We consider the optimal control problem for a linear conditional McKeanVlasov equation with quadratic cost functional.The coefficients of the system and the weighting matrices in the cost functional are allowed to be ...We consider the optimal control problem for a linear conditional McKeanVlasov equation with quadratic cost functional.The coefficients of the system and the weighting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration.Semi closed-loop strategies are introduced,and following the dynamic programming approach in(Pham and Wei,Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics,2016),we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations.We present several financial applications with explicit solutions,and revisit,in particular,optimal tracking problems with price impact,and the conditional mean-variance portfolio selection in an incomplete market model.展开更多
We apply to the concrete setup of a bank engaged into bilateral trade portfolios the XVA theoretical framework of Albanese and Crepey(2017),whereby´so-called contra-liabilities and cost of capital are charged by ...We apply to the concrete setup of a bank engaged into bilateral trade portfolios the XVA theoretical framework of Albanese and Crepey(2017),whereby´so-called contra-liabilities and cost of capital are charged by the bank to its clients,on top of the fair valuation of counterparty risk,in order to account for the incompleteness of this risk.The transfer of the residual reserve credit capital from shareholders to creditors at bank default results in a unilateral CVA,consistent with the regulatory requirement that capital should not diminish as an effect of the sole deterioration of the bank credit spread.Our funding cost for variation margin(FVA)is defined asymmetrically since there is no benefit in holding excess capital in the future.Capital is fungible as a source of funding for variation margin,causing a material FVA reduction.We introduce a specialist initial margin lending scheme that drastically reduces the funding cost for initial margin(MVA).Our capital valuation adjustment(KVA)is defined as a risk premium,i.e.the cost of remunerating shareholder capital at risk at some hurdle rate.展开更多
This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)...This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.展开更多
For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra inte...For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.展开更多
This paper analyzes Conditional Value-at-Risk(CVaR)based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs.A nonlinear partial diffe...This paper analyzes Conditional Value-at-Risk(CVaR)based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs.A nonlinear partial differential equation(PDE)that an option value process inclusive of transaction costs should satisfy is provided.In particular,the closed-form expression of a European call option price is given.Meanwhile,the CVaR-based partial hedging strategy for a call option is derived explicitly.Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility.We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method.Furthermore,our results are implemented to derive target clients’survival probabilities and age of equity-linked life insurance contracts.展开更多
In this work,we propose an alternative to the Pollaczek-Khinchine formula for the ultimate time survival(or ruin)probability calculation in exchange for a few assumptions on the random variables that generate the rene...In this work,we propose an alternative to the Pollaczek-Khinchine formula for the ultimate time survival(or ruin)probability calculation in exchange for a few assumptions on the random variables that generate the renewal risk model.More precisely,we demonstrate the expressibility of the distribution function n P(sup n≥1^(n)∑_(i=1)(X_(i)-cθ_(i))<u),u∈N_(0)using the roots of the probability-generating function,expectation E(X-cθ)X-cθ,and probability mass function of.We assume that the random X_(1),X_(2),...cθ_(1),cθ_(2),...variables of the mutually independent sequences and are cθc>0 X cθindependent copies of X and respectively,wherein,and are independent,θnonnegative,and integer.We also assume that the support of is finite.To illustrate the applicability of the proven theoretical statements we present a few numerical outputs when the mentioned random variables adopt some particular distributions.展开更多
We study an optimal investment problem under default risk where related information such as loss or recovery at default is considered as an exogenous ran-dom mark added at default time.Two types of agents who have dif...We study an optimal investment problem under default risk where related information such as loss or recovery at default is considered as an exogenous ran-dom mark added at default time.Two types of agents who have different levels of information are considered.We first make precise the insider’s information flow by using the theory of enlargement of filtrations and then obtain explicit logarith-mic utility maximization results to compare optimal wealth for the insider and the ordinary agent.展开更多
In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distribu...In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.展开更多
This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations,which improves the results presented by Song[15]and Fang et al.[3].
We are delighted to present this special issue of PUQR in honor of Professor Alain Bensoussan on the occasion of his 80th birthday.While this birthday provides a good opportunity to celebrate the life and the successe...We are delighted to present this special issue of PUQR in honor of Professor Alain Bensoussan on the occasion of his 80th birthday.While this birthday provides a good opportunity to celebrate the life and the successes of an outstanding researcher,the COVID-19 epidemic has made it hard for a normal meeting.We hope that this special issue of collected papers will nevertheless provide a lasting mark for his birthday and express the appreciation and best wishes to Alain Bensoussan,from his colleagues and former students,from his co-authors and co-co-authors around the world,for a long life in good health and creative power.展开更多
We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear cas...We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.展开更多
Fishing quotas are unpleasant but efficient to control the productivity of a fishing site.A popular model has a stochastic differential equation for the biomass on which a stochastic dynamic programming or a Hamilton-...Fishing quotas are unpleasant but efficient to control the productivity of a fishing site.A popular model has a stochastic differential equation for the biomass on which a stochastic dynamic programming or a Hamilton-Jacobi-Bellman algorithm can be used to find the stochastic control–the fishing quota.We compare the solutions obtained by dynamic programming against those obtained with a neural network which preserves the Markov property of the solution.The method is extended to a multi species model and shows that the Neural Network is usable in high dimensions.展开更多
In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence o...In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.展开更多
Consider a single server queueing model which is observed over a continuous time interval(0,T],where T is determined by a suitable stopping rule.Let θ be the unknown parameter for the arrival process and θT be the m...Consider a single server queueing model which is observed over a continuous time interval(0,T],where T is determined by a suitable stopping rule.Let θ be the unknown parameter for the arrival process and θT be the maximum likelihood estimator of θ.The main goal of this paper is to obtain a moderate deviation result of the maximum likelihood estimator for the single server queueing model under certain regular conditions.展开更多
We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite q-variation with q∈[1,2).In contrast to previous work,we apply a direct fixpoint argu...We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite q-variation with q∈[1,2).In contrast to previous work,we apply a direct fixpoint argument and do not rely on any type of flow decomposition.The resulting object is an effective tool to study semilinear rough partial differential equations via a Feynman–Kac type representation.展开更多
The main aim of this paper is to introduce the notion of risk excess measure,to analyze its properties,and to describe some basic construction methods.To compare the risk excess of one distribution Q w.r.t.a given ris...The main aim of this paper is to introduce the notion of risk excess measure,to analyze its properties,and to describe some basic construction methods.To compare the risk excess of one distribution Q w.r.t.a given risk distribution P,we apply the concept of hemi-metrics on the space of probability measures.This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures.Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings.Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the valuesρ(Q),ρ(P)of a risk measureρ.We argue that the differenceρ(Q)−ρ(P)neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure.We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.展开更多
We show that the comparison results for a backward SDE with jumps established in Royer(Stoch.Process.Appl 116:1358–1376,2006)and Yin and Mao(J.Math.Anal.Appl 346:345–358,2008)hold under more simplified conditions.Mo...We show that the comparison results for a backward SDE with jumps established in Royer(Stoch.Process.Appl 116:1358–1376,2006)and Yin and Mao(J.Math.Anal.Appl 346:345–358,2008)hold under more simplified conditions.Moreover,we prove existence and uniqueness allowing the coefficients in the linear growth-and monotonicity-condition for the generator to be random and time-dependent.In the L2-case with linear growth,this also generalizes the results of Kruse and Popier(Stochastics 88:491–539,2016).For the proof of the comparison result,we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure,we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.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
Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps.A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level.Estimation i...Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps.A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level.Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation.The motion is decomposed into a space dependent drift and a space dependent martingale component.Though there is some local mean reversion in the drift,space dependence of the martingale component renders the dynamics to be of the momentum type.Local risk is measured using market calibrated measure distortions that introduce risk charges into the lower and upper prices of two price economies.These risks are compensated by the exponential variation of space dependent arrival rates.Estimations are conducted for the S&P 500 index(SPX),the exchange traded fund for the financial sector(XLF),J.P.Morgan stock prices(JPM),the ratio of JPM to XLF,and the ratio of XLF to SPX.展开更多
This study advances the G-stochastic maximum principle(G-SMP)from a risk-neutral framework to a risk-sensitive one.A salient feature of this advancement is its applicability to systems governed by stochastic different...This study advances the G-stochastic maximum principle(G-SMP)from a risk-neutral framework to a risk-sensitive one.A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under G-Brownian motion(G-SDEs),where the control variable may influence all terms.We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost.Initially,we introduced an auxiliary process to address risk-sensitive performance costs within the G-expectation framework.Subsequently,we established and validated the correlation between the G-expected exponential utility and the G-quadratic backward stochastic differential equation.Furthermore,we simplified the G-adjoint process from a dual-component structure to a singular component.Moreover,we explained the necessary optimality conditions for this model by considering a convex set of admissible controls.To describe the main findings,we present two examples:the first addresses the linear-quadratic problem and the second examines a Merton-type problem characterized by power utility.展开更多
基金work is part of the ANR project CAESARS(ANR-15-CE05-0024)lso supported by FiME(Finance for Energy Market Research Centre)and the“Finance et Developpement Durable-Approches Quantitatives”EDF-CACIB Chair。
文摘We consider the optimal control problem for a linear conditional McKeanVlasov equation with quadratic cost functional.The coefficients of the system and the weighting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration.Semi closed-loop strategies are introduced,and following the dynamic programming approach in(Pham and Wei,Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics,2016),we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations.We present several financial applications with explicit solutions,and revisit,in particular,optimal tracking problems with price impact,and the conditional mean-variance portfolio selection in an incomplete market model.
基金The research of Stephane Cr´epey benefited from the support of the“Chair Markets´in Transition,”Fed´eration Bancaire Franc´¸aise,of the ANR project 11-LABX-0019 and from the EIF grant“Collateral management in centrally cleared trading.”。
文摘We apply to the concrete setup of a bank engaged into bilateral trade portfolios the XVA theoretical framework of Albanese and Crepey(2017),whereby´so-called contra-liabilities and cost of capital are charged by the bank to its clients,on top of the fair valuation of counterparty risk,in order to account for the incompleteness of this risk.The transfer of the residual reserve credit capital from shareholders to creditors at bank default results in a unilateral CVA,consistent with the regulatory requirement that capital should not diminish as an effect of the sole deterioration of the bank credit spread.Our funding cost for variation margin(FVA)is defined asymmetrically since there is no benefit in holding excess capital in the future.Capital is fungible as a source of funding for variation margin,causing a material FVA reduction.We introduce a specialist initial margin lending scheme that drastically reduces the funding cost for initial margin(MVA).Our capital valuation adjustment(KVA)is defined as a risk premium,i.e.the cost of remunerating shareholder capital at risk at some hurdle rate.
基金supported by the National Key R&D Program of China(Grant No.2018YFA0703900)the National Natural Science Foundation of China(Grant No.11671231)+2 种基金the Qilu Young Scholars Program of Shandong Universitysupported by the Tian Yuan Projection of the National Natural Science Foundation of China(Grant Nos.11526205,11626247)the National Basic Research Program of China(973 Program)(Grant No.2007CB814900(Financial Risk)).
文摘This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.
基金Hanxiao Wang would like to thank Chenchen Mou(of City University of Hong Kong)for some useful discussionsJiongmin Yong is supported in part by NSF(Grant No.DMS-1812921)+1 种基金Chao Zhou is supported by NSFC(Grant No.11871364)Singapore MOE AcRF(Grant Nos.A-800453-00-00,R-146-000-271-112 and R-146-000-284-114).
文摘For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.
基金Natural Sciences and Engineering Research Council of Canada(Grant No.RES0043487).
文摘This paper analyzes Conditional Value-at-Risk(CVaR)based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs.A nonlinear partial differential equation(PDE)that an option value process inclusive of transaction costs should satisfy is provided.In particular,the closed-form expression of a European call option price is given.Meanwhile,the CVaR-based partial hedging strategy for a call option is derived explicitly.Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility.We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method.Furthermore,our results are implemented to derive target clients’survival probabilities and age of equity-linked life insurance contracts.
文摘In this work,we propose an alternative to the Pollaczek-Khinchine formula for the ultimate time survival(or ruin)probability calculation in exchange for a few assumptions on the random variables that generate the renewal risk model.More precisely,we demonstrate the expressibility of the distribution function n P(sup n≥1^(n)∑_(i=1)(X_(i)-cθ_(i))<u),u∈N_(0)using the roots of the probability-generating function,expectation E(X-cθ)X-cθ,and probability mass function of.We assume that the random X_(1),X_(2),...cθ_(1),cθ_(2),...variables of the mutually independent sequences and are cθc>0 X cθindependent copies of X and respectively,wherein,and are independent,θnonnegative,and integer.We also assume that the support of is finite.To illustrate the applicability of the proven theoretical statements we present a few numerical outputs when the mentioned random variables adopt some particular distributions.
文摘We study an optimal investment problem under default risk where related information such as loss or recovery at default is considered as an exogenous ran-dom mark added at default time.Two types of agents who have different levels of information are considered.We first make precise the insider’s information flow by using the theory of enlargement of filtrations and then obtain explicit logarith-mic utility maximization results to compare optimal wealth for the insider and the ordinary agent.
基金the National Natural Science Foundation of China(Grant Nos.11731012,12031005)Ten Thousand Talents Plan of Zhejiang Province(Grant No.2018R52042)+1 种基金Natural Science Foundation of Zhejiang Province(Grant No.LZ21A010002)the Fundamental Research Funds for the Central Universities.
文摘In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.
基金This project is supported by National Key R&D Program of China(Grant No.2018YFA0703900)National Natural Science Foundation of China(Grant Nos.11601281,11671231).
文摘This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations,which improves the results presented by Song[15]and Fang et al.[3].
文摘We are delighted to present this special issue of PUQR in honor of Professor Alain Bensoussan on the occasion of his 80th birthday.While this birthday provides a good opportunity to celebrate the life and the successes of an outstanding researcher,the COVID-19 epidemic has made it hard for a normal meeting.We hope that this special issue of collected papers will nevertheless provide a lasting mark for his birthday and express the appreciation and best wishes to Alain Bensoussan,from his colleagues and former students,from his co-authors and co-co-authors around the world,for a long life in good health and creative power.
文摘We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.
文摘Fishing quotas are unpleasant but efficient to control the productivity of a fishing site.A popular model has a stochastic differential equation for the biomass on which a stochastic dynamic programming or a Hamilton-Jacobi-Bellman algorithm can be used to find the stochastic control–the fishing quota.We compare the solutions obtained by dynamic programming against those obtained with a neural network which preserves the Markov property of the solution.The method is extended to a multi species model and shows that the Neural Network is usable in high dimensions.
基金The research leading to these results received funding from the European Research Council under the European Community’s Seventh Framework Program(FP7/2007-2013)/ERC grant agreement 228087.
文摘In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.
文摘Consider a single server queueing model which is observed over a continuous time interval(0,T],where T is determined by a suitable stopping rule.Let θ be the unknown parameter for the arrival process and θT be the maximum likelihood estimator of θ.The main goal of this paper is to obtain a moderate deviation result of the maximum likelihood estimator for the single server queueing model under certain regular conditions.
基金supported by the DAAD P.R.I.M.E.program and NSF grant DMS 1413717.
文摘We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite q-variation with q∈[1,2).In contrast to previous work,we apply a direct fixpoint argument and do not rely on any type of flow decomposition.The resulting object is an effective tool to study semilinear rough partial differential equations via a Feynman–Kac type representation.
文摘The main aim of this paper is to introduce the notion of risk excess measure,to analyze its properties,and to describe some basic construction methods.To compare the risk excess of one distribution Q w.r.t.a given risk distribution P,we apply the concept of hemi-metrics on the space of probability measures.This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures.Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings.Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the valuesρ(Q),ρ(P)of a risk measureρ.We argue that the differenceρ(Q)−ρ(P)neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure.We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.
基金Large parts of this article were written when Alexander Steinicke was member of the Institute of Mathematics and Scientific Computing,University of Graz,Austria,and supported by the Austrian Science Fund(FWF):Project F5508-N26,which is part of the Special Research Program"Quasi-Monte Carlo Methods:Theory and Applications."。
文摘We show that the comparison results for a backward SDE with jumps established in Royer(Stoch.Process.Appl 116:1358–1376,2006)and Yin and Mao(J.Math.Anal.Appl 346:345–358,2008)hold under more simplified conditions.Moreover,we prove existence and uniqueness allowing the coefficients in the linear growth-and monotonicity-condition for the generator to be random and time-dependent.In the L2-case with linear growth,this also generalizes the results of Kruse and Popier(Stochastics 88:491–539,2016).For the proof of the comparison result,we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure,we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.
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.
文摘Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps.A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level.Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation.The motion is decomposed into a space dependent drift and a space dependent martingale component.Though there is some local mean reversion in the drift,space dependence of the martingale component renders the dynamics to be of the momentum type.Local risk is measured using market calibrated measure distortions that introduce risk charges into the lower and upper prices of two price economies.These risks are compensated by the exponential variation of space dependent arrival rates.Estimations are conducted for the S&P 500 index(SPX),the exchange traded fund for the financial sector(XLF),J.P.Morgan stock prices(JPM),the ratio of JPM to XLF,and the ratio of XLF to SPX.
基金supported by PRFU project N(Grant No.C00L03UN070120220004).
文摘This study advances the G-stochastic maximum principle(G-SMP)from a risk-neutral framework to a risk-sensitive one.A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under G-Brownian motion(G-SDEs),where the control variable may influence all terms.We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost.Initially,we introduced an auxiliary process to address risk-sensitive performance costs within the G-expectation framework.Subsequently,we established and validated the correlation between the G-expected exponential utility and the G-quadratic backward stochastic differential equation.Furthermore,we simplified the G-adjoint process from a dual-component structure to a singular component.Moreover,we explained the necessary optimality conditions for this model by considering a convex set of admissible controls.To describe the main findings,we present two examples:the first addresses the linear-quadratic problem and the second examines a Merton-type problem characterized by power utility.
