Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme An operator splitting method for solving a class of Fokker-Planck equations Beatrice Gaviraghi Institut für Mathematik, Universität Würzburg Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme Contents Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme Contents Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme How to characterize a stochastic process We consider a continuous-time stochastic processes X = { X t } t ∈ I with I ⊆ R + and range in R n . How to characterize a stochastic process Stochastic differential equations (SDEs). They describe the evolution of the stochastic processes. Partial(-integro) differential equations (PIDEs). They describe the evolution of the probability density functions (PDFs). Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme How to characterize a stochastic process We consider a continuous-time stochastic processes X = { X t } t ∈ I with I ⊆ R + and range in R n . How to characterize a stochastic process Stochastic differential equations (SDEs). They describe the evolution of the stochastic processes. Partial(-integro) differential equations (PIDEs). They describe the evolution of the probability density functions (PDFs). Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme How to characterize a stochastic process We consider a continuous-time stochastic processes X = { X t } t ∈ I with I ⊆ R + and range in R n . How to characterize a stochastic process Stochastic differential equations (SDEs). They describe the evolution of the stochastic processes. Partial(-integro) differential equations (PIDEs). They describe the evolution of the probability density functions (PDFs). Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme How to characterize a stochastic process We consider a continuous-time stochastic processes X = { X t } t ∈ I with I ⊆ R + and range in R n . How to characterize a stochastic process Stochastic differential equations (SDEs). They describe the evolution of the stochastic processes. Partial(-integro) differential equations (PIDEs). They describe the evolution of the probability density functions (PDFs). Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme An initial value problem We consider an initial-value problem in the interval [ t 0 , T ] with a stochastic differential equation: � dX t = a ( X t , t ) dt + b ( X t , t ) dW t X t 0 = X 0 with drift coefficient a , diffusion coefficient b , Wiener process W , and given initial data X 0 . Under growth and smoothness assumptions on the coefficients, there exists a pathwise-unique solution of this problem. Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme An initial value problem We consider an initial-value problem in the interval [ t 0 , T ] with a stochastic differential equation: � dX t = a ( X t , t ) dt + b ( X t , t ) dW t X t 0 = X 0 with drift coefficient a , diffusion coefficient b , Wiener process W , and given initial data X 0 . Under growth and smoothness assumptions on the coefficients, there exists a pathwise-unique solution of this problem. Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme The discretization of a SDE A sample path of a stochastic process over the interval [ t 0 , T ] can be simulated with The Euler-Maruyama method. Other higher-order methods, which are more computationally expensive due to the evaluation of several stochastic integrals. When a stochastic process is simulated using a numerical method, its values are specified at the points of the discrete time grid t 0 , t 1 , ..., t N = T . Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme Examples of discretized sample paths Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme The PDF of the Ornstein-Uhlenbeck process on a periodic domain Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme Monte Carlo methods Disadvantages of Monte Carlo methods: 1 the rate of convergence in distribution is proportional to N − 1 2 , where N is the number of the samples; 2 the probability density function of the simulated process is not available in closed form. Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme The evolution of the PDF of a stochastic process Given the model � dX t = a ( X t , t ) dt + b ( X t , t ) dW t X t 0 = X 0 , the evolution of the PDF f ( x , t ) is given by the following parabolic PDE n n � � ∂ x i ( a i f ) + 1 � � ∂ 2 ( bb T ) ij f ∂ t f = − x i , x j 2 i = 1 i , j = 1 which is called Fokker-Planck (FP) equation. In the FP equation, the space dimension corresponds to number of components of the stochastic process. Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme The evolution of the PDF of a stochastic process Given the model � dX t = a ( X t , t ) dt + b ( X t , t ) dW t X t 0 = X 0 , the evolution of the PDF f ( x , t ) is given by the following parabolic PDE n n � � ∂ x i ( a i f ) + 1 � � ∂ 2 ( bb T ) ij f ∂ t f = − x i , x j 2 i = 1 i , j = 1 which is called Fokker-Planck (FP) equation. In the FP equation, the space dimension corresponds to number of components of the stochastic process. Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme The evolution of the PDF of a stochastic process Given the model � dX t = a ( X t , t ) dt + b ( X t , t ) dW t X t 0 = X 0 , the evolution of the PDF f ( x , t ) is given by the following parabolic PDE n n � � ∂ x i ( a i f ) + 1 � � ∂ 2 ( bb T ) ij f ∂ t f = − x i , x j 2 i = 1 i , j = 1 which is called Fokker-Planck (FP) equation. In the FP equation, the space dimension corresponds to number of components of the stochastic process. Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
Stochastic processes and the Fokker-Planck equation The Fokker-Planck partial-integro differential equation The numerical scheme The FP problem We define the FP problem as follows � ∂ t f = − � n � n i = 1 ∂ x i ( a i f ) + 1 i , j = 1 ∂ 2 x i , x j ( σ ij f ) 2 f ( x , 0 ) = f X 0 ( x ) , where σ is defined as σ := bb T and the initial condition is given by the density of the initial random variable. The existence and uniqueness of the solution of this problem follows under growth conditions of the coefficients a and σ and under the definite positivity of the matrix σ . Beatrice Gaviraghi An operator splitting method for solving a class of Fokker-Planck
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