Backward Stochastic Differential Equations,g-Expectations And Related Semilinear PDEs

It was mainly during the last almost two decades that the theory of backward stochastic differential equations (BSDEs, in short) took shape as a distinct mathematical discipline. The starting point of the development of general BSDEis the celebrated paper of Pardoux and Peng [117]. Note that, since the boundary condition is given at the terminal time T, it is not really natural for the solution Y_t to be adapted at each time t to the past of the the Brownian motion W before time t. The presence of Z_t seems superfluous. However, we need to point out. that it is the presence of this process that makes it possible to find adapted process Y_t to satisfy (1.1). Hence, a solution of BSDE (1.1) on the probability space of Brownian motion, as mentioned above, is a pair (Y, Z) of adapted processes that satisfies (1.1) almost surely. The role of Z is very different in the theory of BSDE.In [117], Pardoux and Peng established the existence and uniqueness of the solution of equation (1.1) under the uniform Lipschitz condition, i.e., there exists a constant K > 0 such that|g(w,t.y,z) - g(w,t.y',z')|≤K(|y - y'| + |z - z'|) (12)for all y, y'∈Rⁿ, z, z'∈R^n×d, and (w, t)∈Ω×[0, T].From then on, a considerable amount of works has been devoted to study BSDE theory. Generally, these studies about BSDEs involved many aspects. The first important one is the fundamental theory about BSDEs, including to establish the existence uniqueness of solutions for BSDEs under various forms (including BSDEs with reflected barriers, with jumps; FBSDEs; and functional BSDEs), various coefficient conditions and terminal value conditions for extending Pardoux-Peng's initial result. e.g., see Pardoux-Peng [119, 121], El Karoui [49]. Kobylanski [90], Lepeltier-San Martin [98], El Karoui-Kapoudjian-Pardoux-Peng-Quenez [50], Peng-Wu [147], Wu [167]. Hu-Peng [69], Peng-Yang [152]. Wu-Yu [170] Kohlmann-Tang [91. 93], Hu-Yong [71]. Hu [67]. Ma-Yong [105]. Mao [107], Situ [155] and Briand-Delyon-Pardoux-Hu-Stoica[15]. This part also includes the works to study the important properties of solutions of BSDEs detailedly and associated numerical methods, e.g.. see Peng [128], Coquet-Hu-Memin-Peng [38], Briand-Coquet-Hu-Memin-Peng [13], Cao-Yan [21], Wu [168], Chen-Kulperger-Jiang [30], Jiang [81], Ma-Protter-Yong [104], Ma-Zhang [106], Buckdahn-Quincampoix-Rascanu [20], Zhao-Chen-Peng [176], Peng-Xu [148] and Gobet-Lemor-Warin [58]. The second and more important aspect is about the applications of BSDE theory. BSDE theory has found a wider field of applications as in stochastic optimal control and stochastic games (e.g.,see Peng [128, 132], El Karoui-Quenez [52]. Buckdahn-Li [17], Chen-Li-Zhou [23], Lim-Zhou [99], Liu-Peng [103], Yong-Zhou [173], Kolmann-Tang [91, 93] and Kohlmann-Zhou [94]) and at the same time, in mathematical finance, the theory of hedging and nonlinear pricing theory for imperfect, markets (e.g., see El Karoui-Peng-Quenez [51], Chen-Epstein [31], Delbaen-Peng-Rosazza [45], El Karoui-Quenez [52], Barrieu-El Karoui [11], Duffie-Epstein [47], Kohlmann-Tang [91, 92], Yong [172] and Cvitanic-Karatzas [42] et al.) and in nonlinear expectation theory (e.g., see Peng [132-134, 138-144], Delbaen-Peng-Rosazza [45]. Chen-Epstein [31], Coquet-Hu-Memin-Peng [39], Briand-Coquet-Hu-Memin-Peng [13], Chen-Kulperger-Jiang [30], Chen-Peng [32, 33], Hu [68], Jiang-Chen [83] and Rosazza [153] et al.). The theory of BSDE also provides probabilistic formulae for solutions to partial differential equations (e.g., see Peng [127. 132], Pardoux-Peng [120], El Karoui Kapoudjian-Pardoux-Peng-Quenez [50]. Pardoux-Tang [123]. Barles-Buchdahn-Paroux [9], Buckdahn-Hu [16]. Briand-Hu [14], Pardoux [115], Pardoux-Tang [123] and Kobylanski [90] et al.).As a matter of convenience, we denote equation (11) by (g,T,ζ), its solution by (Y_t^g,T,ζ,Z_t^g,T,ζ)_t∈[0,T]. In addition, we denote Y_t^g,T,ζ by E_t,T^g[ζ]. In the this thesis, we always assume that for each (y, z)∈Rⁿ×R^n×d, g(·. y. z)∈L_F²(0, T).We list the contents of this thesis as follows:Chapter 1 Introduction;Chapter 2 BSDEs with continuous coefficients:Chapter 3 BSDEs with uniformly continuous coefficients and related g-expectations: Chapter 4 Probabilistic interpretation of a class of semi-linear parabolic PDEs:Chapter 5 Jensen' s inequality for g-convex function under g-expectation and backward stochastic viability property;Chapter 6 BSDEs with discontinuous coefficients.The thesis study some fundamental problems in BSDE theory.(â… ) In Chapter 2 we establish Kneser's theorem for BSDE; and show an invariant representation theorem for coefficient of BSDE; we prove a continuous dependence theorem for the solution of BSDE with continuous coefficient.As we know, in the case when g is only continuous and of linear growth in (y, z). Lepeltier-San Martin(1997) proved that the solution of the corresponding BSDE exists but may be not unique. All results in this chapter are in this framework.We first, prove a Kneser's theorem for BSDE, i.e.. to study the cardinality of the set of solutions of a BSDE with continuous coefficient.Theorem 2.2.1. Let g be continuous and of linear growth in (y. z). Then the cardinality of the set of solution of equation (1) is either 1 or continuum.This result is not surprising, because Kneser(1923)(see also Hartman [61, pp.15]) and Alexiewicz-Orlicz(1956) got similar results for ODEs and PDEs respectively. But the following result as a consequence of Theorem 2.2.1 is very interesting.Corollary 2.2.4. Let. g satisfy the conditions in Theorem 2.2.1. and let g be independent ofω,ζbe deterministic, i.e.,ζ∈R. Then.(i) The maximal and minimal solution of (g, T,ζ), i.e., ((?)_t,(?)_t)_t∈[0,T] and ((?), (?))_t∈[0,T], are deterministic, namely (?) = (?)≡0 for t 6 [0,T] and, ((?))_t∈[0,T] and ((?))_t∈[0,T] are deterministic processes.(ii) (g,T,ζ) has uncountably many stochastic solutions (and deterministic solutions) whenever its solution is not unique.Moreover all stochastic (and deterministic) solutions are dominated by ((?),(?))_t∈[0,T], and ((?),(?))_t∈[0,T] from below and above, i.e., for any stochastic (deterministic) solution (Y_t, Z_t)_t∈[0,T] of (g, T,ζ), one hasThe above corollary means that there are uncountably many stochastic solutions (deterministic solutions) to a "deterministic" equation, which has deterministic coeffi- cient and deterministic boundary condition, if its solution is not unique. To ray knowledge, this phenomenon hasn't, been being found as yet in any other kind of equations. How to explain this phenomenon? It will be a very interesting problem.In addition. Kneser's theorem for reflected BSDE also holds, the readers can refer to Jia-Xu [77].The second result of this chapter is about the representation theorem of g. For proving converse comparison theorem of BSDE, Briand-Coquet-Hu-Memin-Peng [13] established the following representation theorem of g: (?)(t,y, z)∈[0, T]×R×R^d,for generator g under two additional assumptions that (g(l, y, z))_t∈[0,T] is continuous with respect to t for each (y, z) and E [sup_t∈[0,T] |g(t,0.0)|²] <∞.Recently, Jiang [81] generalized the above result for proving the corresponding converse comparison theorem, he proved that the generator of g of a BSDE could be represented by the solution of corresponding BSDE at point. (t, y, z) if and only ifOf course g in all above works was assumed to be Lipschitz continuous in (y,z). A natural question is, can we have similar representation theorem for g by the solutions of BSDE with continuous coefficient? The answer is positive.Theorem 2.3.7. Let g satisfy the conditions in Theorem 2.2.1, and letdenote any given solution of (g, t +ε, y + z·(W_t+ε - W_t)) where t +ε≤T andε> 0, moreover let p∈[1,2). Then for each (y, z)∈R^1+d, we haveThe third result is about the question whether the uniqueness of solution to a BSDE is equivalent to continuous dependence with respect to coefficient or terminal value or not. As we know, this equivalence holds true for ODEs or PDEs. but we haven't it for BSDEs yet. Actually it is a common, but also fundamental result.Theorem 2.4.7. Letλbelongs to a nonempty set D (?) R, and let g^λsatisfy the following assumptions: (H1'): linear growth: there exists a nonnegative constant A, such that (?) A(1+|y|+|z|),(?)λ,t,w,y,z.(H2'): (g^λ(t, y, z))_t∈[0,T]∈H₁², for each (y, z)∈R×R^d andλ∈D.(H3'): For fixed t,ω,λ, g^λ(ω, t,...) is continuous.(H4'): uniform continuity: g^λis continuous atλ=λ₀ uniformly with respect to(y,z).Then the following statements are equivalent:(i). Uniqueness: there exists a unique solution of (2.25) withλ=λ₀ that is, the solution of (g^λ₀. T.ζ^λ0) is unique.(ii). Continuous dependence with respect to g andζ: for anyζ^λ,ζ^λ0∈L²(F_T), ifζ^λ→ζ^λ₀ in L²(F_T) asλ→λ₀, (y_t^λ,z_t^λ)_t∈[0,T] are any solutions of (g^λT,ζ^λ), (y_t^λ₀, z_t^λ₀)_t∈[0,T] is any solution of (g^λ,T,ζ^λ) atλ=λ₀, then(â…¡) In Chapter 3, we prove Peng's conjecture, i.e., if g is independent of y and uniformly continuous in z uniformly with respect to (ω, t), the solution of corresponding BSDE is unique; we introduce a class of new g-expectation via BSDE with uniformly continuous coefficient, and prove that it is an F_t-consistent nonlinear expectation without strict comparison theorem.In a seminar at Shandong University, Oct. 2005, Prof. Peng gave a conjecture in his lecture that for a BSDE with coefficient which is Holder continuous in z and independent of y, the solution of associated equation is unique. In the first part of this chapter, we will prove this conjecture under a more general and weak condition-uniformly continuity instead of Holder continuity. By this result, we shall get a new existence and uniqueness theorem for BSDEs with coefficients which are uniformly continuous in z and continuous, monotonic in y such as the type Pardoux used in [115] or [116].Theorem 3.3.1. Let g be uniformly continuous in z and independent of y (or dependent of y, but continuous, monotonic in it such as the type Pardoux used in [115] or [116]). Then the solution of (1) is unique.It is worth noting that there is an important difference between the BSDE with coefficient g being Lipschitz continuous in (y,z) and the BSDE in Theorem 3.3.1: although we still have the associated comparwon theorem, the strict comparison result does not hold in general. As a by-product of the proof of Theorem 3.3.1. we also have the following interesting result.Theorem 3.3.8. Let g be uniformly continuous in (y, z) uniformly with respect to (ω, t), and letζ∈L²(Ω. F_T. P: R). Then the set of real numbers c such that the solution of perturbed BSDE with parameter (g + c. T.ζ) is non-unique, is at most countable.For the case of reflected BSDE, the above results also hold, the readers can refer to [77].We now define a new class of g-expectations via BSDEs with uniformly continuous coefficient.Definition 3.5.2. Let g be uniformly continuous in z and Lipschitz continuous in y. and we assume that for each (t, y), g(t, y, 0) = 0. The g-expectation E⁹·[] : L²(Ω, F_T, P)(?) R is defined byE^g[X](?)Y₀=E_0,T^g[X];The conditional g-expectation of X with respect to F_t is defined byE^g[X|F_t] (?)Y_t = E_t,T^g[X],where (Y. Z) is the solution of (g.T.X).Theorem 3.5.5. Let g satisfy the conditions in Definition 3.5.2, and E^g[·|F_t ] be defined as before. Then there exists a sequence of standard g-expectations E^g [·] such that for each X∈L²(Ω, F_T. P) and t∈[0. T],the above convergence is uniform with respect to X.Theorem 3.5.4. The g-expectation E⁹[·|F_t] defined by Definition 3.5.2 is an F_t consistent nonlinear expectation, i.e., for any X,Y∈L²(Ω,F_T, P). we have(i) Monotonicity: E^g[X|F_t]≥E^g[Y|F_t], P-a.s., if X≥Y, P-a.s.;(ii) Constant-preserving: E^g[X|F_t] = X, P-a.s., if X∈L²(Ω,F_t,P);(iii) Consistency: E^g[E^g[X|F_t]|F_s] = E^g[X|F_s], P-a.s., for s≤t≤T;(v) "0-1 Law": for each t∈[0,T], E^g[1_AX|F_t] = 1_AE^g[X|F_t], P-a.s., (?)A∈F_t.In the proofs of the above two theorems, we use continuous dependence theorem of Chapter 2. For this class of g-expectation, we also have the following interesting result. Theorem 3.5.12. Let g satisfy the conditions in Definition 3.5.2. Then the following statements are equivalent.(i) g is independent of y and is Lipschitz continuous in z. and is superhomogencous. i.e.. for anyλ∈R.(?)(t,z). g(t,λz)≥λ(t,z), a,s.;(ii) Jensen's inequality for the new g-expectation holds, i.e.. for any convex function f and X∈L²(Ω,F_T, P) such that f(X)∈L²(Ω,F_T, P), one hasE^g[f(X)|F_t]≥f(E^g[X|F_t]).In other words, if g is uniformly continuous, but not Lipschitz continuous, Jensen's inequality for the new g-expectation in the above sense can not hold any more. This is an interesting phenomenon, which drives us to reconsider Jensen's inequality problem. We need new ideas, we need a new point of view. Frankly, it is also one of motivations to study g-convexity in Chapter 5.We also haveProposition 3.5.9. Let g satisfy the conditions in Definition 3.5.2, and let E^g[·] be defined as in Definition 3.5.2. Then E^g[·] as a nonlinear operator defined on L²(Ω, F_T, P) can be extended continuously to L¹(Ω,F_T, P) whereThis result is a generalization of that in Chen [29]. Although E^g[·] can be extended continuously to L¹(Ω,F_T, P), we can not prove by now that this extension is whether unique with respect to g or not. The main reason is that we have no strict comparison result.As for the definition of new g-martingale(g-supermartingale,g-submartingale) and the associated g-supermartingale decomposition theorem, we all follow the framework of Peng(1999), in particular the proof of g-supermartingale decomposition theorem totally depends on Peng's monotonic limit theorem for Ito process, we do not list them here. the reader can refer to Chapter 3 for details.(â…¢) In Chapter 4, we establish get a probabilistic interpretation of a class of semi-linear parabolic partial differential equations.The well-known Feynman-Kac's formula (see Kac [84. 1951]) expresses the solution of a large class of linear second order partial differential equations of elliptic and parabolic type as the expectation of a functional of a diffusion process. As mentioned in Chapter 1. a probabilistic approach to sentilinear PDEs has been introduced in 1991 by Peng [127. 1991]. which is based on the notion of backward stochastic differential equation. Since 1991. there were many works appearing in this direction, e.g.. see Peng [129. 132], Pardoux-Peng [120], Pardoux [115. 116], Barles-Buckdahn-Pardoux [9], Pardoux-Tang [123], Pardoux-Pradeilles-Rao [122], Barles Lesigne[10]. But all such works required g to be Lipschitz continuous with respect to z. But in classical viscosity solution theory, g is only required to be uniformly continuous inâ–½u (in the BSDE language, z). In other words, there exists a gap yet between PDEs and corresponding probabilistic interpretations-BSDEs. In this chapter we solve this problem partially (Theorem 4.3.2).More importantly, Theorem 4.3.2 firstly gives a probabilistic interpretation for a class of the celebrated generalized deterministic Kardar-Parisi-Zhang (KPZ) equations or Krug-Spohn equations, which have the form as followswhere q are positive constants. Equation (13) with q = 2 was introduced by Kardar. Parisi and Zhang [87, 1986] in connection with the study of growing surfaces and it has since been referred to as the deterministic KPZ equation. For q≠2 it is also called the generalized deterministic KPZ equation or Krug-Spohn equation [95. Krug and Spohn. 1988]. In [96], Krug and Spohn provided a nice and detailed introduction to the physics behind of equation (13); see also Amar-Family [4]. Clearly, Peng in [127] obtained a probabilistic interpretation for the case q = 1, and Kobylanski's work [90] was for the case 1 < q≤2. Theorem 4.3.2 is for the case 0 < q < 1.Here is our main result in this chapter.Let. the continuous functions b : [0, T]×R^d→R^d andσ: [0, T]×R^d→R^d×d satisfy the following conditions:|b(t,x) - b(t,x')| + |σ(t,x) -σ(t,x')|≤L|x- x'| ,(14)where L > 0 is a constant. And let {X_s^t,x: t≤s≤T) be the strong solution of the following SDEWe now consider the following BSDE where H : R^d→R and g : [0,T]×R^d×R×R^d→R satisfy the following conditions:(i) g and H are continuous functions, and there exists constants K,p > 2. such that for any t.x,y,z. we have|H(x)|≤K(1 + |x|^p). |g(t.x,y. z)|≤K(1 + |x|^p + |y| + |z|):(ii) g is uniformly continuous in z. Lipschitz continuous in y uniformly with respectto t,x;(iii) For each R > 0. there exist a constantα> 0 not depending on R and a nondecreasing, positive functionη_R(·) with (?)+η_R(h) = 0, such that for|x|. |x'|, |y|≤R, t∈[0,T], z∈R^d, we have|g(t.x,y, z) - g(t,x', y, z)|≤η_R(|x - x'| (α+α|z|)).Theorem 4.3.2. Let b,σ,g and H satisfy the above conditions. Then u(1,x) = Y_t^t,x is the unique continuous viscosity solution with polynomial growth of the following PDEwhereThis result generalizes the results in Peng [127] and Pardoux-Peng [120] under one-dimensional situation.(â…£) In Chapter 5, we introduce the notion of g-convexity, and establish a necessary and sufficient condition for a function being g-convex; we discover a deep relation between g-convexity and backward stochastic viability property.Jensen's inequality plays an important role in probability theory. As we know, g-expectation is a class of nonlinear expectations. A very interesting problem is whether. for a g-expectation, the following generalized Jensen's inequality holds true:h(E^g[X])≤E^g[h(X)],for each X such that both E^g[X] and E^g[h(X)] are meaningful.This problem is initialed in Briaud, Coquet. Hu, Mémin and Peng [13, 2000] in which a counterexample was given to show that the above generalized Jensen's inequality fails for a very simple convex functions h, and they gave a suffieient condition for a special situation. Chen. Kulperger and Jiang in [30. 2003] obtained a very interesting result: if g does not depend on y, the above generalized Jensen's inequality holds true for each convex function h if and only if g is a super-homogeneous function, i.e.. g(t,λz)≥λg(t,z). dP×dt- a.s. forλ∈R and z∈R^d. This result was improved by Hu [68, 2005] showing that, in fact, g must be independent of y.In this chapter we study this problem from a different point, of view: for a given function g (of course g-expectation is also given), to give an explicit characterization to h satisfying the above generalized Jensen's inequality. More interestingly, we also discover a deep relation between g-convexity and backward stochastic viability property (BSVP, in short) introduced and systematically studied by Buckdahn, Quincampoix and Rascanu in [20].Through out this chapter the function g will satisfy the following conditions.The g-expectation originally introduced in [133] corresponds the case where g satisfies that for each (t, y), g(t, y, 0)≡0. For the simplicity, we call them all g-expectation here whenever g satisfies the above conditions.We first introduce briefly the notion of BSVP. A set K is said to enjoy g-BSVP for (11) if the terminal valueζ∈K P-a.s.. then for each t∈[0, T]. the solution (Y_t)_t∈[0,T] of (1) also belongs to K P-a.s. Here the equation (1) can be multidimensional, i.e., n > 1.Here are our main results in this chapter. To begin with we give the notion of g-convexity.Definition 5.2.1. For a given g-expectation E^g[·], a function h : R→R is said to be g-convex (resp. g-concave) if for each X∈L²(Ω, F_T, P) such that h(X)∈L²(Ω, F_T, P), one hash(E_t,T^g[X])≤E_t,T^g[h(X)], (resp. h(E_t,T^g[X])≤E_t,T^g[h(X)]) P-a.s., t∈[0,T].h is called g-affine if it is both g-convex and g-concave.Theorem 5.2.2. Let g satisfy the above conditions and let h∈C²(R). Then the following two statements are equivalent:(i) h is g-convex (resp. g-concave); (ii) For each y∈R, z∈R^d.1/2h"(y)|z|² + g(t, h(y), h'(y)z) - h'(y)g(t. y. z)≥0.dP×dt a.s. (reap. 1/2h"(y)|z|² + g(t,h(y), h'(y)z) - h'(y)g(t,y,z)≤0 dP×dt-a.s.);Theorem 5.3.2. Let g satisfy the above conditions and h∈C(R) be of polynomial growth. Moreover assume that g is independent ofω. and is continuous with respect to t. Then the the following conditions are equivalent:(i) h is g-convex (resp. g-concave)(ii) for each (t, z)∈[0.T]×R^d, h is a viscosity subsolution (resp. supersolution) of1/2h"(y)|z|² + g(t, h(y), h'(y)z) - h'(y)g(t, y, z) = 0.Corollary 5.3.5. Assume that the same conditions as in the above theorem hold. Then the following condition is equivalent:(i) h is g-convex,(ii) h is convex and for each y such that, h"(y) exists,1/2h"(y)|z|² + g(t,h(y),h'(y)z) - h'(y)g(t.y, z)≥0.Theorem 5.4.7. Let g satisfy the conditions in Theorem 5.2.2. Then the following two statements are equivalent:(i) A continuous function h is g-affine;(ii) h has the form: h(y) = ay + b for some (a, b)∈Π_g^a whereΠ_g^a := {(a, b); g(t, ay + b, az) - ag(t, y. z), dP×dt-a.s.}Theorem 5.2.12. If (Y_t)_t∈[0,T] is a g-martingale, and h is a g-convex function (resp. g-concave function, g-affine function), then (h(Y_t))_t∈[0,T] is a g-suhmartingale (resp. g-supermartingale. g-martingale) provided h(Y_t)∈L²(F_t), t∈[0, T].Moreover its inverse also holds, namely,Theorem 5.2.13. If for each g-martingale (Y_t)_t∈[0,T], (h(Y_t))_t∈[0,T] is a g-submartingale (resp. g-submartingale, g-martingale), then h is a g-convex (resp. g-concave ,g-affine) function.Theorem 5.4.4. Let g satisfy the conditions in Theorem 5.2.2 and h : R→R be a continuous function. Moreover assume that Then the following statements are equivalent:(i). h is g-eonvex;(ii). epi(h) enjoys g-BSVP whereepi(h) = {(x1,x2)∈R²: h(x₁)≤x₂}.Corollary 5.4.6. A continuous g-convex function is convex in the usual sense.In this chapter, we also get many results to describe the relationship between generalized Jensen's inequality and BSVP, and prove some new necessary and sufficient conditions for BSVP. Here we do not list them, the readers can refer to Chapter 5 for details.It is worth noting that for the new g-expectation we define in Chapter 3, all above results except Theorem 5.4.4 hold true.(â…¤) In Chapter 6, we prove a generalized existence theorem for BSDEs without explicit growth constraints.Suppose that an one-dimensional BSDE (g,T,ζ) satisfies the following conditions:(i) For function g :Ω×[0,T]×R×R^d→R, g(t,y,·) is uniformly continuousuniformly respect to (w,t,y). g(t,·,z) is left-continuous, and satisfies left-Lipschitzcondition, i.e., there exists a constant A≥0, such that for each y₁≥y₂∈R. (t. z)∈[0, T]×R^d. we haveg(t,y₁,z) -g(t,y₂,z)≥-A(y₁ - y₂);(ii) there exists two functions g_i :Ω×[0,T]×R×R^d→R (i = 1,2) such that for each (t,y, z)∈[0,T]×R×R^d,g₁(t, y, z)≤g(t, y, z)≤g₂(t, y,z) P- a.s.;(iii) for givenζ∈L²(Ω.F_T. P). (g₁,T,ζ) and (g₂,T,ζ) have at least one solution, which are denoted respectively by ((?),(?)) and ((?),(?)). Moreover we assume that for t∈[0.T], (?)≤(?), P-a.s., andg₁((?),(?))∈L_F²(0,T), and g₂((?),(?))∈L_F²(0,T).Theorem 6.2.2. Let (g,T,ζ) satisfy the above conditions. Then (y,T,ζ) has at least one solution.