In this paper, we establish the existence of the minimal Lp （p 〉 1） solution of backward stochastic differential equations （BSDEs） where the time horizon may be finite or infinite and the generators have a non-un...In this paper, we establish the existence of the minimal Lp （p 〉 1） solution of backward stochastic differential equations （BSDEs） where the time horizon may be finite or infinite and the generators have a non-uniformly linear growth with respect to t. The main idea is to construct a sequence of solutions {（Yn, Zn）} which is a Cauchy sequence in Sp × Mp space, and finally we prove {（Yn, Zn）} converges to the Lp （p 〉 1） solution of BSDEs.展开更多
In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coeffi...In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.展开更多
We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, wher...We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.展开更多
Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with ge...Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.展开更多
The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations(BSDEs).The system coefficients depend on parameter θ,whi...The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations(BSDEs).The system coefficients depend on parameter θ,while the generator of BSDEs exhibits quadratic growth with respect to z.To address the uncertainty present in the model,the variational inequality is derived using weak convergence techniques.Additionally,due to the generator being quadratic with respect to z,the forward adjoint equations are stochastic differential equations with unbounded coefficients,involving mean oscillation martingales.By using the reverse Holder inequality and John-Nirenberg inequality,we demonstrate that the solutions are continuous with respect to parameter θ.Moreover,the necessary and sufficient conditions for robust optimal control are established using the linearization method.展开更多
The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equati...The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations(BSDE)and pre-default reflected backward stochastic differential equations(RBSDE).The goal of this work is twofold.First,we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver A.Second,we study penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a constrained optimal stopping problem or a constrained Dynkin game in which the set of minimizer's admissible exercise times is constrained to the right support of the measure generated by A.展开更多
Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z...Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.展开更多
The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on ti...The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations(SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method,quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K^(-1/4)) rate of gradient convergence with K being the total number of iterative steps.展开更多
This paper is devoted to solving a reflected backward stochastic differential equation(BSDE in short)with one continuous barrier and a quasi-linear growth generator g,which has a linear growth in(y,z),except the upper...This paper is devoted to solving a reflected backward stochastic differential equation(BSDE in short)with one continuous barrier and a quasi-linear growth generator g,which has a linear growth in(y,z),except the upper direction in case of y<0,and is more general than the usual linear growth generator.By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal L^(p)(p>1)solutions for the reflected BSDEs.We also prove that the minimal Lpsolution can be approximated by a sequence of Lpsolutions of certain reflected BSDEs with Lipschitz generators.展开更多
We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher-order interactions.We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attit...We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher-order interactions.We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attitude is influenced by the system.This influence can come in a wide class of choices,including the average system state or average intensity of system interactions.Using Fenchel−Legendre transforms,our main result is a dual representation for the expectation of the risk measure in the convex case.In particular,we exhibit its dependence on the mean-field operator.展开更多
In this paper,we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown z.Using a linearization technique and the BMO martingale theory,we fi...In this paper,we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown z.Using a linearization technique and the BMO martingale theory,we first apply a fixed-point argument to establish the uniqueness and existence result for the case with bounded terminal condition and obstacle.Then,with the help of theθ-method,we develop a successive approximation procedure to remove the boundedness condition on the terminal condition and obstacle when the generator is concave(or convex)with respect to the second unknown.展开更多
We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change...We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change of measure techniques,we obtain stability estimates for the variation of the solutions with different underlying forward processes.In addition,we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.展开更多
We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally mo...We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.展开更多
This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic g...This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z.The existence and uniqueness results are given to these BSDEs.As an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.展开更多
In the context of risk measures,the capital allocation problem is widely studied in the literature where different approaches have been developed,also in connection with cooperative game theory and systemic risk.Altho...In the context of risk measures,the capital allocation problem is widely studied in the literature where different approaches have been developed,also in connection with cooperative game theory and systemic risk.Although static capital allocation rules have been extensively studied in the recent years,only few works deal with dynamic capital allocations and its relation with BSDEs.Moreover,all those works only examine the case of an underneath risk measure satisfying cash-additivity and,moreover,a large part of them focuses on the specific case of the gradient allocation where Gateaux differentiability is assumed.The main goal of this paper is,instead,to study general dynamic capital allocations associated to cash-subadditive risk measures,generalizing the approaches already existing in the literature and motivated by the presence of(ambiguity on)interest rates.Starting from an axiomatic approach,we then focus on the case where the underlying risk measures are induced by BSDEs whose drivers depend also on the yvariable.In this setting,we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations(BSVIEs).展开更多
In this paper,we study mulit-dimensional oblique reflected backward stochastic differential equations(RBSDEs)in a more general framework over finite or infinite time horizon,corresponding to the pricing problem for a ...In this paper,we study mulit-dimensional oblique reflected backward stochastic differential equations(RBSDEs)in a more general framework over finite or infinite time horizon,corresponding to the pricing problem for a type of real option.We prove that the equation can be solved uniquely in L^(p)(1<p≤2)-space,when the generators are uniformly continuous but each component taking values independently.Furthermore,if the generator of this equation fulfills the infinite time version of Lipschitzian continuity,we can also conclude that the solution to the oblique RBSDE exists and is unique,despite the fact that the values of some generator components may affect one another.展开更多
In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explici...In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explicit solutions to an interesting class of nonlinear BSDEs,including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.Moreover,we show its applications in PDEs and contingent pricing in an incomplete market.展开更多
In this paper,we study one-dimensional backward stochastic differential cquations fcaturing two refleccting barricrs.When thc tcrminal timc is not ncccssarily bounded or finite and the generator f(t,y.z)exhibits quadr...In this paper,we study one-dimensional backward stochastic differential cquations fcaturing two refleccting barricrs.When thc tcrminal timc is not ncccssarily bounded or finite and the generator f(t,y.z)exhibits quadratic growth in z,we prove existence and uniqucncss of solutions.In the Markovian casc,we establish thc link with an obstacle problem for quadratic elliptic partial differential equation with Dirichlet boundary conditions.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11371362)the Fundamental Research Funds for the Central Universities(Grant No.2012LWB48)
文摘In this paper, we establish the existence of the minimal Lp （p 〉 1） solution of backward stochastic differential equations （BSDEs） where the time horizon may be finite or infinite and the generators have a non-uniformly linear growth with respect to t. The main idea is to construct a sequence of solutions {（Yn, Zn）} which is a Cauchy sequence in Sp × Mp space, and finally we prove {（Yn, Zn）} converges to the Lp （p 〉 1） solution of BSDEs.
文摘In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.
基金supported by the NSF of China(11071144,11171187,11222110 and 71671104)Shandong Province(BS2011SF010,JQ201202)+4 种基金SRF for ROCS(SEM)Program for New Century Excellent Talents in University(NCET-12-0331)111 Project(B12023)the Ministry of Education of Humanities and Social Science Project(16YJA910003)Incubation Group Project of Financial Statistics and Risk Management of SDUFE
文摘We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.
基金Supported by the National Natural Science Foundation of China(No.10671205,)Youth Foundation of China University of Mining and Technology(No.2006A041,2007A029)
文摘Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.
基金supported by the National Key R&D Program of China(Grant Nos.2022YFA1006101 and 2022YFA1006102)National Natural Science Foundation of China(Grant Nos.72171133,12101291,and 12371445)+3 种基金Natural Science Foundation of Shandong Province(Grant Nos.ZR2024MA039 and ZR2022MA029)Guangdong Basic and Applied Basic Research Foundation(Grant No.2025B1515020091)Shenzhen Fundamental Research General Program(Grant No.JCYJ20230807093309021)Science and Technology Commission of Shanghai Municipality(Grant No.22ZR1407600).
文摘The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations(BSDEs).The system coefficients depend on parameter θ,while the generator of BSDEs exhibits quadratic growth with respect to z.To address the uncertainty present in the model,the variational inequality is derived using weak convergence techniques.Additionally,due to the generator being quadratic with respect to z,the forward adjoint equations are stochastic differential equations with unbounded coefficients,involving mean oscillation martingales.By using the reverse Holder inequality and John-Nirenberg inequality,we demonstrate that the solutions are continuous with respect to parameter θ.Moreover,the necessary and sufficient conditions for robust optimal control are established using the linearization method.
基金supported by the Australian Research Council Discovery Project (Grant No.DP220103106).
文摘The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations(BSDE)and pre-default reflected backward stochastic differential equations(RBSDE).The goal of this work is twofold.First,we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver A.Second,we study penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a constrained optimal stopping problem or a constrained Dynkin game in which the set of minimizer's admissible exercise times is constrained to the right support of the measure generated by A.
基金Supported by National Natural Science Foundation of China (Grant No.10671205)Youth Foundation of CUMT (Grant Nos.2006A041 and 2007A029)
文摘Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.
基金This work was supported by the National Key R&D Program of China(No.2018YFA0703900)the National Natural Science Foundation of China(No.11631004)。
文摘The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations(SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method,quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K^(-1/4)) rate of gradient convergence with K being the total number of iterative steps.
基金supported by National Natural Science Foundation of China(No.12171471)。
文摘This paper is devoted to solving a reflected backward stochastic differential equation(BSDE in short)with one continuous barrier and a quasi-linear growth generator g,which has a linear growth in(y,z),except the upper direction in case of y<0,and is more general than the usual linear growth generator.By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal L^(p)(p>1)solutions for the reflected BSDEs.We also prove that the minimal Lpsolution can be approximated by a sequence of Lpsolutions of certain reflected BSDEs with Lipschitz generators.
文摘We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher-order interactions.We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attitude is influenced by the system.This influence can come in a wide class of choices,including the average system state or average intensity of system interactions.Using Fenchel−Legendre transforms,our main result is a dual representation for the expectation of the risk measure in the convex case.In particular,we exhibit its dependence on the mean-field operator.
基金Ying Hu’s research is supported by the Lebesgue Center of Mathematics“Investissements d’avenir”Program(Grant No.ANR-11-LABX-0020-01),by ANR CAESARS(Grant No.ANR-15-CE05-0024)by ANR MFG(Grant No.ANR-16-CE40-0015-01)+2 种基金Falei Wang’s research is supported by the Natural Science Foundation of Shandong Province for Excellent Youth Scholars(Grant No.ZR2021YQ01)the National Natural Science Foundation of China(Grant Nos.12171280,12031009 and 11871458)the Young Scholars Program of Shandong University.
文摘In this paper,we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown z.Using a linearization technique and the BMO martingale theory,we first apply a fixed-point argument to establish the uniqueness and existence result for the case with bounded terminal condition and obstacle.Then,with the help of theθ-method,we develop a successive approximation procedure to remove the boundedness condition on the terminal condition and obstacle when the generator is concave(or convex)with respect to the second unknown.
基金supported by China Scholarship Council.Gechun Liang is partially supported by the National Natural Science Foundation of China(Grant No.12171169)Guangdong Basic and Applied Basic Research Foundation(Grant No.2019A1515011338)+1 种基金GL thanks J.F.Chassagneux and A.Richou for helpful and inspiring discussions on how to extend to the state dependent volatility case.Shanjian Tang is partially supported by National Science Foundation of China(Grant No.11631004)National Key R&D Program of China(Grant No.2018YFA0703903).
文摘We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change of measure techniques,we obtain stability estimates for the variation of the solutions with different underlying forward processes.In addition,we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.
文摘We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.
基金supported by the National Natural Science Foundation of China(Nos.11631004,12031009).
文摘This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z.The existence and uniqueness results are given to these BSDEs.As an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.
基金financial support of Gnampa Research Project 2024 (Grant No.PRR-20231026-073916-203)funded in part by an Ermenegildo Zegna Founder's Scholarship (Zullino)。
文摘In the context of risk measures,the capital allocation problem is widely studied in the literature where different approaches have been developed,also in connection with cooperative game theory and systemic risk.Although static capital allocation rules have been extensively studied in the recent years,only few works deal with dynamic capital allocations and its relation with BSDEs.Moreover,all those works only examine the case of an underneath risk measure satisfying cash-additivity and,moreover,a large part of them focuses on the specific case of the gradient allocation where Gateaux differentiability is assumed.The main goal of this paper is,instead,to study general dynamic capital allocations associated to cash-subadditive risk measures,generalizing the approaches already existing in the literature and motivated by the presence of(ambiguity on)interest rates.Starting from an axiomatic approach,we then focus on the case where the underlying risk measures are induced by BSDEs whose drivers depend also on the yvariable.In this setting,we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations(BSVIEs).
基金supported by the Natural Science Foundation of Shandong Province(Grant Nos.ZR2022MA079 and ZR2021MG049)the National Social Science Funding of China(Grant No.21CJY027)the TianYuan Special Funds of the National Natural Science Foundation of China(Grant No.11626146)。
文摘In this paper,we study mulit-dimensional oblique reflected backward stochastic differential equations(RBSDEs)in a more general framework over finite or infinite time horizon,corresponding to the pricing problem for a type of real option.We prove that the equation can be solved uniquely in L^(p)(1<p≤2)-space,when the generators are uniformly continuous but each component taking values independently.Furthermore,if the generator of this equation fulfills the infinite time version of Lipschitzian continuity,we can also conclude that the solution to the oblique RBSDE exists and is unique,despite the fact that the values of some generator components may affect one another.
基金This paper was originally exhibited in 2020(arXiv:2006.00222)。
文摘In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explicit solutions to an interesting class of nonlinear BSDEs,including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.Moreover,we show its applications in PDEs and contingent pricing in an incomplete market.
文摘In this paper,we study one-dimensional backward stochastic differential cquations fcaturing two refleccting barricrs.When thc tcrminal timc is not ncccssarily bounded or finite and the generator f(t,y.z)exhibits quadratic growth in z,we prove existence and uniqucncss of solutions.In the Markovian casc,we establish thc link with an obstacle problem for quadratic elliptic partial differential equation with Dirichlet boundary conditions.
