Reflected Backward Stochastic Differential Equations Driven by Countable Brownian Motions with Continuous Coefficients

In this note, we study one-dimensional reflected backward stochastic differential equations (RBSDEs) driven by Countable Brownian Motions with one continuous barrier and continuous generators. Via a comparison theorem, we provide the existence of minimal and maximal solutions to this kind of equations. 1 Motivation Recently, Pengju Duan et al. [10] studied a new class of reflected backward stochastic differential equations driven by countable Brownian motions. That is : Y t = ξ + T t f (s, Y s , Z s)ds+ ∞ j=1 T t g j (s, Y s , Z s) ← − dB j s +K T −K t − T t Z s dW s. Under the global Lipschitz continuity condition, they proved in [10] via Snell envelope and fixed point theorem, the existence and uniqueness of the solution for RBSDEs (1.1). Unfortunately, the global Lipschitz continuity condition can not be satisfied in certain models that limits the scope of the result of Pengju Duan et al. [10] for several applications (finance, stochastic control, stochastic games, SPDEs, etc,...). In finance, this kind of RBSDEs is useful to describe the wealth/strategy of a particular investor who, trading continuously, has countable extra information which can not be detected in the market. In this case, ξ can be regarded as the contingent claim; Y t is the wealth of the investor at time t, Z t is his strategy at time t to make the wealth Y t be measurable to the information inside or outside the market at time t and to allow the realization of the option Y T = ξ, and K t is the subsidy injected by the government in the market at time t to allow the wealths of investors to remain above a threshold price S t at time t; f (t, Y t , Z t) is the appreciation of the wealth/strategy at time t and g j (t, Y t , Z t) is the impact of the j th additional information on the wealth/strategy at time t. For example, consider a particular investor who trades continuously in a financial market. Suppose that this investor has countable additional information which is not

Z s dW s . (1.1) Under the global Lipschitz continuity condition, they proved in [10] via Snell envelope and fixed point theorem, the existence and uniqueness of the solution for RBSDEs (1.1).Unfortunately, the global Lipschitz continuity condition can not be satisfied in certain models that limits the scope of the result of Pengju Duan et al. [10] for several applications (finance, stochastic control, stochastic games, SPDEs, etc,...).In finance, this kind of RBSDEs is useful to describe the wealth/strategy of a particular investor who, trading continuously, has countable extra information which can not be detected in the market.In this case, ξ can be regarded as the contingent claim; Y t is the wealth of the investor at time t, Z t is his strategy at time t to make the wealth Y t be measurable to the information inside or outside the market at time t and to allow the realization of the option Y T = ξ, and K t is the subsidy injected by the government in the market at time t to allow the wealths of investors to remain above a threshold price S t at time t; f (t, Y t , Z t ) is the appreciation of the wealth/strategy at time t and g j (t, Y t , Z t ) is the impact of the j th additional information on the wealth/strategy at time t.For example, consider a particular investor who trades continuously in a financial market.Suppose that this investor has countable additional information which is not available in the market.If the appreciation of his wealth/strategy is Y t 1 {Yt≥0} , then this investor has to solve the following RBSDEs : Since y → y1 {y≥0} is not Lipschitz in y, then we can not apply the existence result in Pengju Duan et al. [10] to get the existence of solution of the above RBSDE.Consequently, the problem of such an investor can not be solved at present.To correct this shortcoming, we relax in this paper the global Lipschitz continuity condition on the coefficients f to a continuity with sub linear condition and derive the existence of minimal and maximal solutions to RBSDEs (1.1).
The paper is organized as follows.In section 2, we give some notations and preliminaries Section 3 is devoted to the main result.

Preliminaries
The scalar product of the space R d (d ≥ 2) will be denoted by < ., .> and the associated Euclidian norm by . .Throughout this paper, T is a positive constant and (Ω, F, P) is a probability space on which, {B j t , 0 ≤ t ≤ T } ∞ j=1 are mutual independent one-dimensional standard Brownian motions and {W t , Let N denote the class of P-null sets of F and define where F η s,t = σ{η r − η s , s ≤ r ≤ t} for any η t , and F η t = F η 0,t .Since {F W t , t ∈ [0, T ]} is an increasing filtration and {F B j t,T , t ∈ [0, T ]} is a decreasing filtration, the collection {F t , t ∈ [0, T ]} is neither increasing nor decreasing so that it does not constitute a filtration.
For any n ∈ N, let M 2 (0, T ; R n ) denote the set of (class of dP ⊗ dt a.e.equal) ndimensional jointly measurable random processes {ϕ t ; 0 ≤ t ≤ T } which satisfy: We denote by S 2 ([0, T ]; R) the set of continuous one-dimensional random processes which satisfy: We set by A 2 (0, T, R + ) the space of a real positive continuous and increasing process, such that ϕ 0 = 0 and (i) ECP 20 (2015), paper 26.Definition 2.1.A solution of a (1.1) is a triplet of (R × R d × R + )-valued process (Y t , Z t , K t ) 0≤t≤T , which satisfies (1.1), and (iii) K is a continuous and increasing process with K 0 = 0 and T 0 (Y t − S t )dK t = 0. Definition 2.2.A triplet of processes (Y , Z, K) (resp.(Y , Z, K)) of E 2 (0, T ) is said to be a minimal (resp.a maximal) solution of (1.1) if for any other solution (Y, Z, K) of (1.1), We consider the following assumptions: where C j > 0 and α j > 0 are constants with where C > 0 is a nonnogative constant; where ϕ ∈ M 2 (0, T, R) is a positive process and M > 0 is a nonnegative constant.

The main results
In this section, our principal aim, is to prove an existence result for reflected BSDEs driven by countable Brownian motions under general continuous conditions.More precisely, we will derive the existence of a minimal and a maximal solution to equation (1.1) under assumptions (H1)-(H3) and (H5).
To this end, we establish first, the following comparison theorem, which extend the comparison theorem due to Aman and Owo [2] with RBSDEs driven by finite Brownian motions.
For n ∈ N and any continuous function f , let consider the following sequences of functions: ECP 20 (2015), paper 26.
Now, we are ready to establish the main result of this paper which is the following theorem.
Proof.We only prove that (1.1) has a minimal solution.The other case can be proved similarly.For fixed (t, ω) ∈ [0, T ] × Ω, it follows from (H5), that (y, z) → f (t, y, z) is continuous and with linear growth.Then, by Lemma 3.2, the associated sequences of functions f n and f n are Lipschitz functions.Therefore, since assumptions (H1) − (H4)(i) and (H5)(ii) hold, we get from Lemma 2.3, that there exists a unique solution T 0 (Y n t − S n t )dK n t = 0.  To complete the proof, it suffices to show that the sequence (Y n , Z n , K n ) converges to a process (Y , Z, K) which is the minimal solution of the RBDSDEs (1.1).
To this end, we will sketch the proof in two steps: Step

1: A priori estimates
There exists a constant C > 0 independent of n such that Indeed, applying Itô's formula to |Y n t | 2 , we have From assumption (H3), the proprieties of f n and Young's inequality, for any θ > 0, we have ECP 20 (2015), paper 26.
Choosing β, θ > 0 such that, β < , we derive that Therefore, we have the existence of a constant c = c(T, M, α) such that which by Burkhölder-Davis-Gundy's inequality provides Step 2: Convergence to the minimal solution We have from (3.3) ECP 20 (2015), paper 26.
Then, from Hölder's inequality, the uniform linear growth condition on the sequence f n and the assumption (H3), we have, Therefore, by virtue of inequality (3.4) and the fact that ϕ ∈ M 2 (0, T, R), we have the existence of a constant C such that, for all n > m ≥ M , Thus from (3.5), {Z n } is a Cauchy sequence in the Banach space M 2 (0, T, R d ), and there exists an F t -jointly measurable process Z such that {Z n } converges to Z as n → ∞.
Similarly, by Itô's formula together with Burkholder-Davis-Gundy inequality, it follows that Obviously, K 0 = 0 and {K t ; 0 ≤ t ≤ T } is a non-decreasing and continuous process.
On the other hand, from the result of Saisho [11] (see p. Finally, passing to the limit in (3.2), we conclude that (Y , Z, K) is a solution of (1.1).
Now, let (Y, Z, K) ∈ E 2 m (0, T ) be any solution of (1.1).By virtue of Theorem 3.1, we have Y n ≤ Y, ∀n ≥ M .Therefore, due to the above step and taking the limit, we have Y ≤ Y .
The proof is complete.
and (3.4) the existence of a process Y such that Y n t Y t a.s.for all t ∈ [0, T ].Hence, it follows from Fatou's lemma together with the dominated convergence theorem that E |Y t | 2 ≤ C and lim , there exists a F t -measurable process K with value in R + such that from which we deduce that P-almost surely, Y n converges uniformly to Y which is continuous, such that E sup0≤t≤T |Y t | 2 ≤ C.On the other hand, since Z n → Z in M 2 (0, T, R d ), along a subsequence which we still denote Z n , Z n → Z, dt ⊗ dP a.e., and there exists Π ∈ M 2 (0, T, R) such that ∀n ≥ M, |Z n | < Π, dt ⊗ dP a.e.. Therefore, by Lemma 3.2, we havef n (t, Y n t , Z n t ) −→ f (t, Y t , Z t ) dt ⊗ dP a.e., as n → ∞. s ) − f (s, Y s , Z s ) 2 ds −→ 0, as n → ∞.T t Z s dW s −→ 0, as n → ∞. 2 −→ 0, as n, m → ∞.Consequently2 −→ 0, as n → ∞.