| This doctoral thesis consists of three main topics:The first one is that of fully cou-pled forward-backward stochastic differential equations with subdifferential operators in both the forward and the backward equations and their associated parabolic partial vari-ational inequalities. The second topic concerns the investigation of backward stochastic differential equations without as well as with subdifferential operator, both driven by a fractional Brownian motion with Hurst parameter H>1/2. Finally, the third topic fo-cuses on a deterministic characterization of viability for stochastic differential equations driven by a fractional Brownian motion.The three research topics mentioned above have in common to study stochastic differential equations with state constraints. The problem of stochastic equations under state constraints can be understood in different ways. One way consists in investigating conditions under which the solution of this equation lives in a given closed set K （?） Rd, whatever its initial point in K is-in this case one speaks about viability. Another approach is to add to the stochastic equation an additional （minimal） force （a process of finite variation）, which guarantees that the solution of the equation, although not satisfying the viability condition, is forced to stay in K-in this case one speaks of a reflected stochastic equation, and if. generalizing the reflection problem, the state constraint K is replaced by that given by a subdifferential operator of a convex proper function φ:Rd'R, then the equation with the additional force is called stochastic variational inequality.While we study forward-backward stochastic variational inequalities in a Brownian framework, our stochastic equations in the viability problem and backward stochastic differential equations with subdifferential operator are governed by a fractional Brownian motion.Let us be a bit more precise and explain the organization of the manuscript.In the part of Introduction, we give an overview of our research topics.In Chapter2we study fully coupled forward-backward stochastic differential equa-tions with subdifferential operators in both the forward and the backward equations. Moreover, a Feymann-Kac type formula will be established through these forward-backward stochastic differential equations, in order to give a stochastic representation for the viscosity solution of a new type of quasilinear partial differential equation with two subdifferential operators, one acting over the state domain and the other over the co-domain. This chapter is mainly based on the work:NIE, T., A stochastic approach to a new type of parabolic variational inequalities. arXiv:1203.4840v2. Submitted for publication.In Chapter3of this dissertation we investigate backward stochastic differential equations with subdifferential operator driven by a fractional Brownian motion with Hurst parameter H>1/2. In order to do this, we first study the wellposedness of fractional backward stochastic differential equations in a rigorous manner. Our approach is based on the ideas of [72]. In the second part of the chapter this approach is extended to fractional backward stochastic differential equations with subdifferential operator. This chapter is mainly based on the paper:MATICIUC, L., NIE, T., Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities. arXiv:1102.3014v3. Submitted for publication.In Chapter4we are concerned with the viability for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H>1/2. In fact, by using direct and inverse images for fractional stochastic tangent sets, we get necessary and sufficient deterministic conditions which keep the solution of a given fractional stochastic differential equation in a given set K （?） Rd. Moreover, we obtain also a comparison theorem for fractional stochastic differential equations. This part of the thesis is mainly based on the paper:NIE, T., RASCANU, A., Deterministic characterization of viability for stochastic differential equation driven by fractional Brownian motion. Accepted for publication by ESAIM:Control, Optimisation and Calculus of Variations, doi:10.1051/cocv/2011188.Now we give an overview over the main results of this dissertation.1Forward-backward stochastic variational inequalities and related partial variational inequalitiesThe results presented in this subsection come from the manuscript Nie [110] submitted for publication.Our objective is to study the viscosity solution of following quasilinear PVI:（15） where the operator （?） is defined by for v∈C1,2（[0, T]×Rn）. The operator （?）ψ （resp.,（?）φ） is the subdifferential of function ψ （resp.,φ） which satisfies:（H’1） The function ψ:Rn'(-∞,+∞] is convex l.s.c. with0∈Int（Domψ） and ψ（z）≥ψ（0）=0,z∈Rn.（H’2） The function φ:R'(-∞,+∞] is convex l.s.c. such that φ（y）≥φ（0）=0, y∈R.We define the viscosity solution for such kind of partial differential equations and prove the uniqueness of the viscosity solution when σ does not depend on y. To prove the existence of a viscosity solution, a stochastic representation formula of Feymann-Kac type will be developed. For this end, we investigate a fully coupled forward-backward stochastic variational inequality.Motivated by [120] and [150], we give the definition of the viscosity solution of PVI （15） in the language of sub-and super-jets as well as by using smooth test function （see Definitions2.1and2.3in Chapter2）. Definition0.1. Let u∈C（[0,T] x Domψ） with u（T,x）=g（x）,∈Domψ. The function u is called a viscosity subsolution （resp., supersolution） of PVI （15）, if for all （t,x）∈[0,T]×Domψ, u（t,x）∈Domφ, and for any （p,q,X）∈P2,+u（t,x）*（resp.,（p,q,X）∈P2,-u（t,x）） we have (resp. Here φ’y（resp., φ’+（y）） denotes the left （resp., right） derivative of φ at point y, and and （?）ψ*（x, q）:=-（?）ψ*（x,-q）.The function u is called a viscosity solution of PVI （15） if it is both a viscosity sub-and super-solution.We mention that we give also an equivalent definition of the viscosity solution of PVI （15） by using smooth test function （see Definition2.3in Chapter2）.Now we can give the following uniqueness result:Theorem0.1. We assume that Domψ （?） Rn is locally compact, and that the functions b, a, f and g are all jointly continuous. Moreover, we suppose that b and f are uniformly Lipschitz continuous in （x,y.z）, and σ（t,x,y） does not depend on y and is uniformly Lipschitz continuous in x. Then, under the assumptions （H’1） and （H’2）, PVI （15） has at most one viscosity solution in the class of continuous functions which are Lipschitz continuous in x uniformly w.r.t. t. To prove this theorem, it is sufficient to show that if u is a subsolution and v is a supersolution, and both satisfy the assumptions of the theorem, then u≤v. Since u and v are continuous, we only need to show that u≤v for all （t,x）∈（0,T）×Int（Domψ）. For this we define where Bd（Domψ） denotes the boundary of Domψ, and we choose r0>0s.t.（Domψ）r0≠（?）. Then it suffices to show that for every0<r≤r0, we have u≤v on （0,T） x（Domψ）r.For this end, we extend and adapt the approaches of Barles, Buckdahn and Pardoux [17] and Cvitanic and Ma [48] and we give the following two lemmas.Lemma0.1. Under the assumptions of Theorem0.1, we suppose u （resp., v） is a con-tinuous sub （resp., super）-solution of （15） which is Lipschitz continuous in x, uniformly w.r.t. t. Then, for all0<r≤r0, the function w:=u-v is a viscosity subsolution of the following equation: where Here K>0is a constant depending only on the Lipschitz constants of the functions b,f, u and v, and we recall thatLemma0.2. For any A>0, there exists C>0such that the function satisfies where t1=（T-A/C）+. Based on the above both lemmas, we prove Theorem0.1by adapting arguments of [17].The remaining part of Chapter2is devoted to study a general kind of FBSVI in order to give a probabilistic interpretation for the viscosity solution of PVI （15）.Let （Ω, F, P） be a complete probability space endowed with an Rd-valued standard Brownian motion{Bt}t≥0. We denote by{Ft}t≥0the filtration generated by the Brown-ian motion B and augmented by the class of P-null sets of F. We consider the following FBSVI: where the adapted processes X, Y, Z take their values in Rn, Rm and Rm×d, respectively, and the functions ψ,φ,b,σ,f and g satisfy standard assumptions （see （H1）-（H5）,（H’4） and （H’5） in subsection4.1of Chapter2）. Moreover, we need compatibility hypotheses which were introduced by Cvitanic and Ma [48]（see （C1）-（C3））.Our first objective is to prove the wellposedness of FBSVI （18）（For the definition of the solution of （18）, see Definition4.1of Chapter2）. The uniqueness is a consequence of the following proposition:Proposition0.1. Under the assumptions （H1）-（H5） and the compatibility hypotheses, we have where and （Xt,x,Yt,x,Zt,x,Vt,x,Ut,x）（resp.,（Xt,x,Yt,x,Zt,x,Vt,x,Ut,x）） denotes a solution of the FBSVI with initial time t and coefficient （x, b. σ, f, g）（resp.,（x,b, σ, f, g））, and C is a constant independent of （t,x,x）. To prove the existence, we study the following penalized FBSDE by using the Yosida approximation for the l.s.c. functions ψ and φ:（19） Here▽φε is the gradient of the Yosida approximation of convex l.s.c. function φ:With the help of the Ito formula and the relations <x1-x2,▽φε（x1）-▽φε（x2）>≥0and <y1-y2,▽φε（y1）-▽φε（y2）)≥0, we establish a priori estimates （see Lemma6.1and Proposition6.2of Chapter2）. Then adapting an argument of [48], we construct a contraction mapping and prove the following theorem:Theorem0.2. Under the assumptions （H1）-（H5） and the compatibility hypotheses, the penalized FBSDE （19） has a unique adapted solution （Xε,Yε, Zε）.In a next step, in order to find a solution of FBSVI （18）, we prove that （Xε, Yε, Zε）, ε>0, is a Cauchy sequence. Indeed, by using a standard calculus as in [48], we haveProposition0.2. Under the assumptions （H1）-（H5） and the compatibility hypotheses, we have that, for all ε1. ε2>0,（20） where C is a constant which does not depend on ε1nor on ε2-To show that （Xε,.Yε,Zε）,ε>0, is a Cauchy sequence, we should estimate the right term of （20）.Motivated by the approach of Pardoux and Rascanu [122], we establish the Lp-estimate of（Xε, Yε, Zε） in order to estimate E∫0T|▽ψε1（Xsε1）‖▽ψε2（Xsε2）|ds. However, the method developed by Cvitanic and Ma [48] provides only L2-estimates. We adopt here the induction method introduced by Delarue [51]. Let us mention that this method requires that a is independent of z （see （H’5）） and some LP integrability conditions for coefficients of FBSVI （18）（see （H’4））.Based on the induction method of Delarue [51], we establish first the Lp-estimates on a small time interval and then we extend them to the whole interval. Indeed, we obtain the following statement:Proposition0.3. We suppose （H1）-（H5）,（H’4） and （H’5） as well as the compatibility hypotheses. Then, for all1≤p≤3+ρ0/2（for the definition of ρ0, see （H’4））, there exists a constant C independent of ε and x, such that where b0（s）:=b（·,s,0,0,0）, σ0（s）:=σ（·,s,0,0,0）,f0（s）:=f（·,s,0,0,0） and g0:=9（·,0）.Under our assumptions and with the help of above Lp-estimates as well as the ideas of Pardoux and Rascanu [120],[122], we get the following two propositions:Proposition0.4. For all0≤ρ≤1+ρ0/4Λ1, there exists a constant C, such that for all ε>0Proposition0.5. Letting1-ρ0/4+4ρ0∨0<ρ≤1+ρ0/4∨1, we have Here C is a constant which depends neither on ε1nor on ε2.By using the above both propositions and Proposition0.1we obtain one of our main results: Theorem0.3. There exists a unique solution （X, Y, Z, V, U） of FBSVI （18）.Finally, we give a probabilistic interpretation for the solution of PVI （15）. For this end, we consider the following FBSVI with （t,x）∈[0, T]×Domψ: where we assume that b, σ, f and g are deterministic continuous functions and the di-mension m=1（see （H7））.Theorem0.3remains true on the interval [t,T], and we denote the unique solution of equation （21） by（Xst,x,Yst,x,Zst,x,Vst,x,Ust,x）. We put Then under （H7）, the function u（t, x） is deterministic. Moreover, by using some suitable argument we can show thatProposition0.6. For all （t,x）∈[0, T]×Domψ, we have u（t,x）∈Domφ and v∈C（[0,T]×Domψ）.With the help of our penalized FBSDEs, by generalizing the approach of Pardoux and Tang [123] and of Pardoux and Rascanu [120], we obtainTheorem0.4. Under our assumptions, the function u（t,x）=Ytt,x,（t,x）∈[0,T]×Domψ, is a viscosity solution of PVI （15）.As a consequence of Theorem0.1and Theorem0.4, we deduce our main result:Theorem0.5. PVI （15） has a unique viscosity solution in the class of continuous func-tions which are Lipschitz continuous in x uniformly w.r.t. t.2Fractional BSDEs and Fractional BSVIs The results in this subsection are based on the manuscript [96], a joint work with Lucian Maticiuc （University of Al. I. Cuza, Iasi, Romania）.Our objectives in Chapter3are twofold. The first one is to study the following backward stochastic differential equation driven by a fractional Brownian motion: where η is a stochastic process η（t）=η（0）+∫0tσ（s）δBH（s）,t∈[0,T],with continuous diffusion coefficient σ∈C（[0,T];R）,and BH is a fractional Brownian motion with Hurst parameter greater than1／2.The stochastic integral is the divergence type integral.The filtration generated by BH is denoted by F={Ft}t≥0.The factional BSDE（23）was first considered by Hu and Peng[72].In their paper they proved the wellposedness of（23）,but with the condition that there exists c0>0s.t. inft∈[0,T]σ（t）/σ（t）≥c0,for σ（t）:=∫0t（?）（t-r）σ（r）dr.Here in our manuscript we work without such a condition.Indeed,we assume only that（see（H2）in Chapter3）σ（t）≠0,for all t∈[0,T],and we use the property that there exists a suitable constant M>0,s.t.1/Mt2H-1≤σ（t）/σ（t）≤Mt2H-1,t∈[0,T]（see Remark3.1of Chapter3）.With the help of this prop erty we develop a rigorous approach by using the ideas of[72].Our approach is based on Malliavin calculus.We denote the Malliavin derivative by DH and define DtHF=∫0T（?）（t-u）DvH Fdv.Let us recall the definition of the divergence type integral.Definition0.2.A process u∈L2（Ω,F,P;H）belongs to the domain Dom（δ）,if there exists δ（u）∈L2（Ω,F,P）,such thatE（Fδ（u））=E（<DH’F,u）T), for all smooth cylindric functions F∈PT,（for H,（·,·）T and PT,see section2of Chapter3）.If u∈Dom（δ）,δ（u）is unique, and we define the diveryence type integral of u∈Dom（δ）w.r.t. the fBm BH6y putting∫0TusδBH（s）:=δ（u）.Let us recall a result about a sufficient eondition for the existence of the divergence type integral（see Proposition6.25,[69]）.Theorem0.6.We denote by LH1,2the space of all stochastic processes u:（Ω,F,P）'H s.t. If u∈LH1,2, then the Ito-Skorohod type stochastic integral∫0T u（s）dBH（s）defined by Proposition6.11[69] exists and coincides with the divergence type integral（see Theorem6.23[69].Moreover,E[∫0Tu（s）dBH（s）]=0and Let us come back now to our fractional BSDE （23）. The main idea to prove the wellposedness of （23） consists in constructing a contraction mapping on a suitable space which is completed under a suitable norm. For this end, in the spirit of [72], we introduce the following space as well as its completion VTα under the following α-norm （α≥1/2） We prove the following properties for our spaces VT and VTα:Lemma0.3. We have VT（?）LH1,2C Dom（δ）.Lemma0.4. If Y∈VTα and ψ is a Lipschitz function, then also ψ（Y）∈VTα.Lemma0.3says that if u∈VT, then∫0T usδBH（s） is well defined. Moreover, moti-vated by Theorem8[5], we prove a general Ito formula of the following form:Theorem0.7. Let ψ be a function of class C1,2（[0, T]×R）. Assume that u∈VT and f∈Cpol0,1（[0,T]×R）. Let Then, for all t∈[0,T], the following formula holds true:We emphasize that by a different approach, we can prove that the above Theorem holds for ψ（s,x）=|x|2（see Theorem3.4in Chapter3）.Besides the above general Ito formula, the notion of quasi-conditional expectation E[·|Ft] is an important tool to solve our backward equation （for the definition of this notion, see subsection3.3of Chapter3）. The investigation of the property of the condi-tional expectation leads to the following both lemmas: Lemma0.5. Let F=f（η{T））, where f: R'R is a continuous function of polynomial growth. Then F∈L2（Ω,F,P） andLemma0.6. Assume that f∈Cpol0,1（[0,T]×R） and put fs=f（s,η（s））, s∈[0,T]. ThenAfter the above preparation we can come back now to our fractional BSDE （23）. First, we give the definition of the solution:Definition0.3. A pair （Y,） is called a solution of BSDE （23）, if the following condi-tions are satisfied:Let us now consider the following equation: where χ,ψ∈Cpol1,3（[0,T]×R） wit （?）x/（?）t,（?）ψ/（?）t∈Cpol0,1[0,]×R).Based on the ideas of Proposition4.5[72], we show the following result in a rigorous manner:Proposition0.7. Under our assumptions （see （H1）-（H4） of Chapter3）, BSDE （24） has a unique solution （Y,Z）∈VT×VT. This solution has the form Y（t）=u（t,η（t）） and Z（t）=v（t,η）（t）) with v（t,x）=σ（t）（?）/（?）xu（t,x）, where u,v∈Cpol1,3（[0,T]×R） s.t.（?）u/（?）t,（?）v/（?）t∈Cpol0,1l（[0,T]×R）Let us consider the mapping Γ:VT×VT'VT×VT given by （U,V）'Γ（U,V）=（Y,Z）, where （Y.Z） is the unique solution in VT×VT for the BSDE First, weremark （see Proposition0.7） that F is well defined. Moreover, Γ is a contraction w.r.t. the norm‖（u,v）‖1/2,H:=‖u‖1/2+‖v‖H, for （u,v）∈VT1/2×VTH. Indeed, the contraction property of Γ can be derived easily from the following statement: Proposition0.8. For （U, V）∈VT×VT,let （Y, Z）∈VT×VT be the unique solution of the BSDE Then, for all β>0, there exists C（β）∈R （depending also on L and T） s.t. Moreover, C（β） can be chosen such that lim C（β）=0.By using that Γ is a contraction as well as the definition of the divergence type integral, we construct a couple （Y, Z） for which we prove that it is a solution of the fractional BSDE （23）.On the other hand, we introduce a space Sf （see subsection4.2of Chapter3） and we obtain the wellposedness of （23） with respect to Sf:Theorem0.8. Under suitable assumptions, fractional BSDE （23） has a unique solution （Y,Z）∈Sf.The second objective of Chapter3is to investigate the fractional backward stochas-tic variational inequality: where （?）φ is the subdifferential operator of the convex function φ, for which we suppose:（H5） φ is a l.s.c. function with φ（x）≥φ（0）=0, for all×R and E|φ（ζ）|<∞. The following theorem is our main result in this part:Theorem0.9. There exists a solution of BSVI （25） which means that there exists a triple （Y, Z, U） satisfying: To prove this theorem, motivated by [120], we first consider the penalized BSDE: Here Δφε is the gradient of the Yosida approximation （for convex l.s.c. function φ）: The first step is to show that BSDE （26） has a solution. We mention that Δφε does not satisfy the assumption （H3）, so we mollify it in order to apply Theorem0.8and then to solve BSDE （26）. Indeed, we obtain the following theorem:Theorem0.10. For all ε>0, BSDE （26） has a solution（Yε, Zε）∈VT1/2×VTH s.t. Moreover, by using （27） we obtain the following estimate for （Yε, Zε）:Proposition0.9. There exists a constant C independent of ε, s.t. for all t∈[0,T],The second step is to use （Yε,Zε）,ε>0, to construct a solution for fractional BSVI （25）. The main idea is to show that （Yε, Zε）, ε>0, is a Cauchy sequence in the following sense:Proposition0.10. There exists a constant C s.t. for all ε,δ>0, In order to show this, we need the following proposition:Proposition0.11. There exists a constant C s.t. for all t∈[0,T], where Let us mention that the following fractional stochastic subdifferential inequality is very useful （to prove Proposition0.11）:Lemma0.7. Let ψ:R'R+be a convex C1function which derivative Δψ is a Lipschitz function. Then, for all t∈（0, T], P-a.s.Finally, from Propositions0.10,0.11and Lemma0.4, we construct a triple （Y, Z, U） which is a solution of fractional BSVI （25）.3Characterization of viability for fractional SDEsThe results of Chapter4are based on the paper Nie and Rascanu [111], accepted for publication by ESAIM:Control, Optimisation and Calculus of Variations.We consider the following stochastic differential equation on Rd where BH={BtH,t≥0} is a k-dimensional fBm with Hurst parameter1/2<H<1. The stochastic integral is the pathwise Riemann-Stieltjes one. The functions b [0,T]×Rd'Rd, σ:[0,T]×Rd'Rd×k are continuous and satisfy the assumptions （H1） and （H2） with β>1-H and δ>1/H-1（see subsection2.1of Chapter4）. From Nualart and Rascanu [115], we know that fractional SDE （28） has a unique solution Xt,ζ∈L0（Ω,F,P; Wα,∞（t, T;Rd））, for all β∈（1-H,α0）（for the notations, see subsection2.1of Chapter4）.In our framework we suppose our standard assumptions:（H1） and （H2） are satisfied with1/2<H<1,1-H<β and δ>1-H/H. Moreover, max{1-H,1-μ}<α<α0.Our work is motivated by Ciotir and Rascanu [42] and our objective is to find easy conditions to characterize the viability for fractional SDE （28）. Moreover, we obtain a comparison theorem for fractional SDEs. The main idea of our work is to investigate stochastic fractional tangent sets to direct images and inverse images in the spirit of Aubin and Da Prato [11].We recall the following well-known definition for viability: Definition0.4. Let K={K（t）: t∈[0,T]} be a family of subsets of Rd.We say that K is viable for equation （28） if, for every t G [0, T] and every starting point x∈K（t）, there exists at least one of its solutions {Xst,x:s∈[t,T]} which satisfiesCiotir and Rascanu [42] studied the viability problem for fractional SDE （28） by introducing the following notions of （1-α）-fractional BH-contingent and tangent set:Definition0.5. Lett∈[0,7], x∈K（t） and1/2<1-α<H. The （1-α）-fractional BH-contingent set to K （t） in x is the set of the pairs （u,v）∈Rd×Rd×k, such that there exist a random variable h=ht,x>0and a stochastic process Q=Qt,x: Ω×[t,t+h]'Rd, and for any R>0with|x|≤R, there exist two random variables Hr,Hr>0and a constant γ=γR（α,β）∈（0,1） such that for all s,τ∈[t,t+h],P-a.s. andDefinition0.6. Let t∈[0,T], x∈K（t） and1/2<1-α<H. The （1-α）-fractional DH-tangent set to K（t） in x, denoted by SK（t）（t,x）, is the set of the pairs （u,v）∈Rd×Rd×k, such that there exists a random variable h=ht.x>0and two stochastic processes and for every R>0with|x|≤R, there exist two random variables DR,DR>0, such that for all s,τ∈[t,t+h],P-a.s. andCiotir and Rascanu [42] gave a criterion for the viability for fractional SDE （28） by using the above notions. Based on their result, if the constraint set K（t） is independent of t, then we obtain the following corollary:Corollary0.1. If K（?） Rd is independent of t, the following assertions are equivalent: （j） K is viable with respect to the fractional SDE （28）.（jj） For all t[0,T] and x∈（?）K,（b（t,x）,σ（t,x）） is （1-α）-fractional BH-contingent to K in x.（jjj） For all t∈[0,T] and x∈（?）K,（b（t,x）,σ（t,x）） is （1-α）-fractional BH-tangent to K in xAfter the above preparation, in the first part of Chapter4, we study stochastic tangent set to direct images and inverse images in the framework of fractional Brownian motion. Indeed, by using the ideas of Aubin and Da Prato [11], we prove the following theorem （concerning stochastic tangent set to direct images）:Theorem0.11. Let K（t）=K（t）（?）Rd,t∈[0,T], and φ be a C2map from Rd to Rm with a bounded second derivative, thenFor stochastic tangent set to inverse images, we need more strict conditions on φ. We introduce a suitable space H （see Section3of Chapter4） and we prove the following theorem:Theorem0.12. If φ∈H, then for all x∈Rd with aφ<|x|<bφ （for aφ and bφ, see the definition of the space H）,（b（t,x）,σ（t,x））∈Sφ-1（φ（K（t）））（t,x）（?）（φ’（x）b（t,x）,φ’（x）σ（t,x）））∈Sφ（K（t））（t,φ（x））In the second part of Chapter4, we suppose that BH is one-dimensional, our ob-jective is to get checkable deterministic sufficient and necessary conditions for viability for fractional SDE （28） when K takes particular form. For this end, we first prove the following lemma by applying Theorem0.12with φ（x）=|x|2:Lemma0.8. Let K={x∈Rd; r≤|x|≤R}. Then for all x, such that|x|=R,（b（t,x）,σ（t,x））∈SK{t,x) if and only if <x,b（t,x）>≤0and... |
PDF Full Text Request