Well-posedness Of Two Types Of Reflected BSDEs With Non-lipschitz Coefficients

In 1990,Pardoux和 Peng[91]first introduced the nonlinear backward stochastic differential equations(BSDEs for short):and established the existence and uniqueness of the solution when the generator f is Lipschitz continuous and ternminal condition ξ is square integrable.From Pardoux and Peng’s seminal work,many mathematican put effort in studying BSDE.Since then,the theory of BSDEs has rapidly grown and has been applied in many areas,such as mathematical finance,theoretical oconomics.stochastic control,stochastic differential games,partial differential equations[84,85,88].Especially.Peng[93]proposed nonlinear Feynman-Kac formula,connecting the decoupled forward and backward stochastic differential equations(FBSDEs)with a class of qnasi-linear parabolic partial differential equations(PDEs),and the solution’s probability representation is given.In order to expand BSDE’s application.and also in order to further study the BSDE theory.Scholars tried to chang the equation form of BSDEs,various deformations of BSDE have been derived,such as reflceted BSDE,anticipated BSDE,BSDE with jump etc.Another important research direction is the study of relaxing BSDE’s coefficients restrictions,as well as the properties of BSDE’s solutions.When studying the extended BSDEs or BSDE with relaxed assumptions,the wellposcdncss(the existence and uniqueness)of the new equation is an evitable problem.Iii practical life,the Lipschitz continuous condition or the square integrability condition may be notsatisfied.And also during theoretical research.with the deepening of the research.more and more demands require the Lipschitz condition of the generator ∫ to be relaxed.In addition,in terms of terminal conditions,there is also a certain p(p>1)integrable,but not square-integrable.In order to deal with these cases,besides the L2 solution,we also need to study the Lp solution sometimes.The mean-constrained BSDE has been widely used due to its complex form,which has attracted more and more interest of scientists.The most widely known is the mean-field BSDE.In 2021,motivated by applications in pricing life insurance contracts with surrender options,Djehiche et al.[41]proposed mean-field reflected BSDE in a quite general form.In 2018,Briand et al.[17]proposed mean reflected BSDE when studying the super-hedging problem under running risk management constraint.In this paper,two kinds of mean-constrained reflected BSDEs with non-Lipschitz coefficients are studied.For the mean-field reflected BSDE,we directly get its Lp solution,while for the mean reflected BSDE,we got L2 solution first and Lp solutions later.The stability of the solutions of the corresponding BSDEs are also discussed.This paper mainly studies the solvability of two kinds of reflected BSDEs.In the first chapter,we first introduce the research background and context of the problem.The research status of BSDE’s solvability problem is given,and the significance of our research is discussed.In the second chapter,we first introduce the various spaces and sets tha will be used in the following chapter.Then we give some assumptions throughout the text,and some frequently used inequalities and several important theorems in this paper.A further introduction of the reflected BSDE studied in this paper and the form of non-Lipschitz conditions used in this paper is given.In the third chapter,for the mean-field reflected BSDE,when the generator f and the reflection boundary h contain both Y and the distribution of Y,By using the Snell envelope representation of BSDE solution,directly making difference without the It? formula,so that we get a more accurate estimate.Then we prove the existence,uniqueness and stability of non-Lipschitz mean-field reflected BSDE solutions.In Chapter 4,for the mean reflected BSDE,we construct a.Picard iterative solution sequence by solving Lipschitz mean reflected BSDEs,and then directly calculate the estimates to prove that the solution sequence converges to the solution we want.In chapter 5,for the Lp solution of non-Lipschitz mean reflected BSDEs,the method we used to obtain the solution of L2 does not work.By applying extended It? formula,the It?-Tanaka formula.with the close connection of mean-reflected BSDE and BSDE,we successfully prove the convergence of the corresponding solution sequence.In the last chapter,I give a review of my study and prospects of future,study.This paper contains six chapters,let’s give an overview of the structure and the main results of my dissertation.Chapter 1 IntroductionI introduce the research background of this dissertation.Both the mean-field reflected BSDE and the mean reflected BSDE have strong financial application backgrounds,They are all inspired and constructed from financial practice.Especially the mean reflected BSDE,due to the high cost of risk-free hedging strategy,some investors willing to take part of risk in order to obtain a risk return,how to measure the price of financial assets at this time.The mean reflected BSDE well characterizes the asset pricing problems under running risk management constraints.Chapter 2 PreliminariesThe main research tools and research objects of this paper are given,various spaces and inequalities,and the "backward Bihari inequality" is particularly important in our later proof.Let ρp:[0,∞)→[0,∞)be a continuous and non-decreasing function satisfyingρ(0)=0,ρ(r)>0 when r>0,and ∫0+dr/ρ(r)=∞.Suppose u(t)is a continuous non-negative function on[0,T]such that where u0 and C are nonnegative constants.If u0=0.then u(t)=0 ?t ∈[0,T].Chapter 3 Mean-field reflected BSDEs with Non-Lipschitz coefficientsWe study well-posedness of mean-field reflected BSDEs with Non-Lipschitz coefficients in this chapter under the following assumptions:(3H1)The terminal condition ξ E Lp satisfies that ξh(T,ξ,Pξ).(3H2)There exists a.constant,λ>0 such that E[(f0T|f(t,0,δ0,0)|2dt)p/2]<∞ and for any t ∈[0.T],y,y’∈ R.v,v’ ∈ P1(R),z,z’∈ Rd.P-a,s.,where ρ:[0,∞)→[0,∞)is a continuous.non-decreasing and concave function satisfying ρ(0)=0,ρ(r)>0 for any r>0,and ∫0-dr/ρ(r)=∞.Without loss of generality assume that L>1 such that ρ(r)≤L(1+|r|).(3H3)The obstacle process h(t,y,v)belongs to Sp for any y ∈,v ∈ P1(R)and there exist two constants γ1,γ2>0 such that for any t ∈[0,T],y1,y2 ∈ R,v1,v2 ∈ P1(R)(3H4)γ1 and γ2 satisfyWe construct a Picard approximation sequence through solving the following mean-field reflected BSDEs with Lipschitz coefficients.for m=1,2,...with Y0≡0。Then via the Snell representation of reflected BSDE’s solutions and BSDE technique,we prove the boundedness and convergence of this solution sequence and furtherly prove its limit is just the solution we wanted.Theorem 0.1.Suppose assumptions(3H1)-(3H4)hold,Then the mean-field reflected BSDE(3.1)with non-Lipschitz coefficients admits a unique solution(Y,Z,K)∈ Sp ×Hp,d × Ap.Later,under some usual assumptions.we prove stability of the solution of meanfield reflected BSDEs with Lipschitz coefficients.(3H5)For each ε≥ 0,ξε and ∫ε satisfies conditions(3H1)and(3H2).(3H6)we got the following stability theorem.Theorem 0.2.Assume assumptions(3H5)-(3H6)are satisfied.Then,Chapter 4 Mean-reflected BSDEs with Non-Lipschitz coefficientsWe study well-posedness of mean-reflected BSDEs with Non-Lipschitz coefficients in this chapter under the following assumptions:(4H1)The terminal condition ξ ∈ L2(FT)satisfies E[l(T,ξ)]≥ 0.(4H2)There exists a constant λ>0 such that E[∫0T|f(t,0,0)|2dt]<∞ and P-a.s.,|f(t,y,z)-f{t,y’,z’)|2 ≤p(|y-y’|2)+λ2|z-z’|2,?t∈[0,T],y,y’∈ R,z,z’∈ Rd,where ρ:[0,∞)→[0,∞)is a continuous,non-decreasing and concave function satisfying ρ(0)=0,ρ(r)>0 for any r>0,and ∫0+dr/ρ(r)=∞.(4H3)There exists a constant L>0 such that,P-a.s.,1.(t,y)→l(t,y)is continuous.2.?t ∈[0,T],y→l(t,y)is strictly increasing;3.?t ∈[0,T],E[(t,∞)]>0,4.?t ∈[0,T],?y ∈ |l(t,y)|≤L(1+|y|).(4H4)There exists a constant μ>0 such that for each t ∈[0,T],|Lt(X)-Lt(Y)|≤μE[|X-Y|],?X,Y∈ L2(FT),where Lt:￡2(FT)→[0,∞),t ∈[0,T]is given by Lt:X→inf{x≥0:E[((t.x+X)]≥0}.We constrtuct a Picard approximation sequence through solving the following mean-reflected BSDEs with Lipschitz coefficients.Set Y0=(Et[ε])0≤t≤T,and then recursively define the stochastic process sequence{(Ytm,Ztm,Ktm)}m=1∞:Then we prove the uniform boundedness and convergence of the solution sequence by careful estimation,and prove that its limit is the solution of the mean-reflected BSDE to be solved.Theorem 0.3.Let assumptions(4H1)-(4H4)hold.Then the mean reflected BSDE(4.1)admits a unique deterministic flat solution(Y,Z,K)∈ S2 × H2,d ∈ AD2.Finally,we prove the stability of the mean-reflected BSDE solution under further assumptions.(4H5)For each ε≥ 0,ξε and fε satisfies conditions(4H1)and(4H2).(4H6)we get the following stability theorem.Theorem 0.4.Assume assumptions(4H5)-(4H6)are satisfied.Then,Chapter 5 Lp-solution of mean-reflected BSDEs with Non-Lipschitz coefficientsWe study well-posedness of mean-reflected BSDE’s Lp solutions with Non-Lipschitz coefficients in this chapter under the following assumptions:(5H1)The terminal condition ξ ∈ Lp(FT)satisfies E[l(T,ξ)]≥ 0.(5H2)There exists a constant λ>0 such that E[(∫0T |∫(t,0,0)|2dt)p/2]<∞ and P-a.s.,|f(t,y,z)-f(t,y’,z’)|p≤ρ(|y-y’p)+λp|z-z’|p.?t ∈[0.T],y,y’ ∈ R,z,z’∈ Rd,where ρ:[0,∞)→[0,∞)is a continuous.non-decreasing and concave function satisfying ρ(0)=0,for any r>0,ρ(r)>0.and ∫0-dr/ρ(r)=∞.(5H3)loss function l:Ω×[0.T]× R→R is a FT × B(R)× B(R)-measurabe mapping and there exists L>0.such that,P-a.s.1.(t,y)→l(t,y)is continuous.2.?l∈[0,T],y→l(l,y)is strictly inereasing,(5H4)There exists a constant μ>0 such that for each t∈[0,T],Lt:￡1(FT)→[0,∞),t∈[0,T]is given bySimilarly:we first set Y0≡0,then recursively define the stochastic process sequence {(Ytm,Ztm,Ktm)}m=1∞:In fact,for each m≥1,from assumption(5H2)we have:So once we get(Ytm-1,Ztm-1,Ktm-1),easily we can prove f(r,Yrm-1,0)∈ Hp,then by the existence and uniqueness theorem 2.3[61],equation(5.3)admits a solution(Ytm,Ztm,Ktm).During the proof of solution seuqence’s convergence,The following representation help us change a mean-reflected BSDE problem to a BSDE problem.Hu et al.[61]gave a new definition of Kt:where,y is the following BSDE’s solution:Compare mean-reflected BSDE(5.3)and the above BSDE(5.5).we get the relation between theire solution(Ytm,Ztm,Ktm)and(y,z):Then we get the main result of this chapter.Theorem 0.5.Let assumptions(5H1)-(5H4)hold.Then the mean reflected BSDE(5.1)admits a unique deterministic flat solution.(Y,Z,K)∈Sp×Hp,d×ADp.Finally,we prove the stability of the mean-reflected BSDE’s Lp solution under further assumptions.(5H5)For each ε≥0,ξε and fε satisfies conditions(5H1)and(5H2).(5H6)Then we get the following stability theorem.Theorem 0.6.Assume assumptions(5H5)-(5H6)are satisfied.Then,Chapter 6 Conclusions and ProspectsIn this chapter,we review the solvability of BSDE under non-Lipschitz conditions,and point out that the equations to be obtained can be approximated by constructing corresponding equation sequences,and then proved by certain BSDE estimation techniques The limit of the solution of this equation sequence is the solution of the equation to be found;not only in the non-Lipschitz conditions we give,but also in other situations such as random Lipschitz conditions,continuous coefficient conditions,etc.,this method is also applicable.After analyzing our current resea.rch results.we believe that the next research direction is to investigate the numerical simulation of mean-restricted backward stochastic differential equations:in order to put the theory into practical use as soon as possible.