We introduce and analyze a class of forward performance criteria in incomplete markets in the presence of model ambiguity.Incompleteness stems from general investment constraints,while model uncertainty is represented...We introduce and analyze a class of forward performance criteria in incomplete markets in the presence of model ambiguity.Incompleteness stems from general investment constraints,while model uncertainty is represented by a convex and compact set of plausible model parameter processes.Following the max-min criteria in traditional(backward)robust control,we formulate similar criteria for the robust forward performance processes and focus on the rich class of time-monotone processes.We provide a novel PDE characterization and a semi-explicit saddle-point construction of the robust forward performance criteria and their optimal policies.Furthermore,we present additional results within the class of homothetic constant relative risk aversion(CRRA)processes.Within this class,we investigate the relationship between forward performance processes on wealth and those on consumption,establishing an interesting dominance through time.展开更多
We introduce a new approach for optimal portfolio choice under model ambiguity by incorporating predictable forward preferences in the framework of Angoshtari et al.[2].The investor reassesses and revises the model am...We introduce a new approach for optimal portfolio choice under model ambiguity by incorporating predictable forward preferences in the framework of Angoshtari et al.[2].The investor reassesses and revises the model ambiguity set incrementally in time while,also,updating his risk preferences forward in time.This dynamic alignment of preferences and ambiguity updating results in time-consistent policies and provides a richer,more accurate learning setting.For each investment period,the investor solves a worst-case portfolio optimization over possible market models,which are represented via a Wasserstein neighborhood centered at a binomial distribution.Duality methods from Gao and Kleywegt[10];Blanchet and Murthy[8]are used to solve the optimization problem over a suitable set of measures,yielding an explicit optimal portfolio in the linear case.We analyze the case of linear and quadratic utilities,and provide numerical results.展开更多
This paper proposes and investigates an optimal pair investment/pension policy for a pay-as-you-go(PAYG)pension scheme.The social planner can invest in a buffer fund in order to guarantee a minimal pension amount.The ...This paper proposes and investigates an optimal pair investment/pension policy for a pay-as-you-go(PAYG)pension scheme.The social planner can invest in a buffer fund in order to guarantee a minimal pension amount.The model aims at taking into account complex dynamic phenomena such as the demographic risk and its evolution over time,the time and age dependence of agents preferences,and financial risks.The preference criterion of the social planner is modeled by a consistent dynamic utility defined on a stochastic domain,which incorporates the heterogeneity of overlapping generations and its evolution over time.The preference criterion and the optimization problem also incorporate sustainability,adequacy and fairness constraints.The paper designs and solves the social planner's dynamic decision criterion,and computes the optimal investment/pension policy in a general framework.A detailed analysis for the case of dynamic power utilities is provided.展开更多
This paper investigates the inverse problem of bi-revealed utilities in a defaultable universe,defined as a standard universe(represented by a filtration F)perturbed by an exogenous defaultable time τ.We assume that ...This paper investigates the inverse problem of bi-revealed utilities in a defaultable universe,defined as a standard universe(represented by a filtration F)perturbed by an exogenous defaultable time τ.We assume that the standard universe does not take into account the possibility of the default,thus τ adds an additional source of risk.The defaultable universe is represented by the filtration G up to time τ(τ included),where G stands for the progressive enlargement of F by T.The basic assumption in force is that τ avoids F-stopping times.The bi-revealed problem consists in recovering a consistent dynamic utility from the observable characteristic of an agent.The general results on bi-revealed utilities,first given in a general and abstract framework,are translated in the defaultable G-universe and then are interpreted in the F-universe.The decomposition of G-adapted processes X^(G) provides an interpretation of a Gcharacteristic X^(G)_(τ) stopped at τ as a reserve process.Thanks to the characterization of G-martingales stopped at τ in terms of F-martingales,we establish a correspondence between G-bi-revealed utilities from characteristic and F-bi-revealed pair of utilities from characteristic and reserves.In a financial framework,characteristic can be interpreted as wealth and reserves as consumption.This result sheds a new light on the consumption in utility criterion:the consumption process can be interpreted as a certain quantity of wealth,or reserves,that are accumulated for the financing of losses at the default time.展开更多
1.Introduction Forward performance measurement concept was developed in the context of investment.The idea grew out of desire to propose a framework for investment performance measurement which can run in parallel to ...1.Introduction Forward performance measurement concept was developed in the context of investment.The idea grew out of desire to propose a framework for investment performance measurement which can run in parallel to another very important idea of modern finance,namely,arbitrage free valuation of derivatives.One may say that the two areas,namely investment and valuation,have little to do with one another.However,in our minds,there were many fundamentally important common ingredients to build on.One of us,the author of this preface,worked before on problems related to arbitrage free valuation.The other,Thaleia Zariphopoulou,worked on questions related to optimization and control.Clearly,between the two of us we had a complementary and relevant set of skills to develop the said idea.展开更多
We introduce a new type of robust forward criterion under model uncertainty,called the G-forward performance process,which extends the classical notion of forward performance process to the G-expectation framework.We ...We introduce a new type of robust forward criterion under model uncertainty,called the G-forward performance process,which extends the classical notion of forward performance process to the G-expectation framework.We then derive the representations of homothetic G-forward performance processes in a single stochastic factor model with uncertainty,building on the well-posedness of ergodic and infinite horizon backward stochastic differential equations driven by G-Brownian motion(G-BSDEs)with quadratic generators.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
The spirit of now in nowcasting suggests expanding the current to include the near future.Decision theory is then developed by incorporating the consequences of actions into the present.With the future falling into th...The spirit of now in nowcasting suggests expanding the current to include the near future.Decision theory is then developed by incorporating the consequences of actions into the present.With the future falling into the present discounting it is no longer permitted.Value functions are then observed to be determinate only up to scale and shift that are then locked down by fixing values arbitrarily in two selected states,much like declaring water to freeze and boil at zero and a hundred degrees celsius.The locked down value functions associated policy functions are seen to exist in decision contexts in where the only time is now.Examples are studied in univariate and multivariate dimensions for the decision state space and the dimension of shocks delivering state transitions.The policy functions are expanded from realisitic training sets to the full state space using Gaussian Process Regression.They are implemented on real data with reported performances.展开更多
We establish existence of Predictable Forward Performance Processes(PFPPs)in conditionally complete markets,which has been previously shown only in the binomial setting.Our market model can be a discrete-time or a con...We establish existence of Predictable Forward Performance Processes(PFPPs)in conditionally complete markets,which has been previously shown only in the binomial setting.Our market model can be a discrete-time or a continuous-time model,and the investment horizon can be finite or infinite.We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation,which is the counterpart of the functional equation found in the binomial case.Although this integral equation has been partially studied in the existing literature,we provide a new solution method using the Fourier transform for tempered distributions.We also provide closedform solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class.We apply our results to two special cases.The first one is the binomial market and is included to relate our work to the existing literature.The second example considers a generalized Black–Scholes model which,to the best of our knowledge,is a new result.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
We investigate the large deviations properties for centered stationary AR(1)and MA(1)processes with independent Gaussian innovations,by giving the explicit bivariate rate functions for the sequence of two-dimensional ...We investigate the large deviations properties for centered stationary AR(1)and MA(1)processes with independent Gaussian innovations,by giving the explicit bivariate rate functions for the sequence of two-dimensional random vectors.Via the Contraction Principle,we provide the explicit rate functions for the sample mean and the sample second moment.In the AR(1)case,we also give the explicit rate function for the sequence of two-dimensional random vectors(W_(n))n≥2=(n^(-1(∑_(k=1)^(n)X_(k),∑_(k=1)^(n)X_(k)^(2))))_(n∈N)n≥2,but we obtain an analytic rate function that gives different values for the upper and lower bounds,depending on the evaluated set and its intersection with the respective set of exposed points.A careful analysis of the properties of a certain family of Toeplitz matrices is necessary.The large deviations properties of three particular sequences of one-dimensional random variables will follow after we show how to apply a weaker version of the Contraction Principle for our setting,providing new proofs for two already known results on the explicit deviation function for the sample second moment and Yule-Walker estimators.We exhibit the properties of the large deviations of the first-order empirical autocovariance,its explicit deviation function and this is also a new result.展开更多
We consider the control of the diserete component n_(t)of a owitching Markov proceaa x_(t)=(z_(t),n_(t))when there ia a running cost and an immediate coat c(i,j)for owitching n_(t)from i to j.We satudy the minimizatio...We consider the control of the diserete component n_(t)of a owitching Markov proceaa x_(t)=(z_(t),n_(t))when there ia a running cost and an immediate coat c(i,j)for owitching n_(t)from i to j.We satudy the minimization of the ergodic(or long-term average)total coat.Eooentially,this paper trento the cnce where,for n_(t)=n fixed,z_(t)ia a reflected diffusion or a reflected diffusion with jumps,nt being,for fixed z,a continuous-time Markov chain.Using the vanishing discount appronch,we exctend existing reoulta dealing with the situation where nt evolvea only by the switching control action and the diffusion is non-degenerate.Moreover,we solve the ergodic problem for a claso of diffusiono which can be degenerate and for an example with aboorbing atate.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
We reexamine the classical linear regression model when it is subject to two types of uncertainty:(i)some covariates are either missing or completely inaccessible,and(ii)the variance of the measurement error is undete...We reexamine the classical linear regression model when it is subject to two types of uncertainty:(i)some covariates are either missing or completely inaccessible,and(ii)the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician.By following the recent theory of sublinear expectation,we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables,which encompass an infinite family of probability distributions for the response variable in the sense of(linear)classical probability theory.The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty.The consistency of the estimators is established under mild conditions in the data generation process.Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.展开更多
In this paper,we study a class of stochastic processes{X_(t)}t∈N,where X_(t)=φ■T_(s)^(t)(X_(0))is obtained from the iterations of the transformation T_(s),invariant for an ergodic probabilityμ_(s)on[0,1]and a cert...In this paper,we study a class of stochastic processes{X_(t)}t∈N,where X_(t)=φ■T_(s)^(t)(X_(0))is obtained from the iterations of the transformation T_(s),invariant for an ergodic probabilityμ_(s)on[0,1]and a certain constant by partial functionφ:[0,1]→R.We consider here the family of transformations T_(s):[0,1]→[0,1]indexed by a parameters>0,known as the Manneville–Pomeau family of transformations.The autocorrelation function of the resulting process decays hyperbolically(or polynomially)and we obtain efficient methods to estimate the parameter s from a finite time series.As a consequence,we also estimate the rate of convergence of the autocorrelation decay of these processes.We compare different estimation methods based on the periodogram function,the smoothed periodogram function,the variance of the partial sum,and the wavelet theory.To obtain our results we analyzed the properties of the spectral density function and the associated Fourier series.展开更多
This study considers an optimal investment and reinsurance problem involving a defaultable security for an insurer in an ambiguous environment.In other words,the insurer is ambiguous about the insurance claim that is ...This study considers an optimal investment and reinsurance problem involving a defaultable security for an insurer in an ambiguous environment.In other words,the insurer is ambiguous about the insurance claim that is exponentially distributed with an uncertain rate parameter.The insurer can purchase proportional reinsurance and invest its wealth in three assets:a risk-free asset,a risky asset,the price process of which satisfies the Heston local-stochastic volatility model,and a defaultable corporate bond.For the optimal investment–reinsurance objective with a smooth ambiguity utility proposed by Klibanoff,P.,Marinacci,M.,and Mukerji,S.[A smooth model of decision making under ambiguity,Econometrica,2005,73(6):1849-1892],the equilibrium strategy is introduced and the extended Hamilton–Jacobi–Bellman equation is established through a stochastic control approach.However,the analytical solution of the strategy under the Heston local-stochastic volatility model cannot be obtained because of the complicated nonlinearity of the partial differential equation.In this study,we employ a perturbation method to derive an asymptotic solution for the post-and pre-default cases.In addition,we present a sensitivity analysis to explain the impact of model parameters on the equilibrium investment–reinsurance strategy.展开更多
Certain Merton type consumption−investment problems under partial information are reduced to the ones of full information and within the framework of a complete market model.Then,specializing to conditionally log−Gaus...Certain Merton type consumption−investment problems under partial information are reduced to the ones of full information and within the framework of a complete market model.Then,specializing to conditionally log−Gaussian diffusion models,concrete analysis about the optimal values and optimal strategies is performed by using analytical tools like Feynman−Kac formula,or HJB equations.The explicit solutions to the related forward-backward equations are also given.展开更多
This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries.The fluid is driven by the noisy movement of the bottom boundary,where the no...This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries.The fluid is driven by the noisy movement of the bottom boundary,where the noise is given by a Lévy process.After establishing existence of a martingale solution,we use the background flow method to derive an upper bound on the turbulent energy dissipation rate.Our estimate recovers one of the basic scaling ideas of turbulence theory,namely,that the dissipation rate is independent of the viscosity at high Reynolds number.展开更多
In this study,we delve into the optimal stopping problem by examining the p(ϕ(τ),τ∈T_(0)^(p))case in which the reward is given by a family of nonnegative random variables indexed by predictable stopping times.We ai...In this study,we delve into the optimal stopping problem by examining the p(ϕ(τ),τ∈T_(0)^(p))case in which the reward is given by a family of nonnegative random variables indexed by predictable stopping times.We aim to elucidate various properties of the value function family within this context.We prove the existence of an optimal predictable stopping time,subject to specific assumptions regarding the reward functionϕ.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
文摘We introduce and analyze a class of forward performance criteria in incomplete markets in the presence of model ambiguity.Incompleteness stems from general investment constraints,while model uncertainty is represented by a convex and compact set of plausible model parameter processes.Following the max-min criteria in traditional(backward)robust control,we formulate similar criteria for the robust forward performance processes and focus on the rich class of time-monotone processes.We provide a novel PDE characterization and a semi-explicit saddle-point construction of the robust forward performance criteria and their optimal policies.Furthermore,we present additional results within the class of homothetic constant relative risk aversion(CRRA)processes.Within this class,we investigate the relationship between forward performance processes on wealth and those on consumption,establishing an interesting dominance through time.
文摘We introduce a new approach for optimal portfolio choice under model ambiguity by incorporating predictable forward preferences in the framework of Angoshtari et al.[2].The investor reassesses and revises the model ambiguity set incrementally in time while,also,updating his risk preferences forward in time.This dynamic alignment of preferences and ambiguity updating results in time-consistent policies and provides a richer,more accurate learning setting.For each investment period,the investor solves a worst-case portfolio optimization over possible market models,which are represented via a Wasserstein neighborhood centered at a binomial distribution.Duality methods from Gao and Kleywegt[10];Blanchet and Murthy[8]are used to solve the optimization problem over a suitable set of measures,yielding an explicit optimal portfolio in the linear case.We analyze the case of linear and quadratic utilities,and provide numerical results.
基金The authors's research is part of the ANR project DREAMeS(ANR-21-CE46-0002)The research of Sarah Kaakai is Funded by the European Union(ERC,SINGER,101054787)。
文摘This paper proposes and investigates an optimal pair investment/pension policy for a pay-as-you-go(PAYG)pension scheme.The social planner can invest in a buffer fund in order to guarantee a minimal pension amount.The model aims at taking into account complex dynamic phenomena such as the demographic risk and its evolution over time,the time and age dependence of agents preferences,and financial risks.The preference criterion of the social planner is modeled by a consistent dynamic utility defined on a stochastic domain,which incorporates the heterogeneity of overlapping generations and its evolution over time.The preference criterion and the optimization problem also incorporate sustainability,adequacy and fairness constraints.The paper designs and solves the social planner's dynamic decision criterion,and computes the optimal investment/pension policy in a general framework.A detailed analysis for the case of dynamic power utilities is provided.
基金This work is with the financial support of the“Chaire Risque Financier”of the“Fondation du Risque”,the Labex MME-DII.The authors's research is part of the ANR project DREAMeS(ANR-21-CE46-0002).
文摘This paper investigates the inverse problem of bi-revealed utilities in a defaultable universe,defined as a standard universe(represented by a filtration F)perturbed by an exogenous defaultable time τ.We assume that the standard universe does not take into account the possibility of the default,thus τ adds an additional source of risk.The defaultable universe is represented by the filtration G up to time τ(τ included),where G stands for the progressive enlargement of F by T.The basic assumption in force is that τ avoids F-stopping times.The bi-revealed problem consists in recovering a consistent dynamic utility from the observable characteristic of an agent.The general results on bi-revealed utilities,first given in a general and abstract framework,are translated in the defaultable G-universe and then are interpreted in the F-universe.The decomposition of G-adapted processes X^(G) provides an interpretation of a Gcharacteristic X^(G)_(τ) stopped at τ as a reserve process.Thanks to the characterization of G-martingales stopped at τ in terms of F-martingales,we establish a correspondence between G-bi-revealed utilities from characteristic and F-bi-revealed pair of utilities from characteristic and reserves.In a financial framework,characteristic can be interpreted as wealth and reserves as consumption.This result sheds a new light on the consumption in utility criterion:the consumption process can be interpreted as a certain quantity of wealth,or reserves,that are accumulated for the financing of losses at the default time.
文摘1.Introduction Forward performance measurement concept was developed in the context of investment.The idea grew out of desire to propose a framework for investment performance measurement which can run in parallel to another very important idea of modern finance,namely,arbitrage free valuation of derivatives.One may say that the two areas,namely investment and valuation,have little to do with one another.However,in our minds,there were many fundamentally important common ingredients to build on.One of us,the author of this preface,worked before on problems related to arbitrage free valuation.The other,Thaleia Zariphopoulou,worked on questions related to optimization and control.Clearly,between the two of us we had a complementary and relevant set of skills to develop the said idea.
基金The research of Falei Wang is supported by the National Natural Science Foundation of China(Grant Nos.12171280 and 12031009)the Natural Science Foundation of Shandong Province(Grant Nos.ZR2021YQ01 and ZR2022JQ01)the National Key Research&Development Program of China(Grant No.2018YFA0703900).
文摘We introduce a new type of robust forward criterion under model uncertainty,called the G-forward performance process,which extends the classical notion of forward performance process to the G-expectation framework.We then derive the representations of homothetic G-forward performance processes in a single stochastic factor model with uncertainty,building on the well-posedness of ergodic and infinite horizon backward stochastic differential equations driven by G-Brownian motion(G-BSDEs)with quadratic generators.
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The spirit of now in nowcasting suggests expanding the current to include the near future.Decision theory is then developed by incorporating the consequences of actions into the present.With the future falling into the present discounting it is no longer permitted.Value functions are then observed to be determinate only up to scale and shift that are then locked down by fixing values arbitrarily in two selected states,much like declaring water to freeze and boil at zero and a hundred degrees celsius.The locked down value functions associated policy functions are seen to exist in decision contexts in where the only time is now.Examples are studied in univariate and multivariate dimensions for the decision state space and the dimension of shocks delivering state transitions.The policy functions are expanded from realisitic training sets to the full state space using Gaussian Process Regression.They are implemented on real data with reported performances.
基金supported by the National Science Foundation(Grant No.DMS-1929348).
文摘We establish existence of Predictable Forward Performance Processes(PFPPs)in conditionally complete markets,which has been previously shown only in the binomial setting.Our market model can be a discrete-time or a continuous-time model,and the investment horizon can be finite or infinite.We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation,which is the counterpart of the functional equation found in the binomial case.Although this integral equation has been partially studied in the existing literature,we provide a new solution method using the Fourier transform for tempered distributions.We also provide closedform solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class.We apply our results to two special cases.The first one is the binomial market and is included to relate our work to the existing literature.The second example considers a generalized Black–Scholes model which,to the best of our knowledge,is a new result.
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.
基金M.J.Karling was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior(CAPES)-Brazil(Grant No.1736629)Conselho Nacional de Desenvolvimento Científico e Tecnológico(CNPq)-Brazil(Grant No.170168/2018-2)+1 种基金A.O.Lopes’research was partially supported by CNPq-Brazil(Grant No.304048/2016-0)S.R.C.Lopes’research was partially supported by CNPq-Brazil(Grant No.303453/2018-4).
文摘We investigate the large deviations properties for centered stationary AR(1)and MA(1)processes with independent Gaussian innovations,by giving the explicit bivariate rate functions for the sequence of two-dimensional random vectors.Via the Contraction Principle,we provide the explicit rate functions for the sample mean and the sample second moment.In the AR(1)case,we also give the explicit rate function for the sequence of two-dimensional random vectors(W_(n))n≥2=(n^(-1(∑_(k=1)^(n)X_(k),∑_(k=1)^(n)X_(k)^(2))))_(n∈N)n≥2,but we obtain an analytic rate function that gives different values for the upper and lower bounds,depending on the evaluated set and its intersection with the respective set of exposed points.A careful analysis of the properties of a certain family of Toeplitz matrices is necessary.The large deviations properties of three particular sequences of one-dimensional random variables will follow after we show how to apply a weaker version of the Contraction Principle for our setting,providing new proofs for two already known results on the explicit deviation function for the sample second moment and Yule-Walker estimators.We exhibit the properties of the large deviations of the first-order empirical autocovariance,its explicit deviation function and this is also a new result.
文摘We consider the control of the diserete component n_(t)of a owitching Markov proceaa x_(t)=(z_(t),n_(t))when there ia a running cost and an immediate coat c(i,j)for owitching n_(t)from i to j.We satudy the minimization of the ergodic(or long-term average)total coat.Eooentially,this paper trento the cnce where,for n_(t)=n fixed,z_(t)ia a reflected diffusion or a reflected diffusion with jumps,nt being,for fixed z,a continuous-time Markov chain.Using the vanishing discount appronch,we exctend existing reoulta dealing with the situation where nt evolvea only by the switching control action and the diffusion is non-degenerate.Moreover,we solve the ergodic problem for a claso of diffusiono which can be degenerate and for an example with aboorbing atate.
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supported by the National Key R&D program of China(Grant Nos.2018YFA0703900 and ZR2019ZD41)the National Natural Science Foundation of China(Grant No.11701330)Taishan Scholar Talent Project Youth Project.
文摘We reexamine the classical linear regression model when it is subject to two types of uncertainty:(i)some covariates are either missing or completely inaccessible,and(ii)the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician.By following the recent theory of sublinear expectation,we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables,which encompass an infinite family of probability distributions for the response variable in the sense of(linear)classical probability theory.The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty.The consistency of the estimators is established under mild conditions in the data generation process.Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.
基金supported by CNPq-Brazil(Grant No.310053/2020-0)Silvia Regina Lopes was partially supported by CNPq-Brazil(Grant No.303453/2018-4).
文摘In this paper,we study a class of stochastic processes{X_(t)}t∈N,where X_(t)=φ■T_(s)^(t)(X_(0))is obtained from the iterations of the transformation T_(s),invariant for an ergodic probabilityμ_(s)on[0,1]and a certain constant by partial functionφ:[0,1]→R.We consider here the family of transformations T_(s):[0,1]→[0,1]indexed by a parameters>0,known as the Manneville–Pomeau family of transformations.The autocorrelation function of the resulting process decays hyperbolically(or polynomially)and we obtain efficient methods to estimate the parameter s from a finite time series.As a consequence,we also estimate the rate of convergence of the autocorrelation decay of these processes.We compare different estimation methods based on the periodogram function,the smoothed periodogram function,the variance of the partial sum,and the wavelet theory.To obtain our results we analyzed the properties of the spectral density function and the associated Fourier series.
基金isupported by the National Natural Science Foundation of China(Grant Nos.11871010 and 11971040)the Fundamental Research Funds for the Central Universities(Grant No.2019XD-A11)The work of Weilin Xiao is supported by the Humanities and Social Sciences of Ministry of Education Planning Fund of China(Grant No.23YJA630102).
文摘This study considers an optimal investment and reinsurance problem involving a defaultable security for an insurer in an ambiguous environment.In other words,the insurer is ambiguous about the insurance claim that is exponentially distributed with an uncertain rate parameter.The insurer can purchase proportional reinsurance and invest its wealth in three assets:a risk-free asset,a risky asset,the price process of which satisfies the Heston local-stochastic volatility model,and a defaultable corporate bond.For the optimal investment–reinsurance objective with a smooth ambiguity utility proposed by Klibanoff,P.,Marinacci,M.,and Mukerji,S.[A smooth model of decision making under ambiguity,Econometrica,2005,73(6):1849-1892],the equilibrium strategy is introduced and the extended Hamilton–Jacobi–Bellman equation is established through a stochastic control approach.However,the analytical solution of the strategy under the Heston local-stochastic volatility model cannot be obtained because of the complicated nonlinearity of the partial differential equation.In this study,we employ a perturbation method to derive an asymptotic solution for the post-and pre-default cases.In addition,we present a sensitivity analysis to explain the impact of model parameters on the equilibrium investment–reinsurance strategy.
文摘Certain Merton type consumption−investment problems under partial information are reduced to the ones of full information and within the framework of a complete market model.Then,specializing to conditionally log−Gaussian diffusion models,concrete analysis about the optimal values and optimal strategies is performed by using analytical tools like Feynman−Kac formula,or HJB equations.The explicit solutions to the related forward-backward equations are also given.
基金supported by the Research Fund of Indiana University.
文摘This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries.The fluid is driven by the noisy movement of the bottom boundary,where the noise is given by a Lévy process.After establishing existence of a martingale solution,we use the background flow method to derive an upper bound on the turbulent energy dissipation rate.Our estimate recovers one of the basic scaling ideas of turbulence theory,namely,that the dissipation rate is independent of the viscosity at high Reynolds number.
文摘In this study,we delve into the optimal stopping problem by examining the p(ϕ(τ),τ∈T_(0)^(p))case in which the reward is given by a family of nonnegative random variables indexed by predictable stopping times.We aim to elucidate various properties of the value function family within this context.We prove the existence of an optimal predictable stopping time,subject to specific assumptions regarding the reward functionϕ.
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.