We study linear stochastic evolution equations driven by various infinite-dimensional Gaussian pr... more We study linear stochastic evolution equations driven by various infinite-dimensional Gaussian processes, some of which are more irregular in time than fractional Brownian motion (fBm) with any Hurst parameter H, while others are comparable to fBm with H < 1 2 . Sharp necessary and sufficient conditions for the existence and uniqueness of solutions are presented. Specializing to stochastic heat equations on compact manifolds, especially on the unit circle, sharp Gaussian regularity results are used to determine sufficient conditions for a given fixed function to be an almost-sure modulus of continuity for the solution in space; these sufficient conditions are also proved necessary in highly irregular cases, and are nearly necessary (logarithmic corrections are given) in other cases, including the Hölder scale.
We consider the class of self-similar Gaussian stochastic volatility models, and compute the smal... more We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter H, its first Karhunen-Lo\`{e}ve eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for H result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1/2 and 3/2, and the estimator for H sees...
This paper considers the class of stochastic processes X defined on [0, T ] by X (t) = T 0 G (t, ... more This paper considers the class of stochastic processes X defined on [0, T ] by X (t) = T 0 G (t, s) dM (s) where M is a square-integrable martingale and G is a deterministic kernel. When M is Brownian motion, X is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let m be an odd integer. Under the asumption that the quadratic variation
Stochastics An International Journal of Probability and Stochastic Processes, 1998
We study the almost-sure large time exponential growth of the solution to a linear stochastic par... more We study the almost-sure large time exponential growth of the solution to a linear stochastic parabolic equation with continuous space parameter, and a Gaussian-correlated potential that is white noise in time and homogeneous in space. We use the evolution form of that equation,for which existence and uniqueness are known. We establish a Feynman-Kac formula for the solution. By using a method of discretization of time and space, we prove that for small diffusion parameter κ, there is a deterministic constant c such that almost surely
ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anom... more ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anomalies back to 1000AD, based on a model that includes external climate forcings and accounts for any long-memory features. Our reconstruction is based on two linear models, with the first linking the latent temperature series to three main external forcings (solar irradiance, greenhouse gas concentration, and volcanism), and the second linking the observed temperature proxy data (tree rings, sediment record, ice cores, etc.) to the unobserved temperature series. Uncertainty is captured with additive noise, and a rigorous statistical investigation of the correlation structure in the regression errors motivates the use of long memory fractional Gaussian noise models for the error terms. We use Bayesian estimation to fit the model parameters and to perform separate reconstructions of land-only and combined land-and-marine temperature anomalies. We quantify the effects of including the forcings and long memory models on the quality of model fits, and find that long memory models can result in more precise uncertainty quantification, while the external climate forcings substantially reduce the squared bias and variance of the reconstructions. Finally, we use posterior samples of model parameters to arrive at an estimate of the transient climate response to greenhouse gas forcings of $2.56^{\circ}$C (95% credible interval of [$2.20$, $2.95$]$^{\circ}$C), in line with previous, climate-model-based estimates.
We consider the parameter estimation problem for the Ornstein-Uhlenbeck process X driven by a fra... more We consider the parameter estimation problem for the Ornstein-Uhlenbeck process X driven by a fractional Ornstein-Uhlenbeck process V , i.e. the pair of processes defined by the non-Markovian continuous-time long-memory dynamics dX t = −θX t dt + dV t ; t 0, with dV t = −ρV t dt + dB H t ; t 0, where θ > 0 and ρ > 0 are unknown parameters, and B H is a fractional Brownian motion of Hurst index H ∈ ( 1 2 , 1). We study the strong consistency as well as the asymptotic normality of the joint least squares estimator θ T , ρ T of the pair (θ, ρ), based either on continuous or discrete observations of {X s ; s ∈ [0, T ]} as the horizon T increases to +∞. Both cases qualify formally as partial-hbobservation questions since V is unobserved. In the latter case, several discretization options are considered. Our proofs of asymptotic normality based on discrete data, rely on increasingly strict restrictions on the sampling frequency as one reduces the extent of sources of observation. The strategy for proving the asymptotic properties is to study the case of continuous-time observations using the Malliavin calculus, and then to exploit the fact that each discrete-data estimator can be considered as a perturbation of the continuous one in a mathematically precise way, despite the fact that the implementation of the discrete-time estimators is distant from the continuous estimator. In this sense, we contend that the continuous-time estimator cannot be implemented in practice in any naïve way, and serves only as a mathematical tool in the study of the discrete-time estimators' asymptotics.
In this paper, we study central and non-central limit theorems for partial sum of functionals of ... more In this paper, we study central and non-central limit theorems for partial sum of functionals of general stationary Gaussian fields. We apply our result to study drift parameter estimation problems for some stochastic differential equations related to stationary Gaussian processes.
We consider a stochastic volatility stock price model in which the volatility is a non-centered c... more We consider a stochastic volatility stock price model in which the volatility is a non-centered continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the stock price density and implied volatility in these models in the limit of large or small strikes. Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility, based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the L 2 norm of the projection of the mean function on the top eigenspace. Strategies for using this expansion for calibration purposes are discussed.
ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anom... more ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anomalies back to 1000AD, based on a model that includes external climate forcings and accounts for any long-memory features. Our reconstruction is based on two linear models, with the first linking the latent temperature series to three main external forcings (solar irradiance, greenhouse gas concentration, and volcanism), and the second linking the observed temperature proxy data (tree rings, sediment record, ice cores, etc.) to the unobserved temperature series. Uncertainty is captured with additive noise, and a rigorous statistical investigation of the correlation structure in the regression errors motivates the use of long memory fractional Gaussian noise models for the error terms. We use Bayesian estimation to fit the model parameters and to perform separate reconstructions of land-only and combined land-and-marine temperature anomalies. We quantify the effects of including the forcings and long memory models on the quality of model fits, and find that long memory models can result in more precise uncertainty quantification, while the external climate forcings substantially reduce the squared bias and variance of the reconstructions. Finally, we use posterior samples of model parameters to arrive at an estimate of the transient climate response to greenhouse gas forcings of $2.56^{\circ}$C (95% credible interval of [$2.20$, $2.95$]$^{\circ}$C), in line with previous, climate-model-based estimates.
ABSTRACT We investigate optimal portfolio selection problems with mispricing and model ambiguity ... more ABSTRACT We investigate optimal portfolio selection problems with mispricing and model ambiguity under a financial market which contains a pair of mispriced stocks. We assume that the dynamics of the pair satisfies a “cointegrated system” advanced by Liu and Timmermann in a 2013 manuscript. The investor hopes to exploit the temporary mispricing by using a portfolio strategy under a utility function framework. Furthermore, she is ambiguity-averse and has a specific preference for model ambiguity robustness. The explicit solution for such a robust optimal strategy, and its value function, are derived. We analyze these robust strategies with mispricing in two cases: observed and unobserved mean-reverting stochastic risk premium. We show that the mispricing and model ambiguity have completely distinct impacts on the robust optimal portfolio selection, by comparing the utility losses. We also find that the ambiguity-averse investor who ignores the mispricing or the model ambiguity, suffers a substantially larger utility loss if the risk premium is unobserved, compared to when it is observed.
We study linear stochastic evolution equations driven by various inflnite-dimensional Gaussian pr... more We study linear stochastic evolution equations driven by various inflnite-dimensional Gaussian processes, some of which are more irregular in time than fractional Brownian motion (fBm) with any Hurst parameter H, while others are comparable to fBm with H < 1 2 . Sharp necessary and su-- cient conditions for the existence and uniqueness of solutions are presented. Specializing to stochastic heat equations on compact manifolds, especially on the unit circle, sharp Gaussian regularity results are used to determine suf- flcient conditions for a given flxed function to be an almost-sure modulus of continuity for the solution in space; these su-cient conditions are also proved necessary in highly irregular cases, and are nearly necessary (logarithmic cor- rections are given) in other cases, including the Holder scale.
We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can... more We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.
Discrete and Continuous Dynamical Systems - Series B, 2006
We study the time-regularity of the paths of solutions to stochastic partial differential equatio... more We study the time-regularity of the paths of solutions to stochastic partial differential equations (SPDE) driven by additive infinite-dimensional fractional Brownian noise. Sharp sufficient conditions for almost-sure Hölder continuity, and other, more irregular levels of uniform continuity, are given when the space parameter is fixed. Additionally, a result is included on time-continuity when the solution is understood as a spatially Hölder-continuous-function-valued stochastic process. Tools used for the study include the Brownian representation of fractional Brownian motion, and sharp Gaussian regularity results.
We study linear stochastic evolution equations driven by various infinite-dimensional Gaussian pr... more We study linear stochastic evolution equations driven by various infinite-dimensional Gaussian processes, some of which are more irregular in time than fractional Brownian motion (fBm) with any Hurst parameter H, while others are comparable to fBm with H < 1 2 . Sharp necessary and sufficient conditions for the existence and uniqueness of solutions are presented. Specializing to stochastic heat equations on compact manifolds, especially on the unit circle, sharp Gaussian regularity results are used to determine sufficient conditions for a given fixed function to be an almost-sure modulus of continuity for the solution in space; these sufficient conditions are also proved necessary in highly irregular cases, and are nearly necessary (logarithmic corrections are given) in other cases, including the Hölder scale.
We consider the class of self-similar Gaussian stochastic volatility models, and compute the smal... more We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter H, its first Karhunen-Lo\`{e}ve eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for H result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1/2 and 3/2, and the estimator for H sees...
This paper considers the class of stochastic processes X defined on [0, T ] by X (t) = T 0 G (t, ... more This paper considers the class of stochastic processes X defined on [0, T ] by X (t) = T 0 G (t, s) dM (s) where M is a square-integrable martingale and G is a deterministic kernel. When M is Brownian motion, X is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let m be an odd integer. Under the asumption that the quadratic variation
Stochastics An International Journal of Probability and Stochastic Processes, 1998
We study the almost-sure large time exponential growth of the solution to a linear stochastic par... more We study the almost-sure large time exponential growth of the solution to a linear stochastic parabolic equation with continuous space parameter, and a Gaussian-correlated potential that is white noise in time and homogeneous in space. We use the evolution form of that equation,for which existence and uniqueness are known. We establish a Feynman-Kac formula for the solution. By using a method of discretization of time and space, we prove that for small diffusion parameter κ, there is a deterministic constant c such that almost surely
ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anom... more ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anomalies back to 1000AD, based on a model that includes external climate forcings and accounts for any long-memory features. Our reconstruction is based on two linear models, with the first linking the latent temperature series to three main external forcings (solar irradiance, greenhouse gas concentration, and volcanism), and the second linking the observed temperature proxy data (tree rings, sediment record, ice cores, etc.) to the unobserved temperature series. Uncertainty is captured with additive noise, and a rigorous statistical investigation of the correlation structure in the regression errors motivates the use of long memory fractional Gaussian noise models for the error terms. We use Bayesian estimation to fit the model parameters and to perform separate reconstructions of land-only and combined land-and-marine temperature anomalies. We quantify the effects of including the forcings and long memory models on the quality of model fits, and find that long memory models can result in more precise uncertainty quantification, while the external climate forcings substantially reduce the squared bias and variance of the reconstructions. Finally, we use posterior samples of model parameters to arrive at an estimate of the transient climate response to greenhouse gas forcings of $2.56^{\circ}$C (95% credible interval of [$2.20$, $2.95$]$^{\circ}$C), in line with previous, climate-model-based estimates.
We consider the parameter estimation problem for the Ornstein-Uhlenbeck process X driven by a fra... more We consider the parameter estimation problem for the Ornstein-Uhlenbeck process X driven by a fractional Ornstein-Uhlenbeck process V , i.e. the pair of processes defined by the non-Markovian continuous-time long-memory dynamics dX t = −θX t dt + dV t ; t 0, with dV t = −ρV t dt + dB H t ; t 0, where θ > 0 and ρ > 0 are unknown parameters, and B H is a fractional Brownian motion of Hurst index H ∈ ( 1 2 , 1). We study the strong consistency as well as the asymptotic normality of the joint least squares estimator θ T , ρ T of the pair (θ, ρ), based either on continuous or discrete observations of {X s ; s ∈ [0, T ]} as the horizon T increases to +∞. Both cases qualify formally as partial-hbobservation questions since V is unobserved. In the latter case, several discretization options are considered. Our proofs of asymptotic normality based on discrete data, rely on increasingly strict restrictions on the sampling frequency as one reduces the extent of sources of observation. The strategy for proving the asymptotic properties is to study the case of continuous-time observations using the Malliavin calculus, and then to exploit the fact that each discrete-data estimator can be considered as a perturbation of the continuous one in a mathematically precise way, despite the fact that the implementation of the discrete-time estimators is distant from the continuous estimator. In this sense, we contend that the continuous-time estimator cannot be implemented in practice in any naïve way, and serves only as a mathematical tool in the study of the discrete-time estimators' asymptotics.
In this paper, we study central and non-central limit theorems for partial sum of functionals of ... more In this paper, we study central and non-central limit theorems for partial sum of functionals of general stationary Gaussian fields. We apply our result to study drift parameter estimation problems for some stochastic differential equations related to stationary Gaussian processes.
We consider a stochastic volatility stock price model in which the volatility is a non-centered c... more We consider a stochastic volatility stock price model in which the volatility is a non-centered continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the stock price density and implied volatility in these models in the limit of large or small strikes. Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility, based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the L 2 norm of the projection of the mean function on the top eigenspace. Strategies for using this expansion for calibration purposes are discussed.
ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anom... more ABSTRACT We produce new reconstructions of Northern Hemisphere annually averaged temperature anomalies back to 1000AD, based on a model that includes external climate forcings and accounts for any long-memory features. Our reconstruction is based on two linear models, with the first linking the latent temperature series to three main external forcings (solar irradiance, greenhouse gas concentration, and volcanism), and the second linking the observed temperature proxy data (tree rings, sediment record, ice cores, etc.) to the unobserved temperature series. Uncertainty is captured with additive noise, and a rigorous statistical investigation of the correlation structure in the regression errors motivates the use of long memory fractional Gaussian noise models for the error terms. We use Bayesian estimation to fit the model parameters and to perform separate reconstructions of land-only and combined land-and-marine temperature anomalies. We quantify the effects of including the forcings and long memory models on the quality of model fits, and find that long memory models can result in more precise uncertainty quantification, while the external climate forcings substantially reduce the squared bias and variance of the reconstructions. Finally, we use posterior samples of model parameters to arrive at an estimate of the transient climate response to greenhouse gas forcings of $2.56^{\circ}$C (95% credible interval of [$2.20$, $2.95$]$^{\circ}$C), in line with previous, climate-model-based estimates.
ABSTRACT We investigate optimal portfolio selection problems with mispricing and model ambiguity ... more ABSTRACT We investigate optimal portfolio selection problems with mispricing and model ambiguity under a financial market which contains a pair of mispriced stocks. We assume that the dynamics of the pair satisfies a “cointegrated system” advanced by Liu and Timmermann in a 2013 manuscript. The investor hopes to exploit the temporary mispricing by using a portfolio strategy under a utility function framework. Furthermore, she is ambiguity-averse and has a specific preference for model ambiguity robustness. The explicit solution for such a robust optimal strategy, and its value function, are derived. We analyze these robust strategies with mispricing in two cases: observed and unobserved mean-reverting stochastic risk premium. We show that the mispricing and model ambiguity have completely distinct impacts on the robust optimal portfolio selection, by comparing the utility losses. We also find that the ambiguity-averse investor who ignores the mispricing or the model ambiguity, suffers a substantially larger utility loss if the risk premium is unobserved, compared to when it is observed.
We study linear stochastic evolution equations driven by various inflnite-dimensional Gaussian pr... more We study linear stochastic evolution equations driven by various inflnite-dimensional Gaussian processes, some of which are more irregular in time than fractional Brownian motion (fBm) with any Hurst parameter H, while others are comparable to fBm with H < 1 2 . Sharp necessary and su-- cient conditions for the existence and uniqueness of solutions are presented. Specializing to stochastic heat equations on compact manifolds, especially on the unit circle, sharp Gaussian regularity results are used to determine suf- flcient conditions for a given flxed function to be an almost-sure modulus of continuity for the solution in space; these su-cient conditions are also proved necessary in highly irregular cases, and are nearly necessary (logarithmic cor- rections are given) in other cases, including the Holder scale.
We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can... more We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.
Discrete and Continuous Dynamical Systems - Series B, 2006
We study the time-regularity of the paths of solutions to stochastic partial differential equatio... more We study the time-regularity of the paths of solutions to stochastic partial differential equations (SPDE) driven by additive infinite-dimensional fractional Brownian noise. Sharp sufficient conditions for almost-sure Hölder continuity, and other, more irregular levels of uniform continuity, are given when the space parameter is fixed. Additionally, a result is included on time-continuity when the solution is understood as a spatially Hölder-continuous-function-valued stochastic process. Tools used for the study include the Brownian representation of fractional Brownian motion, and sharp Gaussian regularity results.
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Papers by Frederi Viens