In this paper,the distributed stochastic model predictive control(MPC)is proposed for the noncooperative game problem of the discrete-time multi-player systems(MPSs)with the undirected Markov jump graph.To reflect the...In this paper,the distributed stochastic model predictive control(MPC)is proposed for the noncooperative game problem of the discrete-time multi-player systems(MPSs)with the undirected Markov jump graph.To reflect the reality,the state and input constraints have been considered along with the external disturbances.An iterative algorithm is designed such that model predictive noncooperative game could converge to the socalledε-Nash equilibrium in a distributed manner.Sufficient conditions are established to guarantee the convergence of the proposed algorithm.In addition,a set of easy-to-check conditions are provided to ensure the mean-square uniform bounded stability of the underlying MPSs.Finally,a numerical example on a group of spacecrafts is studied to verify the effectiveness of the proposed method.展开更多
This paper presents a mechanical model of jumping robot based on the biological mechanism analysis of frog. By biological observation and kinematic analysis the frog jump is divided into take-offphase, aerial phase an...This paper presents a mechanical model of jumping robot based on the biological mechanism analysis of frog. By biological observation and kinematic analysis the frog jump is divided into take-offphase, aerial phase and landing phase. We find the similar trajectories of hindlimb joints during jump, the important effect of foot during take-off and the role of forelimb in supporting the body. Based on the observation, the frog jump is simplified and a mechanical model is put forward. The robot leg is represented by a 4-bar spring/linkage mechanism model, which has three Degrees of Freedom (DOF) at hip joint and one DOF (passive) at tarsometatarsal joint on the foot. The shoulder and elbow joints each has one DOF for the balancing function of arm. The ground reaction force of the model is analyzed and compared with that of frog during take-off. The results show that the model has the same advantages of low likelihood of premature lift-off and high efficiency as the frog. Analysis results and the model can be employed to develop and control a robot capable of mimicking the jumping behavior of frog.展开更多
Wavelets are applied to detect the jumps in a heteroscedastic regression model. It is shown that the wavelet coefficients of the data have significantly large absolute values across fine scale levels near the jump poi...Wavelets are applied to detect the jumps in a heteroscedastic regression model. It is shown that the wavelet coefficients of the data have significantly large absolute values across fine scale levels near the jump points. Then a procedure is developed to estimate the jumps and jump heights. All estimators are proved to be consistent.展开更多
Wavelets are applied to detection of the jump points of a regression function in nonlinear autoregressive model x(t) = T(x(t-1)) + epsilon t. By checking the empirical wavelet coefficients of the data,which have signi...Wavelets are applied to detection of the jump points of a regression function in nonlinear autoregressive model x(t) = T(x(t-1)) + epsilon t. By checking the empirical wavelet coefficients of the data,which have significantly large absolute values across fine scale levels, the number of the jump points and locations where the jumps occur are estimated. The jump heights are also estimated. All estimators are shown to be consistent. Wavelet method ia also applied to the threshold AR(1) model(TAR(1)). The simple estimators of the thresholds are given,which are shown to be consistent.展开更多
This paper presents a modified half-sine-squared load model of the jumping impulses for a single person. The model is based on a database of 22,921 experimentally measured single jumping load cycles from 100 test subj...This paper presents a modified half-sine-squared load model of the jumping impulses for a single person. The model is based on a database of 22,921 experimentally measured single jumping load cycles from 100 test subjects. Threedimensional motion capture technology in conjunction with force plates was employed in the experiment to record jumping loads. The variation range and probability distribution of the controlling parameters for the load model such as the impact factor, jumping frequency and contact ratio, are discussed using the experimental data. Correlation relationships between the three parameters are investigated. The contact ratio and jumping frequency are identified as independent model parameters, and an empirical frequency-dependent function is derived for the impact factor. The feasibility of the proposed load model is established by comparing the simulated load curves with measured ones, and by comparing the acceleration responses of a single-degree-of-freedom system to the simulated and measured jumping loads. The results show that a realistic individual jumping load can be generated by the proposed method. This can then be used to assess the dynamic response of assembly structures.展开更多
This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geomet...This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.展开更多
This paper is concerned with the distributed model predictive control(MPC) problem for a class of discrete-time Markovian jump linear systems(MJLSs) subject to actuator saturation and polytopic uncertainty in system m...This paper is concerned with the distributed model predictive control(MPC) problem for a class of discrete-time Markovian jump linear systems(MJLSs) subject to actuator saturation and polytopic uncertainty in system matrices. The global system is decomposed into several subsystems which coordinate with each other. A set of distributed controllers is designed by solving a min-max optimization problem in terms of the solutions of linear matrix inequalities(LMIs). An iterative algorithm is developed to achieve the online computation. Finally,a simulation example is employed to show the effectiveness of the proposed algorithm.展开更多
Using Fourier inversion transform, P.D.E. and Feynman-Kac formula, the closedform solution for price on European call option is given in a double exponential jump-diffusion model with two different market structure ri...Using Fourier inversion transform, P.D.E. and Feynman-Kac formula, the closedform solution for price on European call option is given in a double exponential jump-diffusion model with two different market structure risks that there exist CIR stochastic volatility of stock return and Vasicek or CIR stochastic interest rate in the market. In the end, the result of the model in the paper is compared with those in other models, including BS model with numerical experiment. These results show that the double exponential jump-diffusion model with CIR-market structure risks is suitable for modelling the real-market changes and very useful.展开更多
This paper proposes and makes a study of a new model(called the 3/2 plus jumps model) for VIX option pricing. The model allows the mean-reversion speed and volatility of volatility to be highly sensitive to the actual...This paper proposes and makes a study of a new model(called the 3/2 plus jumps model) for VIX option pricing. The model allows the mean-reversion speed and volatility of volatility to be highly sensitive to the actual level of VIX. In particular, the positive volatility skew is addressed by the 3/2 plus jumps model. Daily calibration is used to prove that the proposed model preserves its validity and reliability for both in-sample and out-of-sample tests.The results show that the models are capable of fitting the market price while generating positive volatility skew.展开更多
This paper studies the critical exercise price of American floating strike lookback options under the mixed jump-diffusion model. By using It formula and Wick-It-Skorohod integral, a new market pricing model estab...This paper studies the critical exercise price of American floating strike lookback options under the mixed jump-diffusion model. By using It formula and Wick-It-Skorohod integral, a new market pricing model established under the environment of mixed jumpdiffusion fractional Brownian motion. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given, then the American floating strike lookback options factorization formula is obtained, the results is generalized the classical Black-Scholes market pricing model.展开更多
As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimen...As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimensional integral, numerical calculation is time-consuming. In the current studies, some scholars just obtained theoretical derivation, or gave some simulation calculations. Others impose underlying assets on some strong assumptions, for example, a lot of calculations are based on the Black-Scholes model. This thesis considers Merton jump diffusion model as the basic model to derive the pricing formula of discrete double barrier option;numerical calculation method is used to approximate the continuous convolution by calculating discrete convolution. Then we compare the results of theoretical calculation with simulation results by Monte Carlo method, to verify their efficiency and accuracy. By comparing the results of degeneration constant parameter model with the results of previous models we verified the calculation method is correct indirectly. Compared with the Monte Carlo simulation method, the numerical results are stable. Even if we assume the simulation results are accurate, the time consumed by the numerical method to achieve the same accuracy is much less than the Monte Carlo simulation method.展开更多
A stochastic susceptible-infective-recovered(SIR)epidemic model with jumps was considered.The contributions of this paper are as follows.(1) The stochastic differential equation(SDE)associated with the model has a uni...A stochastic susceptible-infective-recovered(SIR)epidemic model with jumps was considered.The contributions of this paper are as follows.(1) The stochastic differential equation(SDE)associated with the model has a unique global positive solution;(2) the results reveal that the solution of this epidemic model will be stochastically ultimately bounded,and the non-linear SDE admits a unique stationary distribution under certain parametric conditions;(3) the coefficients play an important role in the extinction of the diseases.展开更多
This paper considers the pricing problem of collateralized debt obligations tranches under a structural jump-diffusion model, where the asset value of each reference entity is generated by a geometric Brownian motion ...This paper considers the pricing problem of collateralized debt obligations tranches under a structural jump-diffusion model, where the asset value of each reference entity is generated by a geometric Brownian motion and jump with an asymmetric double exponential distribution. Conditioned on the common factor of individual entity, this paper gets the conditional distribution, and further obtains the loss distribution of the whole reference portfolio. Based on the semi-analytic approach, the fair spreads of collateralized debt obligations tranches, i.e., the prices of collateralized debt obligations tranches, are derived.展开更多
In order to characterize large fluctuations of the financial markets and optimize financial portfolio, a new dynamic asset control strategy was proposed in this work. Firstly, a random process item with variable jump ...In order to characterize large fluctuations of the financial markets and optimize financial portfolio, a new dynamic asset control strategy was proposed in this work. Firstly, a random process item with variable jump intensity was introduced to the existing discrete microstructure model to denote large price fluctuations. The nonparametric method of LEE was used for detecting jumps. Further, the extended Kalman filter and the maximum likelihood method were applied to discrete microstructure modeling and the estimation of two market potential variables: market excess demand and liquidity. At last, based on the estimated variables, an assets allocation strategy using evolutionary algorithm was designed to control the weight of each asset dynamically. Case studies on IBM Stock show that jumps with variable intensity are detected successfully, and the assets allocation strategy may effectively keep the total assets growth or prevent assets loss at the stochastic financial market.展开更多
Quasi-elastic neutron scattering(QENS) has many applications that are directly related to the development of highperformance functional materials and biological macromolecules, especially those containing some water. ...Quasi-elastic neutron scattering(QENS) has many applications that are directly related to the development of highperformance functional materials and biological macromolecules, especially those containing some water. The analysis method of QENS spectra data is important to obtain parameters that can explain the structure of materials and the dynamics of water. In this paper, we present a revised jump-diffusion and rotation-diffusion model(rJRM) used for QENS spectra data analysis. By the rJRM, the QENS spectra from a pure magnesium-silicate-hydrate(MSH) sample are fitted well for the Q range from 0.3 ^(-1) to 1.9 ^(-1) and temperatures from 210 K up to 280 K. The fitted parameters can be divided into two kinds. The first kind describes the structure of the MSH sample, including the ratio of immobile water(or bound water) C and the confining radius of mobile water a_0. The second kind describes the dynamics of confined water in pores contained in the MSH sample, including the translational diffusion coefficient Dt, the average translational residence timeτ0, the rotational diffusion coefficient D_r, and the mean squared displacement(MSD) u^2. The r JRM is a new practical method suitable to fit QENS spectra from porous materials, where hydrogen atoms appear in both solid and liquid phases.展开更多
This article discusses the kinematics of a parachutist making a very-high-altitude jump. The effect of altitude on the density of air, on the gravitational field strength of the Earth, and on the atmosphere’s tempera...This article discusses the kinematics of a parachutist making a very-high-altitude jump. The effect of altitude on the density of air, on the gravitational field strength of the Earth, and on the atmosphere’s temperature has been taken into account in our analysis. The well-known equations of classical mechanics governing the selected topic have been solved numerically by using the mathematical software Mathcad. Especially, the possibility of a person exceeding the speed of sound during their fall has been considered in our analysis. The effect of the sound barrier is taken into account so that the shape factor of the falling body is given as a speed-dependent function, which reaches its maximum value at Mach 1.0. The obtained results have been found to be highly consistent with the available experimental data on some high-altitude jumps. The data published on the famous jump of Captain Joseph Kittinger has been analyzed very carefully, and although our calculations reproduced the reported values for most parts, some interesting inconsistencies were also discovered. Kittinger jumped from a gondola attached to a helium-filled balloon from a record-high altitude of 102,800 ft, or 31,330 m, in August 1960. We also made numerical analysis on the high-altitude jump of Felix Baumgartner. He bailed out from his gondola at the record-high altitude of 39.0 km in October 2012.展开更多
基金This work was supported by the National Natural Science Foundation of China(62122063,62073268,U22B2036,11931015)the Young Star of Science and Technology in Shaanxi Province(2020KJXX-078)+1 种基金the National Science Fund for Distinguished Young Scholars(62025602)the XPLORER PRIZE。
文摘In this paper,the distributed stochastic model predictive control(MPC)is proposed for the noncooperative game problem of the discrete-time multi-player systems(MPSs)with the undirected Markov jump graph.To reflect the reality,the state and input constraints have been considered along with the external disturbances.An iterative algorithm is designed such that model predictive noncooperative game could converge to the socalledε-Nash equilibrium in a distributed manner.Sufficient conditions are established to guarantee the convergence of the proposed algorithm.In addition,a set of easy-to-check conditions are provided to ensure the mean-square uniform bounded stability of the underlying MPSs.Finally,a numerical example on a group of spacecrafts is studied to verify the effectiveness of the proposed method.
基金the National High Technology Research and Development Program of China (No.2006AA04Z245)Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (IRT0423)
文摘This paper presents a mechanical model of jumping robot based on the biological mechanism analysis of frog. By biological observation and kinematic analysis the frog jump is divided into take-offphase, aerial phase and landing phase. We find the similar trajectories of hindlimb joints during jump, the important effect of foot during take-off and the role of forelimb in supporting the body. Based on the observation, the frog jump is simplified and a mechanical model is put forward. The robot leg is represented by a 4-bar spring/linkage mechanism model, which has three Degrees of Freedom (DOF) at hip joint and one DOF (passive) at tarsometatarsal joint on the foot. The shoulder and elbow joints each has one DOF for the balancing function of arm. The ground reaction force of the model is analyzed and compared with that of frog during take-off. The results show that the model has the same advantages of low likelihood of premature lift-off and high efficiency as the frog. Analysis results and the model can be employed to develop and control a robot capable of mimicking the jumping behavior of frog.
文摘Wavelets are applied to detect the jumps in a heteroscedastic regression model. It is shown that the wavelet coefficients of the data have significantly large absolute values across fine scale levels near the jump points. Then a procedure is developed to estimate the jumps and jump heights. All estimators are proved to be consistent.
文摘Wavelets are applied to detection of the jump points of a regression function in nonlinear autoregressive model x(t) = T(x(t-1)) + epsilon t. By checking the empirical wavelet coefficients of the data,which have significantly large absolute values across fine scale levels, the number of the jump points and locations where the jumps occur are estimated. The jump heights are also estimated. All estimators are shown to be consistent. Wavelet method ia also applied to the threshold AR(1) model(TAR(1)). The simple estimators of the thresholds are given,which are shown to be consistent.
基金the National Natural Science Foundation of China under Grant Nos.51178338 and 51478346State Key Laboratory of Disaster Reduction in Civil Engineering under Grant No.SLDRCE14-B-16
文摘This paper presents a modified half-sine-squared load model of the jumping impulses for a single person. The model is based on a database of 22,921 experimentally measured single jumping load cycles from 100 test subjects. Threedimensional motion capture technology in conjunction with force plates was employed in the experiment to record jumping loads. The variation range and probability distribution of the controlling parameters for the load model such as the impact factor, jumping frequency and contact ratio, are discussed using the experimental data. Correlation relationships between the three parameters are investigated. The contact ratio and jumping frequency are identified as independent model parameters, and an empirical frequency-dependent function is derived for the impact factor. The feasibility of the proposed load model is established by comparing the simulated load curves with measured ones, and by comparing the acceleration responses of a single-degree-of-freedom system to the simulated and measured jumping loads. The results show that a realistic individual jumping load can be generated by the proposed method. This can then be used to assess the dynamic response of assembly structures.
基金Supported by The National Natural Science Foundation of China(71261015)Humanity and Social Science Youth Foundation of Education Ministry in China(10YJC630334)Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region
文摘This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.
基金supported by National Natural Science Foundation of China(61403254,61374039,61203143)Shanghai Pujiang Program(13PJ1406300)+2 种基金Natural Science Foundation of Shanghai City(13ZR1428500)Innovation Program of Shanghai Municipal Education Commission(14YZ083)Hujiang Foundation of China(C14002,B1402/D1402)
文摘This paper is concerned with the distributed model predictive control(MPC) problem for a class of discrete-time Markovian jump linear systems(MJLSs) subject to actuator saturation and polytopic uncertainty in system matrices. The global system is decomposed into several subsystems which coordinate with each other. A set of distributed controllers is designed by solving a min-max optimization problem in terms of the solutions of linear matrix inequalities(LMIs). An iterative algorithm is developed to achieve the online computation. Finally,a simulation example is employed to show the effectiveness of the proposed algorithm.
基金Supported by the NNSF of China(40675023)the PHD Foundation of Guangxi Normal University.
文摘Using Fourier inversion transform, P.D.E. and Feynman-Kac formula, the closedform solution for price on European call option is given in a double exponential jump-diffusion model with two different market structure risks that there exist CIR stochastic volatility of stock return and Vasicek or CIR stochastic interest rate in the market. In the end, the result of the model in the paper is compared with those in other models, including BS model with numerical experiment. These results show that the double exponential jump-diffusion model with CIR-market structure risks is suitable for modelling the real-market changes and very useful.
基金Supported by the National Natural Science Foundation of China(71371168,11571310)
文摘This paper proposes and makes a study of a new model(called the 3/2 plus jumps model) for VIX option pricing. The model allows the mean-reversion speed and volatility of volatility to be highly sensitive to the actual level of VIX. In particular, the positive volatility skew is addressed by the 3/2 plus jumps model. Daily calibration is used to prove that the proposed model preserves its validity and reliability for both in-sample and out-of-sample tests.The results show that the models are capable of fitting the market price while generating positive volatility skew.
基金Supported by the Fundamental Research Funds of Lanzhou University of Finance and Economics(Lzufe2017C-09)
文摘This paper studies the critical exercise price of American floating strike lookback options under the mixed jump-diffusion model. By using It formula and Wick-It-Skorohod integral, a new market pricing model established under the environment of mixed jumpdiffusion fractional Brownian motion. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given, then the American floating strike lookback options factorization formula is obtained, the results is generalized the classical Black-Scholes market pricing model.
文摘As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimensional integral, numerical calculation is time-consuming. In the current studies, some scholars just obtained theoretical derivation, or gave some simulation calculations. Others impose underlying assets on some strong assumptions, for example, a lot of calculations are based on the Black-Scholes model. This thesis considers Merton jump diffusion model as the basic model to derive the pricing formula of discrete double barrier option;numerical calculation method is used to approximate the continuous convolution by calculating discrete convolution. Then we compare the results of theoretical calculation with simulation results by Monte Carlo method, to verify their efficiency and accuracy. By comparing the results of degeneration constant parameter model with the results of previous models we verified the calculation method is correct indirectly. Compared with the Monte Carlo simulation method, the numerical results are stable. Even if we assume the simulation results are accurate, the time consumed by the numerical method to achieve the same accuracy is much less than the Monte Carlo simulation method.
基金Natural Science Foundation of Hunan University of Technology,China(No.2012HZX08)the Special Foundation of National Independent Innovation Demonstration Area Construction of Zhuzhou(Applied Basic Research),China
文摘A stochastic susceptible-infective-recovered(SIR)epidemic model with jumps was considered.The contributions of this paper are as follows.(1) The stochastic differential equation(SDE)associated with the model has a unique global positive solution;(2) the results reveal that the solution of this epidemic model will be stochastically ultimately bounded,and the non-linear SDE admits a unique stationary distribution under certain parametric conditions;(3) the coefficients play an important role in the extinction of the diseases.
基金Supported by the National Natural Science Foundation of China (70771018)the Natural Science Foundation of Shandong Province (2009ZRB019AV)Mathematical Subject Construction Funds and the Key Laboratory of Financial Information Engineering of Ludong University (2008)
文摘This paper considers the pricing problem of collateralized debt obligations tranches under a structural jump-diffusion model, where the asset value of each reference entity is generated by a geometric Brownian motion and jump with an asymmetric double exponential distribution. Conditioned on the common factor of individual entity, this paper gets the conditional distribution, and further obtains the loss distribution of the whole reference portfolio. Based on the semi-analytic approach, the fair spreads of collateralized debt obligations tranches, i.e., the prices of collateralized debt obligations tranches, are derived.
基金Projects(71271215,71221061) supported by the National Natural Science Foundation of ChinaProject(2011DFA10440) supported by the International Science&Technology Cooperation Program of ChinaProject(CX2012B067) supported by Hunan Provincial Innovation Foundation for Postgraduate,China
文摘In order to characterize large fluctuations of the financial markets and optimize financial portfolio, a new dynamic asset control strategy was proposed in this work. Firstly, a random process item with variable jump intensity was introduced to the existing discrete microstructure model to denote large price fluctuations. The nonparametric method of LEE was used for detecting jumps. Further, the extended Kalman filter and the maximum likelihood method were applied to discrete microstructure modeling and the estimation of two market potential variables: market excess demand and liquidity. At last, based on the estimated variables, an assets allocation strategy using evolutionary algorithm was designed to control the weight of each asset dynamically. Case studies on IBM Stock show that jumps with variable intensity are detected successfully, and the assets allocation strategy may effectively keep the total assets growth or prevent assets loss at the stochastic financial market.
文摘Quasi-elastic neutron scattering(QENS) has many applications that are directly related to the development of highperformance functional materials and biological macromolecules, especially those containing some water. The analysis method of QENS spectra data is important to obtain parameters that can explain the structure of materials and the dynamics of water. In this paper, we present a revised jump-diffusion and rotation-diffusion model(rJRM) used for QENS spectra data analysis. By the rJRM, the QENS spectra from a pure magnesium-silicate-hydrate(MSH) sample are fitted well for the Q range from 0.3 ^(-1) to 1.9 ^(-1) and temperatures from 210 K up to 280 K. The fitted parameters can be divided into two kinds. The first kind describes the structure of the MSH sample, including the ratio of immobile water(or bound water) C and the confining radius of mobile water a_0. The second kind describes the dynamics of confined water in pores contained in the MSH sample, including the translational diffusion coefficient Dt, the average translational residence timeτ0, the rotational diffusion coefficient D_r, and the mean squared displacement(MSD) u^2. The r JRM is a new practical method suitable to fit QENS spectra from porous materials, where hydrogen atoms appear in both solid and liquid phases.
文摘This article discusses the kinematics of a parachutist making a very-high-altitude jump. The effect of altitude on the density of air, on the gravitational field strength of the Earth, and on the atmosphere’s temperature has been taken into account in our analysis. The well-known equations of classical mechanics governing the selected topic have been solved numerically by using the mathematical software Mathcad. Especially, the possibility of a person exceeding the speed of sound during their fall has been considered in our analysis. The effect of the sound barrier is taken into account so that the shape factor of the falling body is given as a speed-dependent function, which reaches its maximum value at Mach 1.0. The obtained results have been found to be highly consistent with the available experimental data on some high-altitude jumps. The data published on the famous jump of Captain Joseph Kittinger has been analyzed very carefully, and although our calculations reproduced the reported values for most parts, some interesting inconsistencies were also discovered. Kittinger jumped from a gondola attached to a helium-filled balloon from a record-high altitude of 102,800 ft, or 31,330 m, in August 1960. We also made numerical analysis on the high-altitude jump of Felix Baumgartner. He bailed out from his gondola at the record-high altitude of 39.0 km in October 2012.