wanyu/IteraTeR_full_sent huggingface dataset:
Original:  The existence and uniqueness theorem for the Hamilton-Jacobi-Bellman (HJB) equation is shown. We construct an iterative sequence of functions converging to the solutionof this equation. An optimal convergence rate for this sequence is found and sharp computable upper bounds for the approximation accuracy of the optimal consumption - investment strategies are obtained.
Revised:  The existence and uniqueness theorem for the Hamilton-Jacobi-Bellman (HJB) equation which is a highly non linear partial differential equation (PDE) of the second oder. By using the Feynman - Kac representation we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate for this sequence is found and sharp computable upper bounds for the approximation accuracy of the optimal consumption - investment strategies are obtained.
Edit intent: meaning-changed
Original:  An optimal convergence rate for this sequence is found and sharp computable upper bounds for the approximation accuracy of the optimal consumption - investment strategies are obtained. It turns out that the optimal  convergence rate in this case is super geometrical, i.e. is more rapid than any geometrical rate .
Revised:  An optimal convergence rate of the iterative numerical schemes for both the value function and the optimal convergence rate in this case is super geometrical, i.e. is more rapid than any geometrical rate .
Edit intent: clarity
Original:  Our results show that a class of rebate barrier options can be used for this approximation , when its rebate and barrier are chosen appropriately.
Revised:  Our results show that a class of rebate barrier options can be used for this approximation . Among of them, a speci?c rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation.
Edit intent: meaning-changed
Original:  For such models, there is typically a wide variation in scales in that the different species and reaction rates vary over several orders of magnitude. Quantifying how different numerical approximation techniques behave in this settingtherefore requires that these scalings be taken into account in an appropriate manner .
Revised:  For such models, there is typically a wide variation in temporal and other quantitative scales. In this multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time. In this paper we provide a general framework for understanding the weak error of numerical approximation techniques behave in this settingtherefore requires that these scalings be taken into account in an appropriate manner .
Edit intent: meaning-changed
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind.
Revised:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind.
Edit intent: meaning-changed
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind.
Revised:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that the leading order of the error process scales with the square of the time discretization, making it the first of its kind.
Edit intent: clarity
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind. In the multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify or reduce a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time .
Revised:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first second order method in this setting .
Edit intent: coherence
Original: A chemical reaction network is a chemical system involving multiple reactions and chemical species. The simpleststochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain.
Revised: The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain.
Edit intent: clarity
Original:  The simpleststochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. For such models , there is typically a wide variation in temporal and other quantitative scales. In this multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify a system. Therefore, numerical methodsoftentimes are the only reasonable means by which such models can be understood in real time .
Revised:  The simpleststochastic models of such networks treat the system as a continuous time Markov chains. Simulation of such models is often relatively straightforward as there are easily implementable methods for the generation of exact sample paths. However, when using ensemble averages to approximate expected values, the computational complexity can become prohibitive as the number of computations per path scales linearly with the number of jumps, or reactions, of the process. When such methods become computationally intractable, approximate methods, which introduce a bias, can become advantageous .
Edit intent: coherence
Original:  In this paper  we provide a general framework for understanding the weak error of numerical approximation techniques in the multi-scale setting. We quantify how the error of three different methods depends upon both the natural scalings within a given system , and with the step-size of the numerical method.
Revised:  In this paper we provide a general framework for understanding the weak error of numerical approximation techniques . The analysis takes into account both the natural scalings within a given system , and with the step-size of the numerical method.
Edit intent: clarity
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting  .
Revised:  Further, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting .
Edit intent: coherence
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting  .
Revised:  Further, we introduce a new algorithm in this setting, the weak trapezoidal method is introduced in the current setting, and is proven to be second order accurate in a weak sense , making it the first second order method in this setting .
Edit intent: meaning-changed
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting  .
Revised:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first higher order method in this setting .
Edit intent: meaning-changed
Original:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting  .
Revised:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting . Examples are provided demonstrating both the main analytical results, and the reduction in computational complexity achieved with the approximate methods .
Edit intent: meaning-changed
Original:  Here, we investigate the dynamics of switching between metastable states of a feedback based on/off switch , by employing the WKB theory to treat the underlying master equations.
Revised:  Here, we investigate the dynamics of switching between metastable states of a feedback based on/off switch . Unlike protein-only models, we explicitly account for stochastic fluctuations of mRNA, which have a dramatic impact on switch dynamics. Employing the WKB theory to treat the underlying master equations.
Edit intent: meaning-changed
Original:  Here, we investigate the dynamics of switching between metastable states of a feedback based on/off switch , by employing the WKB theory to treat the underlying master equations. This technique, applicable for any generic feedback function, yields accurate results for the quasi-stationary distributions of mRNA and protein copy numbers and  mean switching time, starting from either state.
Revised:  Here, we investigate the dynamics of switching between metastable states of a feedback based on/off switch , by employing the WKB theory to treat the underlying chemical master equations, we obtain accurate results for the quasi-stationary distributions of mRNA and protein copy numbers and mean switching time, starting from either state.
Edit intent: meaning-changed
Original:  Importantly, the approach may be used to study the effect of varying biological parameters on the stability of the switch states, and we use it to show that in some cases promoter kinetics, not just thermodynamic stability, can influence switching .
Revised:  Importantly, one can use the approach to study the effect of varying biological parameters on the stability of the switch states, and we use it to show that in some cases promoter kinetics, not just thermodynamic stability, can influence switching .
Edit intent: clarity
Original:  We analyze the stability of the algorithm when the number of exercise dates increases and prove that  if the underlying process for the stock price is continuous  then the regression problem is ill-conditioned for small values of parameter t, time .
Revised:  We analyze the stability of the algorithm when the number of exercise dates increases and prove that , if the underlying process for the stock price is continuous then the regression problem is ill-conditioned for small values of parameter t, time .
Edit intent: fluency
Original:  We analyze the stability of the algorithm when the number of exercise dates increases and prove that  if the underlying process for the stock price is continuous  then the regression problem is ill-conditioned for small values of parameter t, time .
Revised:  We analyze the stability of the algorithm when the number of exercise dates increases and prove that if the underlying process for the stock price is continuous , then the regression problem is ill-conditioned for small values of parameter t, time .
Edit intent: fluency
Original:  We analyze the stability of the algorithm when the number of exercise dates increases and prove that  if the underlying process for the stock price is continuous  then the regression problem is ill-conditioned for small values of parameter t, time .
Revised:  We analyze the stability of the algorithm when the number of exercise dates increases and prove that if the underlying process for the stock price is continuous then the regression problem is ill-conditioned for small values of the time parameter .
Edit intent: clarity
Original:  With this service framework , we develop a guaranteed QoS model, such as the upper bounds on buffer queue length , buffer queue delay, end-to-end delay/jitter and end-to-end effective bandwidth.
Revised:  With the system skeleton , we develop a guaranteed QoS model, such as the upper bounds on buffer queue length , buffer queue delay, end-to-end delay/jitter and end-to-end effective bandwidth.
Edit intent: clarity
Original:  With this service framework , we develop a guaranteed QoS model, such as the upper bounds on buffer queue length , buffer queue delay, end-to-end delay/jitter and end-to-end effective bandwidth.
Revised:  With this service framework , we develop a guaranteed QoS model, such as the upper bounds on buffer queue length /delay/effective bandwidth, and single-hop/ multi-hops delay/jitter and end-to-end effective bandwidth.
Edit intent: meaning-changed
Original:  With this service framework , we develop a guaranteed QoS model, such as the upper bounds on buffer queue length , buffer queue delay, end-to-end delay/jitter and end-to-end effective bandwidth.
Revised:  With this service framework , we develop a guaranteed QoS model, such as the upper bounds on buffer queue length , buffer queue delay, end-to-end delay/jitter / effective bandwidth.
Edit intent: fluency
Original:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Revised:  Numerical results show the system skeleton and the guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Edit intent: clarity
Original:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Revised:  Numerical results show that the service framework and guaranteed QoS model are scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Edit intent: clarity
Original:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Revised:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Edit intent: fluency
Original:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Revised:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar traffic flows, and they are only affected by the parameters of service curves and flow regulators .
Edit intent: clarity
Original:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Revised:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and the parameters of service curves and flow regulators .
Edit intent: coherence
Original:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of service curves and flow regulators .
Revised:  Numerical results show that the service framework and guaranteed QoS model are highly scalable for different types of input flows, including the self-similar input flows, and they are only affected by the parameters of flow regulators and service curves of sensor nodes affect them .
Edit intent: meaning-changed
Original:  Motivated by a recent experiment reporting chiral transitions between right-handed B-DNA and left-handed Z-DNA  ,   we analyze the free energy for the particular case of linear DNA with an externally applied torque.
Revised:  Motivated by a recent experiment reporting chiral transitions between right-handed B-DNA and left-handed Z-DNA M. Lee, et. al., Proc. Natl. Acad. Sci. (USA) 107 , we analyze the free energy for the particular case of linear DNA with an externally applied torque.
Edit intent: meaning-changed
Original:  Motivated by a recent experiment reporting chiral transitions between right-handed B-DNA and left-handed Z-DNA  ,   we analyze the free energy for the particular case of linear DNA with an externally applied torque.
Revised:  Motivated by a recent experiment reporting chiral transitions between right-handed B-DNA and left-handed Z-DNA , 4985 (2010) we analyze the free energy for the particular case of linear DNA with an externally applied torque.
Edit intent: meaning-changed
Original:  Motivated by a recent experiment reporting chiral transitions between right-handed B-DNA and left-handed Z-DNA  ,   we analyze the free energy for the particular case of linear DNA with an externally applied torque.
Revised:  Motivated by a recent experiment reporting chiral transitions between right-handed B-DNA and left-handed Z-DNA , , we analyze the free energy for the particular case of linear DNA with an externally applied torque.
Edit intent: fluency
Original:  This measure allows comparing different products and models (pricing hypothesis) under a homogeneous framework and conclude which one is the best. The model risk measure is defined in terms of the expected value and standard deviation of the loss given by the hedging strategy at a given time horizon .
Revised:  This measure allows comparing different products and models (pricing hypothesis) under a homogeneous framework which allows concluding which model is the closest to the reference. The relative model risk measure is defined in terms of the expected value and standard deviation of the loss given by the hedging strategy at a given time horizon .
Edit intent: meaning-changed
Original:  The model risk measure is defined in terms of the expected value and standard deviation of the loss given by the hedging strategy at a given time horizon . It has been assumed that the market volatility surface is driven by Heston 's dynamics calibrated to market for a given time horizon  .
Revised:  The model risk measure is defined in terms of the expected value and standard deviation of the loss given by the hedging strategy at a given time horizon for a chosen significance level. The reference model has been chosen to be Heston calibrated to market for a given time horizon .
Edit intent: meaning-changed
Original: We consider a structural model for the estimation of credit risk based on Merton's original model . By using Random Matrix Theory we demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the correlations are not identically zero.
Revised: We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. This ensemble is set up by Random Matrix Theory we demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the correlations are not identically zero.
Edit intent: meaning-changed
Original: The risk minimizing problem E[l((H-X_T^{x,\pi})^{+})]\pi{\longrightarrow}\min in the  Black-Scholes framework with correlation is studied.
Revised: The risk minimizing problem E[l((H-X_T^{x,\pi})^{+})]\pi{\longrightarrow}\min in the multidimensional Black-Scholes framework with correlation is studied.
Edit intent: meaning-changed
Original: The risk minimizing problem E[l((H-X_T^{x,\pi})^{+})]\pi{\longrightarrow}\min in the  Black-Scholes framework with correlation is studied.
Revised: The risk minimizing problem E[l((H-X_T^{x,\pi})^{+})]\pi{\longrightarrow}\min in the Black-Scholes framework is studied.
Edit intent: coherence
Original:  General formulas for the minimal risk function and the cost reduction function for the option H depending on multiple underlying are derived. The case of a linear and a strictly convex loss function l are examined. Explicit computation for l(x)=x and l(x)=x^p, with p>1 for digital, quantos, outperformance and spread options are presented.
Revised:  General formulas for the minimal risk function and the cost reduction function for basket derivatives are shown. Explicit integral representations for the risk functions for l(x)=x and l(x)=x^p, with p>1 for digital, quantos, outperformance and spread options are presented.
Edit intent: clarity
Original:  Explicit computation for l(x)=x and l(x)=x^p, with p>1 for digital, quantos, outperformance and spread options are presented. The method is based on the quantile hedging approach presented in \mbox{%DIFAUXCMD FL10pt%DIFAUXCMD , \mbox{%DIFAUXCMD FL2 }0pt%DIFAUXCMD and developed for the multidimensional options in \mbox{%DIFAUXCMD Barski}0pt%DIFAUXCMD }  .
Revised:  Explicit computation for l(x)=x and l(x)=x^p, with p>1 for digital, quantos, outperformance and spread options are 0pt%DIFAUXCMD , \mbox{%DIFAUXCMD FL2 }0pt%DIFAUXCMD and developed for the multidimensional options in \mbox{%DIFAUXCMD Barski}0pt%DIFAUXCMD } .
Edit intent: clarity