| This dissertation studies the multiple players nonzero-sum stochastic differential games (NZSDG) in the Markovian framework and their connection with multiple dimensional back-ward stochastic differential equations (BSDEs). There are three problems that we are focused on. Firstly, we consider a NZSDG where the drift coefficient is not bound but is of linear growth. Some particular case with the unbounded diffusion process is also considered. The existence of Nash equilibrium point is proved under the generalized Isaacs condition via the existence of the associated BSDE. The novelty is that the generator of the BSDE is of stochas-tic linear growth with respect to the volatility process. The second problem is a risk-sensitive case with the exponential type of payoff where the coefficients are unbounded. The associ-ated BSDE is of multi-dimension whose generator is quadratic on the volatility. We show the existence of Nash equilibria. The last problem that we treat, is a bang-bang game where the payoff is not continuous. In this case, Nash equilibria exists and is of bang-bang type which is not continuous and the value of the control will jump between the border of the domain with respect to the sign of the derivative of the value function. The BSDE in this case is a coupled multi-dimensional system, whose generator is discontinuous on the volatility process. The last problem that we considered is a recursive game model. It involves instantaneous util-ity depending not only on instantaneous consumption rate but also on the future utility. The main tool is the notion of backward stochastic differential equations which, in this case, are multidimensional with continuous coefficient. The generator is of stochastic linear growth on the volatility process and stochastic monotonic on the value process. We give the existence of the Nash equilibrium point.In this thesis, there are mainly four results, all related to NZSDG problem. We summarize briefly, those four different frameworks and their main results.1. Nonzero-sum Stochastic Differential Games with Unbounded CoefficientsChapter 2 in this thesis is a published cowork with Hamadene (ref.[64]).In Chapter 2, we study the nonzero-sum stochastic differential game of type control against control with the diffusion process a independent of controls, in the same line as in the paper by Hamadene et al. [58] in Markovian framework. The general formulation about NZSDG on multiple players framework is introduced in Subsection 1.1.3. We summarize the setting of this problem here in two-player case for simplicity. Notice that all those results and techniques in this Chapter can be generalized into the multiple players without any difficulties. Let us now recall the following:Setting for a NZSDG:As shown by [58], the setting in literature concerns only the case when the coefficients f and σ of the diffusion in the weak formulation (0.0.13) are bounded. According to our knowledge the setting where those coefficients are not bounded and of linear growth is not considered yet. Therefore the main objective of Chapter 2 is to relax as much as possible the boundedness of the coefficients f (mainly) and σ (which is not bounded as stated in the final extension in Chapter 2).The specific hypotheses that we impose are stated as follows.Assumption 0.1.The novelty of the results in Chapter 2 is that we show the existence of a Nash equilib-rium point for the NZSDG when f is no longer bounded but only satisfies the linear growth condition. The formulation is analogous as in Hamadene, Lepeltier and Peng (1997) [58]. But in the framework of [58], the coefficient f is bounded. Since the work depends heavily on Gir-sanov’s probability transformation:one has to deal with Doleans-Dade exponential of σ-1 f. When the coefficients σ-1 and f are bounded, obviously, Doleans-Dade exponential is a prob-ability density. However, when f is of linear growth, this conclusion is not so straightforward, meanwhile, some good estimates and properties of solutions to the corresponding BSDE will be invalid. This is the main difficulty in our work.An efficient tool that we applied to overcome this difficulty is a result by Haussmann (1986) (see Theorem 2.1 which related to the integrability of Doleans-Dade exponential with f in linear growth. Following from this assertion, we know that Doleans-Dade exponential corresponding to σ-1 f belongs to Lp space with some constant p located between 1 and 2, even f is of linear growth. With this integrability property in hand, Girsanov’s transformation can be carried out smoothly. Besides, a good link between the expectation under original probability and the one under the new probability is provided due to Haussmann’s result. This plays an important role in latter techniques for our scheme.As in [58] our approach is based on backward SDEs. As shown by Proposition 2.1, the payoffs for players are coincide with the initial values of solutions for the following BSDEs whose integrability is not standard. Then we are able to show that a NEP exists (see Theorem 2.1) with the help of BSDE (0.0.14) and the following BSDE (0.0.15) where the Hamiltonian type driver depend on Borel feedback controls. The proof is built by a localization scheme. Once we show that, there exist processes with proper integrabilities satisfying BSDE (0.0.15), then we can conclude that a NEP exists, which is exactly the pair of control processes ((u*, v*)(t, xt0,x, Zs1, Zs2))t≤T, for this NZSDG. Therefore, basically the problem turns into studying the following BSDE:This specific BSDE is multiple dimensional and each dimension is coupled mutually by terms of volatility processes. The main difficulty to solve this BSDE consists in the fact that its driver involve the termziσ-1(t,x)f(t,x,(u*,v*)(t,x, z1, z2)), i= 1,2, where f is not bounded but of linear growth in x. As a consequence of that, in Markovian framework, the driver of BSDE (0.0.15) is actually of linear growth in volatility term ω by ω. Alternatively, we may refer the driver is of stochastic linear growth or of stochastic Lipschitz (see [8]). In addition, the Hamiltonian with the feedback type controls, which plays a driver’s role for this BSDE, is continuous in (z1, z2). Results of BSDEs with stochastic Lipschitz condition includes the one by Briand [15] for case of BSDE with irregular generator involving BMO martingale.We finally show that this specific BSDE has a solution which then provides a NEP for the NZSDG when the generalized Isaacs condition is fulfilled and the laws of the dynamics of the non-controlled system satisfy the so-called Lq-domination condition. This latter is especially satisfied when the diffusion coefficient σ satisfies the well-known uniform ellipticity condition.Our method is based on:(ⅰ) the introduction of an approximating scheme of BSDEs which is well-posed since the coefficients are Lipschitz. In this markovian framework of randomness, the solutions (Yn, Zn),n≥ 1, of this scheme can be represented via deterministic functions (ωn,vn), n≥ 1, and the Markov process as well; (ⅱ) sharp estimates for (Yn,Zn) and (ωn, vn) and the Lq-domination condition enable us to obtain the strong convergence of a subsequence (ωnk)k≥1 from a weak convergence in an appropriate space. This yields the strong convergence of the corresponding subsequences (Ynk)k≥1 and (Znk)k≥1; (ⅲ) we finally show that the limit of (Ynk, Znk)k≥1 is a solution for the BSDE associated with the NZSDG.To summarize, there are three main points that we require in our approach:(ⅰ) Haussmann’s result:there exists a p ∈ (1,2) such that for any admissible control (u,v), E[(ζT(σ-1（·, xt,x)f(·, xt,x, u,v)))p]< ∞ where ζT is defined as in (0.0.13);(ii) The Lq-domination property or its adaptation;(iii) The generalized Isaacs condition.At the end of this chapter we provide an example which illustrates our result. We also discuss possible extensions of our findings to the case when both the drift f and diffusion coefficient σ of the state process are not bounded.2. Risk-sensitive Nonzero-sum Stochastic Differential Game with Unbounded Coeffi-cientsChapter 3 is a joint work with S. Hamadene.Chapter 3 deals with the risk-sensitive NZSDG, as presented in Abstract, which is a game problem taking into account the attitudes of the players. For the details about the three different cases including risk averse, risk seeking and risk neutral, readers are referred to Abstract. In Chapter 3, we focus on the risk averse situation below. Besides, for notation’s simplicity, we discuss under two-player framework. However, the generalization to multiple players case is formal and can be carried out in the same spirit.The setting is analogous to the one for a standard NZSDG, see (0.0.13) for details. We also start with a state process which is a diffusion. We later set up our problem on the weak formulation, i.e. transform the original probability P into the new one Pu,v by Girsanov’s transformation. Then the law of this state diffusion process under probability P maintains the same as the one under Pu,v. The weak formulation of state process is given by:dxst,x= f(s, xst,x, us, vs)ds+σ(s, xst,x)dBsu,v for s ∈ [t, T] and xst,x= x for s< t.The unique distinguish is that the payoffs are more complicated and are of exponential types which are natural and popular especially in the economic field. The payoffs for a risk-sensitive NZSDG is stated as follows:for player i= 1,2 and for each pair of admissible controls (u, v), the payoffs areThe objective of the risk-sensitive NZSDG is to find a NEP, which is a pair of admissible controls (u*, v*), such that J1(u*, v*)≤ J1(u, v*) and J2(u*, v*)≤J2(u*, v).We emphasis that, all the assumptions under Chapter 3 is the same as Assumption 0.1:(i) diffusion σ is uniformly Lipschitz, invertible, bounded and its inverse is bounded. It follows from above that such a a satisfies uniform ellipticity condition; (ii) drift function f is of linear growth in x; (ⅲ) both the running payoff hi and the terminal payoff gi are of polynomial growth in x for i= 1,2; (ⅳ) Isaacs condition is fulfilled; (ⅴ) the Hamiltonians with the feedback type controls are continuous.About the risk-sensitive stochastic differential game problem, including nonzero-sum, zero-sum and mean-field cases, there are some previous works. Readers are referred to [7,36,43,44,67,95] for further acquaintance. Among those results, a particular popular approach is partial differential equation, such as [7,43,44,67,95] with various objectives. Another method is through backward stochastic differential equation (BSDE) theory, see [36]. In Chapter 3, we also deal with this risk-sensitive game through BSDE tools in the same line as article by El-Karoui and Hamadene (2003) [36].However in [36], the setting of game problem concerns only the case when the drift coefficient f in diffusion dynamic is bounded. This constrain is too strict to some extent. Therefore, our motivation is to relax as much as possible the boundedness of the coefficient f. We assume, like Subsection (see Assumption 0.1) that f is not bounded any more but instead, it has a linear growth condition. It is the main novelty of this work. To our knowledge, this general case has not been studied in the literature for a risk-sensitive NZSDG.As illustrated in the previous part for a standard NZSDG, the payoff is related to the initial value of a corresponding BSDE. For a risk-sensitive case, this fact is still true which tells us that the payoff coincides with the exponential of the initial value of some specific BSDE. The equivalence is proved in Chapter 3 in the main text (see Proposition 3.1). Therefore, to find a NEP for the risk-sensitive NZSDG is reduced to study the existence of solutions for the following BSDE:This BSDE is multiple-dimensional with continuous generator involving both linear and quadratic terms of z. The difficulties to solve this BSDE rely on two perspectives:(ⅰ) The first difficulty is the quadratic term of z which is involved in the driver. In compare with the standard NZSDG as Subsection, we need to carry out some techniques to deal with this quadratic term specially.(ⅱ) The second one is the following:since the driver has two components, one is a linear part of the volatility process which is Hamiltonian including the feedback type controls, the other one is a quadratic term of the volatility process. It takes the form of Hi(s, x, zi, (u*, v*)(t, x, z1,z2))+1/2|zi|2= zif(s, x, z1,z2)+hi(s, x, z1,z2)+1/2|zi|2 for i= 1,2 where f is of linear growth in x. Similar as in Subsection, the first linear term of z is of linear growth ω by ω in Markovian framework due to the linear growth of f. The case when f is bounded has been dealt by [36].Respect to these two difficulties, our strategies to overcome them are the following:(i)’To deal with the quadratic term of z, we apply the classical exponential transform (see M.kobylanski et al (2000) [72]):Yi= eYi; Zi= YiZi for i= 1,2. By this technique, the quadratic term can be eliminated. However, as a cost, the value process get involved into the driver.(ii)’The difficulty, specially in making apply the Girsanov’s transformation, brought by the linear growth of f is overcome by a Haussmann’s result. As illustrated in Subsection, the Doleans-Dade exponential local martingale on σ-1 f is integrable in Lp with somep ∈ (1,2). This result enable us to carry out Girsanov’s transformation in order to move out the volatility term from the driver, which then provide us an access to obtain the integrability of the value process.Under the generalized Isaacs hypothesis and domination property related to the law of diffusion process, which holds when the uniform ellipticity condition on σ is satisfied, we finally show that the associated BSDE (0.0.16) has a solution which then provides the NEP for this risk-sensitive NZSDG.The method is summarized as follows:(ⅰ) We firstly take an exponential transformation to eliminate the quadratic term of volatility process. The original BSDE is replaced by a new one which is multiple dimensional with a driver involving both parts of Y and Z, being of linear growth ω by ω in these two components; (ⅱ)We then introduce an approximate scheme which smooth the generator of BSDEs by the mollifier technique. The new sequence of BSDEs is well-posed since the coefficients are Lipschitz. Besides, under Markovian framework of randomness, the solutions (Yn, Zn), n≥ 1 of this scheme can be represented by deterministic functions (ζn,δn), n≥ 1. An exponential type growth property of ζn is provided which then yields the sharp estimates of (Yn, Zn). (ⅲ) We may subtract a subsequence (ζnk)k≥1 which is proved to be a strong convergent sequence from a weak convergence in an appropriate space. This yields the strong convergence of the corresponding subsequences (Ynk, Znk)k≥1;(iv) We finally show that the limit of subsequence (Ynk, Znk)k≥1 is a solution for the transformed BSDE. By taking the inverse exponential transform, the solution for BSDE (0.0.16) exists.3. Bang-Bang Type Equilibrium Point for Nonzero-sum Stochastic Differential GameChapter4 in this dissertation is a published work with Hamadene (ref. [62]).The motivation of this work is the following. Notice that in the previous results about NZSDG, such as [58,54,53,76], the authors concern only about the smooth feedback controls as well as the Hamiltonian functions. The same as our results in Chapter 2 and 3. The proofs rely heavily on the continuous assumption on Hamiltonian (see Assumption 0.1:Continuity). The case of discontinuous controls is not fully explored. Indeed, the discontinuous controls are naturally exist and reasonable, especially in economic and engineering fields.Therefore, the main goal of this chapter is to study a special type of NZSDG in Markovian framework. We show the existence of Nash equilibrium point which is discontinuous and of bang-bang type under natural conditions. The main tool is the notion of BSDEs which, in our case, are multidimensional with discontinuous generator with respect to the volatility processWe now briefly introduce the game model in two players and one dimensional case for simplicity. The general multiple players and high dimensional situation is a straightforward adaption. The dynamic of this game system is given by a stochastic differential equation (SDE for short) as follows, for any fixed (t, x) ∈ [0, T] x R, Vs≤ T, Xst,x= x+(Bs∨t-Bt). (0.0.17) Each player has his own control. Let us denote by U and V two bounded subsets on R and M1 (resp.M2) be the set of admissible controls which is the set of Ρ-measurable process u=(ut)t≤T (resp. v= (vt)t<T) on [0,T] × Ω with value on U (resp. V). M= M1 × M2. Let Γ:(t, x, u, v) ∈ [0, T] × R × U × V→R be the dynamic function for the game problem. The precise assumptions on the coefficients are stated below.For any admissible pair of controls (u., v.) ∈ M, let Pu,v be the positive measure on (Ω, F) as Allows, dPt,xu,v= ζT(Γ(.,X.t,x,u.,v.))dP with ζt(θ):= 1+∫0tθsζsdBs, t≤ T for any measurable Ft-adapted process θ:= (θt)t≤T.Under appropriate assumptions on Γ, it follows that Pt,xu,v is a probability on (Ω, F). Then, the process Bu,v= (Bs-∫0sΓ(r, Xrt,x,ur,vr)dr)s≤T is a (Fs, Pu,v)-Brownian motion and (Xst,x)s≤T satisfies the fol-lowing SDE,dXst,x= Γ(s, Xst,x, us, vs)ds+dBsu,v (?)s ∈ [t, T] and Xst,x= x, s ∈ [0, t]. (0.0.18) We denote the terminal payoff function by gi:x ∈ R→ R for player i= 1,2For fixed (0, x), Let us define the payoffs for players as following, for (u, v) G M, Ji(u,v):= Eu,v[g1(XT0,x)] and J2(u,v):= Eu,v[g2(XT0,x)],where Euu,v is the expectation under probability Pu,v for fixed (0, x). We concern about the existence of Nash equilibrium point, i.e. a couple of controls (u*, v*) ∈ M satisfying, for all (u, v) ∈ M, J1(u*,v*)≥ J1(u,v*) and J2(u*,v*)≥ J2(u*,v). Our assumptions are listed below:Assumption 0.2. (i) The value sets for the admissible controls (u, v) are two bounded subsets U and V on R associate U= [0,1] and V= [-1,1];(ii) The dynamic function Γ is an affine combination of controls which has form Γ(t,x,u,v)= f(t,x)+u+v where f:(t,x) ∈ [0, T] ×R→R be a Borelian function. The function f is of linear growth w.r.t. x, Therefore, Γ is also of linear growth on x uniformly w.r.t(u,v) ∈U×V(ⅲ) The terminal values gi,i= 1,2 are of polynomial growth on x.There are several properties in the setting of this NZSDG that we would like to emphasis: (i) The dynamic function Γ is not bounded as the previous results related to NZSDG (see [58,54,53,76]), instead, it is of linear growth with respect to x. As illustrated in Subsection and, the difficulty brought by the linear growth property of Γ can be overcome by a result of Haussmann (see Lemma 4.1) which related to the integrability of Doleans-Dade exponential. Indeed, the Girsanov’s transformation can be carried out smoothly which yields the weak formulation of the state process (0.0.18). (ⅱ) The value sets U and V for the admissible controls are two specific bounded subsets on R. Besides, the dynamic Γ is also of specific affine form. In additional, there are only terminal values getting involved in the payoffs J1 and J2, however, without the instantaneous payoffs. Therefore, Nash equilibrium point, if exists, should be in general bang-bang type.The notion of bang-bang type control comes from the classical stochastic control theory. Consider in our case, when Γ does not depend on v, the stochastic differential game problem will be reduced to a stochastic control problem. By bang-bang control, we refer to a discontin-uous control which will jump at a certain point between the border of the domain depending on the sign of the gradient of the value function. A common form is expressed by Heaviside function as presented in our work. Let us explain in the following, the exact form of bang-bang type candidate Nash equilibrium point for this NZSDG.Since the dynamic coefficient Γ is known as a specific function of u and v clearly, there-fore, the candidate optimal control for us can be worked out directly by the generalized Isaacs’ condition. Let H1 and H2 be the Hamiltonian functions of this game, i.e, the functions which are not depend on ω defined from [0,T] ×R×R×U×V into R by: H1(t,x,p,u,v):= pΓ(t,x,u,v)= p(f(t,x)+u+v); H2(t,x,q,u,v):= qΓ(t,x,u,v)= q(f(t,x)+u+v).Now, the controls u and v which defined on R × U and R × V, valued on U and V respectively, as follows:(?)p, q ∈ R, ε ∈ U, ε ∈ will exactly satisfy the generalized Isaacs’condition as follows:For all (t,x,p,q,u,v) ∈ [0,T]×R×R×U×V and(ε,ε)∈U×V, we have, H1*(t,x,p,q,ε):= H1(t,x,p,u(p,ε),v(q,ε))> H1(t,x,p,u,v(q,ε)), HZ(t,x,p,q,ε):= H2(t,x,q,u(p,ε),v(q,ε))> H2(t,x,q,u(p,ε),v). (0.0.20)We should point out that, the function H1*(resp.H2*) does not depend on ε (resp.ε), since, pu(p, ε)= p ∨0 (resp. qv(q, ε)=|q|) does not depend on ε (resp. ε). Besides, the Hamiltonian function here is discontinuous w.r.t. (p, q). It follows from (0.0.19) that, the pair of control (u, v) is of bang-bang type. However, it is not a feedback one since it is also depend on some constants. Similar nonzero-sum differential game of unsmooth type has been studied by G.J. Olsder [85] in the deterministic case. Recent works on this subject, also in deterministic case, include papers by P. Cardaliaguet and S. Plaskacz [24], P. Cardaliaguet [23] which show that there exists a unique Nash equilibrium payoff of feedback form. But this equilibrium payoff depend in a very unstable way on the terminal data. Besides, it is not obvious to generalize the result in [24] to higher dimensions. The stochastic case has been analyzed by P. Mannucci [80] with the help of a system of Hamilton-Jacobi equations and related parabolic PDE techniques. Notice that the state process in [80] belongs to a bounded domain. However, some techniques of PDE in the global domain are not so straightforward.The main novelty of Chapter 4 is that we show the existence of Nash equilibrium point of bang-bang type to a nonzero-sum stochastic differential game in a global domain. Moreover, the results and the techniques can be generalized to the multiple dimensions directly. However, the existence of NEP of feedback form is still an open problem.As in [58], we apply the BSDE approach. This game problem finally reduces to solving a multiple-dimensional BSDE with a discontinuous generator with respect to z component and of linear growth in z ω by ω. Under the generalized Isaacs’hypothesis, we show that the associated BSDE has a solution which then provides a bang-bang type NEP for the NZSDG. The main result is summarized as follows:Theorem 0.1 (Existence of bang-bang type NEP). Let us suppose Assumption 0.2 and gener-alized Isaacs’condition (0.0.20) are fulfilled. Then, there exists η1,η2, (Y1,Z1), (Y2, Z2) and θ,v such that:(ⅰ) η1 and η2 are two deterministic measurable functions with polynomial growth from [0, T] x R to R;(ⅱ) (Y1, Z1) and (Y2, Z2) are two couples of Ρ-measurable processes with values on R1+1:(ⅲ) θ (resp.v) is a Ρ-measurable process valued on U(resp. V),and satisfy:(a) P-a.s., Vs≤ T, Ysi= ηi(s, Xs0,x) and Zi(ω):= (Zsi(ω))s≤T is ds-square integrable;(b) For all s< T,Besides, Y0i= Ji(u,v), i= 1,2 and the pair of controls (u(Zs1,θs),v(Zs2,vs))s≤T is a bang-bang type Nash equilibrium point of the nonzero-sum stochastic differential game.The link between NZSDG and BSDEs is obtained in a standard way as Subsection which tells us that the initial value of the associated BSDE is coincide with the payoff for the game problem. In additional, once we show that BSDE (0.0.21) has solutions with appropriate properties, then, the existence of NEP naturally holds true which obtained by comparing the solutions of BSDEs after change probability and in using the fact that (u, v) verifies the gener-alized Isaacs’ condition (0.0.20). Therefore, the main task of this work is thus dedicated to the proof of the solvability of the system of BSDE (0.0.21) which is multiple dimensional with a discontinuous generator with respect to the volatility z. Apparently, the discontinuity is a main difficulty of this work.The way we solve BSDE (0.0.21) is the following:(ⅰ) We first construct an approximation scheme by smoothing the discontinuous functions (u, v) via Lipschitz continuous applications (un,un), n≥ 1. The sequence of BSDEs is well-posed since the generators are uniformly Lipschitz which followed by the existence of solutions. Besides, the solutions (Yn, Zn) can be expressed by deterministic functions (ηn,ζn). (ⅱ) Sharp estimates on those solutions are provided in appropriated spaces, as well as the polynomial growth property of functions ηn. (ⅲ) Sequence ηn is proved as a Cauchy sequence following from a result of weak convergence. This yields the strong convergence of (Yn, Zn) in some proper spaces. (ⅳ) In order to verify that the limit process which comes from the strong convergence result is indeed the solution of the original BSDE, it remains to show that the sequence of the approximate drivers converge. We finally obtain a weak convergence of the subsequence of the drivers toward Hamiltonian when the arbitrary constants (ε, ε) are replaced by some process (θ,v). The difficulty of the discontinuity for Hamiltonian is overcome in this step by the weak convergence arguments.4. Recursive Nonzero-sum Stochastic Differential Game with Unbounded CoefficientsChapter 5 in this dissertation is a joint work with professor Zhen Wu.In this chapter, we discuss a recursive nonzero-sum stochastic differential game (NZSDG for short) under Markovian framework. Generally speaking, stochastic differential game the-ory deals with conflict or cooperate problems in a dynamic system which is influenced by multiple players. Let us introduce the setting of the problem briefly. Assume that we have a system which is described as follows: dxt= σ(t, xt)dBt for t≤T and x0= x. (0.0.22)This system can also be controlled by two players which we represent by a weak formu-lation of stochastic differential equation: dxt= f(t, xt, ut, vt)dt+σ(t, xt)dBtu,v for t≤T and x0= x. (0.0.23)The process Bu’v is a new Brownian motion generated from B by applying Girsanov’s transformation. Processes u= (ut)t≤T and v= (vt)t≤T represent the control actions of the two players imposed on this system. Indeed, the controls are not free, which bring some costs for players. What we discussed is a recursive type of cost functional, which is defined by the initial value of the following backward stochastic differential equation (BSDE for short):for i= 1,2, (0.0.24)The costs that we concerned are:Ji(u, v)= y0,z,u,v for players i= 1,2. The objective of this game model is to find a Nash equilibrium point (u*, v*) such that, J1(u*,v*)≤ J1(u,v*) and J2(u*,v*)≤ J2(u*,v) for any admissible control (u, v). This is actually say that both of the two players would like to minimize their costs and no one can cut more by unilaterally changing his own control.The concept of stochastic differential recursive utility has been considered by Duffie and Epstein [35] which extends the classical utility. The recursive one involves instantaneous utility depending not only on instantaneous consumption rate but also on the future utility. The manner of using solutions of BSDEs to describe cost functionals of stochastic differential game is initially inspired by El Karoui et al. [39], where some formulations of recursive utilities and their properties are also discussed. If we consider functions hi are independent on parameters yi,then the costs Ji will be reduced to the classical structure, i.e., Eu,v[gi(xT)+ ∫0T hi(s,xs,us,vs)], which is the accumulate of instantaneous cost and usually guaranteed with a terminal cost. Some recursive optimal control problems are studied by Wang and Wu [96]. Works associated with stochastic recursive game problem include [97] which is a zero-sum case. Readers are referred to a series of works by Hamadene for research on classical NZSDGs without the recursive part, say [53,58] and the references therein.In the present paper, we study the recursive NZSDG through BSDE’s technique in the same line as article by Wei and Wu [97]. However in [97], the drift function f of the state process is bounded or almost equivalent to bounded one. This boundedness is important when we consider the related BSDEs since this guarantee the good Liptsitz property of the generator of BSDE with respect to z component. However, this restriction is too strict to some extent. Therefore, the motivation of our work is to relax this limitation on f. To instead, we consider a drift f which is of linear growth on the state process x. This has already been considered in classical problem without recursive part by [64] and [62]. To our knowledge, this general recursive case has not been studied in literatures. The existence of Nash equilibrium point turns into studying its associated multiple dimensional BSDE with generator which is of linear growth on z, ω by ω and stochastic monotonic on y. Under the generalized Isaacs condition, we show the existence of solution for the later BSDE which provides a NEP for this recursive NZSDG.Finally, we point out that this work establishes a model involving only two players, how-ever, it can be generalized to multiple players case following the same way without any diffi-culty.For the existence of solutions of BSDE (0.0.24), our idea is to take a partition of interval [0, T] and firstly solve this BSDE in a small interval [T-δ,T], then, extend it backwardly to the whole interval. |
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