ([0, T ]), this random variable T 0 X t dK t can be verified as an element in L 1 G (Ω T ) or not.In general, the answer is negative.That is because the integrability of X and K can not ensure the quasi-continuity of T 0 X t dK t (cf.Definition 2.5 and Theorem 2.7).More precisely, the pathwise convergence (3.3) is not necessarily uniform in ω outside a polar set A, and it is hard to verify directly the convergence in the sense of L 1 G (Ω T ) due to the lack of dominated convergence theorem in G-framework.But in some special cases, a proper sequence {V N [0,T ] (X, K)} N ∈N approximating to T 0 X t dK t can be found, so that the quasi-continuity is inherited during the approximation.
G (Ω T ).Proof: Consider a sequence of refining partitions {π N [0,T ] } N ∈N mentioned in Remark 3.7 and define the sequence of approximation: for N ∈ N, From the explanation in Remark 2.11, we can always assume that at the points of division, On the other hand, it is easy to verify by Theorem 2.7 that for all we deduce the desired result.Remark 3.12.To verify that for all Proposition 3.13.Let X be a q.s.continuous G-Itô process such that (3.4) where f , h and g are elements in , where 1/p + 1/q = 1.Then, T 0 X t dK t is an element in L 1 G (Ω T ).Proof: Given a sequence of refining partitions {π N [0,T ] } N ∈N , we construct the sequence (3.2).By the definitions of stochastic integrals and the BDG type inequalities, one can verify that for each t ∈ [0, T ], From the integrability of f , h and g, we have Ē[ sup For N ∈ N, we calculate The desired result follows.

3.2.
An extension of G-Itô's formula.For 0 ≤ s ≤ t ≤ T , consider a sum of a G-Itô process and an increasing process K: Lemma 3.14.Let Φ ∈ C 2 (R) be a real function with bounded and Lipschitz derivatives.Let f , h and g be bounded processes in M 2 G ([0, T ]), and The proof of this lemma is based on some previous results in Peng [14] (cf.Lemma 6.1 and Proposition 6.3 in Chapter III).To avoid redundancy, we first prove a reduced lemma when f = h = g ≡ 0 to show how the increasing process K plays a role in this dynamic, and then we give sketch to indicate some key points to combine the simple lemma with the previous results in Peng [14].
Lemma 3.15.Let Φ ∈ C 2 (R) be a real function with bounded and Lipschitz derivatives, and Proof: Consider a sequence of refining partitions {π N [s,t] } N ∈N .For N ∈ N, from the second order Taylor expansion, we have where ξ N k satisfies K t N k ≤ ξ N k ≤ K t N k+1 , q.s..For the first part, similar to that in Remark 3.7, we obtain lim For the second part, since d 2 Φ dx 2 is bounded and the quadratic variation of K on [0, T ] is q.s.0, then, The proof is complete.
Sketch of the proof of Lemma 3.14: To combine the result above with the ones in Peng [14], we decompose X into M X + K, where M X denotes the G-Itô part of X.Given a sequence of refining partitions {π 2 N [s,t] } N ∈N : for N ∈ N, we have from the second order Taylor expansion where q.s..
A key point in the proof is to verify the following convergence in M 2 G ([0, T ]): For the G-Itô part M X , we deduce by the BDG type inequalities For the increasing process K, thanks to assumption (3.6), for each u ∈ [s, t], By dominated convergence theorem to the integral on [s, t], (3.12) lim Combining (3.10) and (3.12), (3.8) and (3.9) are readily obtained by Lipschitz continuity of dΦ dx and d 2 Φ dx 2 .Then, we can proceed similarly to that in Peng [14] to treat with I N 1 and I N 2 .
On the other hand, due to the boundedness of d 2 Φ dx 2 and the boundedness and uniformly continuity of paths M X • (ω) and K • (ω) on [0, T ], for ω ∈ A c , we can easily get that I N 3 and I N 4 are q.s.vanished.
For I N 5 , we calculate where (ξ , q.s..The result in Peng [14] shows that the first part converges to 0 in while the second part is vanished as a result of the uniformly continuity of paths K • (Ω) on [0, T ], for ω ∈ A c , and the q.s.boundedness of the quadratic variation of the G-Itô part M X .For I N 6 , it converges to t s dΦ dx (X u )dK u , q.s. by Definition 3.5.
Remark 3.16.In the proof of classical Itô's formula, (3.8) and (3.9) can be verified directly by the pathwise continuity of X and the dominated convergence theorem on the product space [s, t] × Ω.But in G-framework, we are short of such a theorem.In general, given ).Thus, (3.6) is needed to ensure that (3.11) holds true.
In fact, the left side of (3.7), particularly the term G (Ω t ), which can be verified by choosing a sequence such that t n → t and for n ∈ N, X tn ∈ L 2 G (Ω tn ) (Remark 2.11 ensures the existence of this sequence), and by deducing from assumption (3.6).
Similar to Theorem 6.5 of Peng [14], we can extend G-Itô's formula in Lemma 3.14 to those Φ whose second derivates d 2 Φ dx 2 has polynomial growth.Unfortunately, this extension is at a cost of more restriction on the increasing process K. Theorem 3.17.Let Φ ∈ C 2 (R) be a real function such that d 2 Φ dx 2 satisfies polynomial growth condition.Let f , h and g be bounded processes in M 2 G ([0, T ]), and Proof: By the same argument in the proof of Theorem 6.5 of Peng [14], we can choose a sequence of functions Φ N ∈ C 2 0 (R) such that for any x ∈ R, where C and k are positive constants independent of N .Obviously, Φ N satisfies the conditions in Lemma 3.14.Therefore, Borrowing the notation in the proof of Lemma 3.14 and using the BDG type inequalities, we have Then, from (3.14) and (3.16), we deduce that as N → +∞, ).We can proceed as in Peng [14] to show that the terms on right side of (3.15), except t s dΦ N dx (X u )dK u , converge to their corresponding terms in (3.13).To complete the proof, it is sufficient to show that for ω ∈ A c , by the continuity and boundedness of paths X • (ω) and Remark 3.19.Following exactly the procedure above, we can have similar result when a bounded variation process K 1 − K 2 appears in the dynamic.

Reflected G-Brownian motion
Before moving to the main result of this paper, we first consider a reduced RGSDE, that is, taking f = h ≡ 0 and g ≡ 1, only a G-Brownian motion and an increasing process drive the dynamic on the right side of (1.1).In what follows, we establish the solvability to the RGSDE of this type, i.e., the existence and uniqueness of the reflected G-Brownian Motion.
Let y be a real valued continuous function on [0, T ] with y 0 ≥ 0. It is well-known that, there exists a unique pair (x, k) of functions on [0, T ] such that x = y + k, where x is positive, k is an increasing and continuous function starting from 0, moreover, the Riemann-Stieltjes integral 0 x t dk t = 0.The solution to this Skorokhod problem on [0, T ] is given by (4.1) x t = y t + k t ; k t = sup s≤t x − s , which is explicit and unique.Theorem 4.1.For any p ≥ 1, there exists a unique pair of processes and (a) K 0 = 0; (b) X is positive; and (c) T 0 X t dK t = 0, q.s.. Proof: With the help of (4.1), we define a pair of processes (X, K) pathwisely on [0, T ]: ) and (a), (b) and (c) are satisfied.Therefore, to complete the proof, we only need to verify that K ∈ M p G ([0, T ]).
Since for all 1 , we can assume that p > 2 without loss of generality.Given a sequence of partitions {π N [0,T ] } N ∈N , we set We observe that both (( then, letting f = h ≡ 0 and g ≡ 1 in (3.5), we obtain which shows that (sup 0≤s≤t (B − s ) N ) 0≤t≤T converges to K in M p G ([0, T ]).On the other hand, the uniqueness of such pair (X, K) is inherited from the solution to Skorokhod problem pathwisely.The proof is complete.Remark 4.2.We call the process X in Theorem 4.1 a G-reflected Brownian motion on the half line [0, +∞).
Furthermore, if the G-Brownian motion B is replaced by some G-Itô process, we have the following statement similar to Theorem 4.1.
Theorem 4.3.For some p > 2, consider a q.s.continuous G-Itô process Y defined in the form of (3.4) whose coefficients are all elements in M p G ([0, T ]).Then, there exists a unique pair of processes and (a) X is positive; (b) K 0 = 0; and (c) T 0 X t dK t = 0, q.s..We omit the proof, since it is an analogue to the proof above and deduced mainly by the integrability of the coefficients of Y and (3.5).

Scalar valued RGSDEs
We state our main result in this section by giving the existence and uniqueness of the solutions to the scalar valued RGSDEs with Lipschitz coefficients.Besides, a comparison theorem is given at the end of this paper.5.2.Some a priori estimates and the uniqueness result.Let (X, K) be a pair of solution to (5.1).Replacing Y t by x and X t by X t − S t in (4.4), we have the following representation of K on [0, T ]: We now give a priori estimate on the uniform norm of the solution.
Proposition 5.1.Let (X, K) be a solution to (5.1).Then, there exists a constant C > 0 such that Proof: As X is the solution to (5.1), we obtain Similarly to (5.3), from the representation of K (5.2), we have (5.4) Then, we have Gronwall's lemma gives the desired result.
We deduce immediately the following uniqueness result by taking Proof: We set X 0 = x and K 0 = 0.For each n > 0, X n+1 is given by recurrence: Substituting X n+1 by Xn+1 + S t on the left side of (5.8), we know that ( )) by Theorem 4.3.Firstly, we establish a priori estimates uniform in n for { Ē[sup 0≤t≤T |X n t | p ]} n∈N .Similarly to (5.5), we have By recurrence, it is easily to verify that, for n ∈ N, where p(•) is the solution to the following ordinary differential equation: Secondly, for n and m ∈ N, we define Following the procedures in the proof of Theorem 5.2, we have Taking supremum on the left side, we obtain Finally, we define It is easy to find that v n t ≤ Cp(t), where C is independent of n.By classical Fatou's Lemma, we have Gronwall's lemma gives α t = 0, 0 ≤ t ≤ T, which implies that {X n } n∈N is a Cauchy sequence in M p G ([0, T ]).We denote the limit by X and set Obviously, the pair of processes (X, K) satisfies (A5) -(A7).We notice that Ē sup Then, one can verify that K n converges to K in M p G ([0, T ]) following the steps of (5.7).We conclude that the pair of processes (X, K), well defined in M p G ([0, T ]) × (M I ([0, T ]) ∩ M p G ([0, T ])), is a solution to (5.1).
Remark 5.5.Unlike in a classical RSDE, the constraint process S here is assumed to be a G-Itô process instead of a continuous process with Ē[sup 0≤t≤T (S + t ) 2 ] ≤ +∞ (cf.El Karoui et al. [3]).In fact, this is a sufficient condition to ensure that K n+1 is still a M p G ([0, T ]) process in (5.8) by Theorem 4.3, which may be weakened to: Remark 5.6.Using the approach in Soner et al. [20], this existence result still holds from a classical argument under each P ∈ P, that is, a pair of process (X P , K P ) can be found to satisfy (5.1) and (A5)-(A7) in the sense of P-a.s..However, it is difficult to determine a universal (X, K) which is the aggregation of these solutions (X P , K P ), P ∈ P.
Remark 5.7.In contrast with the fact mentioned in Remark 3.3 of Matoussi et al. [12], our results can be directly applied to the symmetrical problem i.e., the RGSDE with an upper barrier.This conclusion is due to the facts that the proof is only based on a pathwise construction and a fixed-point iteration.

Comparison principle.
In this subsection, we establish a comparison principle for RGSDEs.At first, we assume additionally a bounded condition on the coefficients f , h and g and the obstacle process S, and then we remove it in the second step.
s)ds , and p(•) is continuous and thus bounded on [0, T ].