Backward Doubly Stochastic Differential Equations with Markov Chains and a Comparison Theorem

: In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the Itô’s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.


Introduction
Backward stochastic differential equations (BSDEs) were first introduced by Pardoux and Peng [1]. Then a class of BDSDEs were introduced by Pardoux and Peng [2] in 1994, with two different directions of stochastic integrals, i.e., the equations involve both a forward stochastic integral dW t and a backward stochastic integral dB t . They have proved the existence and uniqueness of solutions for BDSDEs under uniformly Lipschitz conditions. Since then many efforts have been done in relaxing the Lipschitz conditions of the coefficient. For instance, Lepeltier and San Martin [3] have proved the existence of a solution for one-dimensional BSDEs when the coefficient is only continuous with linear growth in 1997. In 2002, Bahlali [4] dealt with multi-dimensional BSDEs with locally Lipschitz and sublinear growth coefficient. Then in 2004, Bahlali et al. [5] studied the existence and uniqueness of reflected BSDEs with both monotone and locally monotone coefficient. Inspired by the results above, Wu and Zhang [6] obtained the existence and uniqueness result of the solutions to BDSDEs with locally monotone and locally Lipschitz coefficient. On the other hand, Lepeltier and Martín [7], Kobylanski [8] studied BSDEs with generators of quadratic growth. In 2010, Zhang and Zhao [9] proved the existence and uniqueness of the L 2 ρ (R d ; R 1 ) ⊗ L 2 ρ (R d ; R d )-valued solutions for BDSDEs with linear growth and the monotonicity condition.
The comparison theorem is one of the important properties of the solutions of BDSDEs. It is not only important in basic fields, but also in stochastic control and financial mathematics. For example, it can be used to study viscosity solutions of the associated stochastic partial differential equations. For this, there have been many works. In 1997, Karoui et al. [10] studied the comparison theorem for one-dimension BSDEs, then Briand et al. [11] gave a converse comparison theorem for one-dimension BSDEs. After that, Hu and Peng [12] proved the comparison theorem for multidimensional BSDEs. Moreover, Yin and Situ [13] proved the comparison theorem of forward-backward SDEs with jumps and with random terminal time. In 2005, Shi et al. [14] firstly gave the comparison theorem for one-dimensional BDSDEs and by this, they showed the existence of the minimal solution of BDSDEs under linear growth conditions. Wu and Xu [15] proved some comparison theorems for forward-backward SDEs in one-dimension or multi-dimension by the probabilistic method and duality technique in 2009. In this paper, We are going to prove the comparison theorem for BDSDEs with Markov chains under locally Lipschitz and monotone conditions.

Preliminaries
We set (Ω, F , P) as a probability space. Let {W t , 0 ≤ t ≤ T} and {B t , 0 ≤ t ≤ T} be two mutually independent standard Brownian motions defined on (Ω, F , P), with values respectively in R d and R l . Let {α t , 0 ≤ t ≤ T} be a finite-state Markov chain with the state space I = {1, 2, . . . , m}, for some positive integer m. The transition intensities are λ ij (t) for i = j with λ ij nonnegative and bounded and λ ii = − ∑ j∈I\{i} λ ij . Assume that W, B and α are independent. Let N be the class of P-null sets of F .
where for any process η t , F η

Remark 1.
The collection {F t , t ∈ [0, T]} does not constitute a filtration, for it is neither increasing nor decreasing.
Let |x| be the Euclidean norm of a vector x ∈ R k . For a d × d matrix A, we define A = √ Tr AA * . For any n ∈ N, let M 2 ([t, T]; R n ) denote the set of (classes of dP × dt a.e. equal) n dimensional jointly measurable random processes {ϕ s ; s ∈ [t, T]} which satisfy: We denote similarly by S 2 ([t, T]; R n ) (resp. N 2 ([t, T] × I; R k )) the set of continuous n dimensional random processes which satisfy: Let V t (j) denote the number of jumps of {α s , 0 ≤ s ≤ T} from any state in I to state j between time 0 and t and let V denote the corresponding integer-valued random measure on ([0, T] × I, B([0, T] ⊗ B I )). The compensator of V t (j) is given by 1 α t− =j λ α t− ,j dt, i.e., is a martingale (compensated measure). We set λ t (j) = 1 α t− =j λ α t− ,j . Then the canonical special semimartingale representation for α (see [18,19]) is given by We need the following lemma, which is an extension of the well-known Itô's formula. The proof is a combination of Theorem 5.1 in Chapter 2 in [20] and Lemma 1.3 in [2]. So we give the following lemma without proof.

BDSDEs with Lipschitz Conditions
At the beginning of our study, we are going to study the BDSDEs with the Lipschitz condition. Let be jointly measurable such that for any (y, z, i) ∈ R k × R k×d × I, Moreover, we assume that there exist constants c > 0 and 0 < β < 1 such that for any (ω, Given ξ ∈ L 2 Ω, F T , P; R k , we consider the following backward doubly stochastic differential equation: where the integral with respect to {B t } is a backward Itô integral and the integral with respect to {W t } is a standard forward Itô integral. One can refer to Nualart and Pardoux [21] for more details.
The main objective of this section is to prove: Before we start proving the theorem, let us establish the same result in the case when f and g do not depend on Y and Z. Given f ∈ N 2 ([0, T] × I; R k ) and g ∈ N 2 ([0, T] × I; R k ) and ξ as above, consider the BSDE: Hence by orthogonality Then Y t ≡ 0 P a.s., Z t ≡ 0 and U t ≡ 0 dtdP a.e., Existence. We define the filtration (G t ) 0≤t≤T by by Burkholder-Davis-Gundy's inequality and Hölder's inequality, then there exists a constant 0 < C < ∞, such that By virtue of condition (H2), we have Similarly, we have E T 0 |g(s, α s )| 2 ds < +∞, so E|M t | 2 < +∞. Then by the martingale representation theorem (see Crépey and Matoussi [18]) there exist some Z ∈ M 2 ([0, T]; R k×d ) and Replacing M T and M t by their defining formulas and subtracting t 0 f (s, α s )ds + t 0 g(s, α s )dB s from both sides of the equality yields Applying Lemma 1 toȲ yields: Hence from (H1) and (H2) and the inequality ab ≤ 1 2(1−β) a 2 + 1−β 2 b 2 , where 0 < β < 1 is the constant appearing in (H1). Consequently Existence. We define recursively a sequence There exists c, γ > 0 such that Now choose θ = γ + 2c 1+β , and definec = 2c 1+β . It follows immediately that
It is easy to check that Let β = µ, then the transformed processes (Ȳ,Z,Ū) is the solution of a BDSDE with the generator f satisfying y − y f (t, y, z, i) −f t, y , z, i ≤ 0.
Before proving the theorem, we first recall the following definition and lemma from [6].

Definition 1.
Let F : R n → R n be a continuous function such that Then for any α > 0 and y ∈ R n , there exists a unique x = J α (y) such that x − αF(x) = y. We define the Yosida approximations F α , α > 0, of F, by setting Lemma 2. Let F be a continuous and monotone function, and F γ , γ > 0, be its Yosida approximations. Then we have (iii) ∀γ, β > 0,

Proposition 2.
For any V ∈ M 2 0, T; R k×d , there exists a unique triple of F t measurable processes Proof. Uniqueness. If Y 1 , Z 1 , U 1 and Y 2 , Z 2 , U 2 are two solutions of BDSDE (5), then by Itô's formula applied to Then the uniqueness can be concluded from Gronwall's inequality.
Existence. For any V ∈ M 2 [0, T]; R k×d , set f v (s, y, i) = f (s, y, V s , i) and g v (s, y, i) = g (s, y, V s , i) . Then f v is continuous and globally monotone in y, and g v is globally Lipschitz in y. Let f γ v , γ > 0, be the Yosida approximations of f v . Then from Theorem 1 we conclude that, for any γ > 0, the following BDSDE , from properties (i), (ii) of Lemma 2, Gronwall's inequality and the Burkholder-Davis-Gundy's inequality , we can obtain that there exists c > 0 which is independent of γ, such that ∀γ > 0, applying Gronwall's inequality and the Burkholder-Davis-Gundy's inequality gives Hence, {(Y γ , Z γ , U γ ) , γ > 0} is a Cauchy sequence in S 2 0, T; R k × M 2 0, T; R k×d × N 2 ([0, T] × I; R k ), and it has a limit denoted by (Y, Z, U). Passing to the limit on γ, as γ → 0, in (6), from the dominated convergence theorem we obtain that (Y, Z, U) satisfies BDSDE (5).
Proof of Theorem 2. By Proposition 2 we can construct a mapping Θ from M 2 0, T; R k×d into itself as follows. For any V ∈ M 2 , Z = Θ(V) can be uniquely determined by BDSDE (5). Let V, V ∈ M 2 , (Y, Z, U) and (Y , Z , U ) be the solutions introduced by V and V respectively. We will use the notations Then according to the proof of the first case, we get that Y 1 t ≥ Y 2 t , a.s. , ∀t ∈ [0, T] and we conclude the proof.

Discussion
It is well-known that the distribution of Brownian motion is symmetric and has rotational symmetry itself. Moreover, under the duality hypotheses, Brownian motion and Markov chains will have some time symmetry, see e.g., [22]. Therefore, our study can provide a theoretical basis for the establishment of symmetric models. Compared with [18], our innovation is to add a backward Brownian motion to the equation and relax the conditions of the equation, but the equation in this paper does not contain obstacles. Moreover, compared with [14], we studied the BDSDEs deriven by both Brownian motion and Markov chains while [14] studied the BDSDEs deriven only by Brownian motion.
In this paper, we studied the BDSDEs with Markov chains. Firstly, the existence and uniqueness results of the solutions to the BDSDEs were given under the Lipschitz condition. Then, we extended this result to the monotonicity condition. Finally, we proved the comparison theorem, which is very helpful for us to study the viscosity solution of the associated stochastic partial differential equation and the corresponding control problem. If the coefficient only satisfies the local monotonic condition, the study of the problem will be more difficult. We shall come back to this case in future work.
Author Contributions: All authors contributed equally to this article. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Natural Science Foundation of China (11831010, 61961160732), and Shandong Provincial Natural Science Foundation (ZR2019ZD42).