Contains exact results for the solution of the stochastic differential equation

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

where $x(t)$ is proportional to the magnetic momentum, the constant $C>0$ is related to the linear approximation of the diffusion coeeficient !coefficient $D(x)$ by the local approximation near $x=0$ by

$D(x)=2Cx+O(x^2)$

The solution of this SDE is the square of the Bessel process of order 0 and can be represented as

$x(t)=(w_1(Ct)+\sqrt(x_0))^2+(w_2(Ct))^2$

where

$w_1(t), \dots w_2(t)$

are independent standard Wiener processes and

$x_0=x(t=0)$

. More...

List of all members.

Public Member Functions
subroutine	Bessel0 (x_in, delta_t, C_dif, x_fin)
	Subroutine Bessel0.
subroutine	exact_moments (x_in, delta_t, C_dif, moments)
	For test of the subroutine Bessel0 only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif The tested subroutine is Bessel0 from module BesselProcess, same notations.
subroutine	Bessel0 (x_in, delta_t, C_dif, x_fin)
	Subroutine Bessel0 Integration near x=0.
subroutine	Bessel0_General (x_in, delta_t, C_dif, D_0, x_fin)
	Integration, with local approximation of the diffusion coefficient $D(x)=2*C_dif+D_0$ .
subroutine	exact_moments (x_in, delta_t, C_dif, moments)
	For test of the subroutine Bessel0 only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif The tested subroutine is Bessel0 from module BesselProcess, same notations.
subroutine	exact_momentsG (x_in, delta_t, C_dif, D_0, moments)
	For test of the subroutine Bessel0_General only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif, D_0 The tested subroutine is Bessel0_General from module BesselProcess, same notations.
subroutine	Bessel0 (x_in, delta_t, C_dif, x_fin)
	Integration near x=0, when $D(x)=2C_dif x+O(x^2)$ INPUT:
subroutine	Bessel0_General (x_in, delta_t, C_dif, D_0, x_fin)
	Integration, with local approximation of the diffusion coefficient $D(x)=2*C_dif+D_0$ The updates are performed according to exact solution of the squared Bessel process of order 0 INPUT:
subroutine	exact_moments (x_in, delta_t, C_dif, moments)
	For test of the subroutine Bessel0 only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif The tested subroutine is Bessel0 from module BesselProcess, same notations INPUT:
subroutine	exact_momentsG (x_in, delta_t, C_dif, D_0, moments)
	For test of the subroutine Bessel0_General only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif, D_0 The tested subroutine is Bessel0_General from module BesselProcess, same notations INPUT:
subroutine	RFOF_MeanExitTime0 (x_0, x_min, C_dif, Time_exit, ErrorFlag)
	INPUT.
subroutine	RFOF_MeanExitTime1 (a, x_0, b, relative_tolerance, C_dif, D_0, mean_first_exit_time, Error_Flag)
	INPUT.
subroutine	RFOF_MeanExitTime2 (a, x_0, b, relative_tolerance, C_dif, D_0, mean_first_exit_time)
	INPUT.
subroutine	RFOF_Bessel_0_move (a, x_init, b, max_duration, relative_tolerance, C_dif, D_0, duration, x_final, Error_Flag)
	Main subroutine It performs repeated updates according to subroutine Bessel0_General,until x(t) exits the interval (a,b) with sufficient small time step such that the final point is not too far from the interval (a,b), so the linear approximation of the diffusion coefficient is still valid INPUT:
subroutine	RFOF_Bessel_0_move_test (a, x_0, b, max_duration, relative_tolerance, C_dif, D_0, duration, x_final, ErrorFlag)
	INPUT:

Detailed Description

Contains exact results for the solution of the stochastic differential equation

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

where $x(t)$ is proportional to the magnetic momentum, the constant $C>0$ is related to the linear approximation of the diffusion coeeficient !coefficient $D(x)$ by the local approximation near $x=0$ by

$D(x)=2Cx+O(x^2)$

The solution of this SDE is the square of the Bessel process of order 0 and can be represented as

$x(t)=(w_1(Ct)+\sqrt(x_0))^2+(w_2(Ct))^2$

where

$w_1(t), \dots w_2(t)$

are independent standard Wiener processes and

$x_0=x(t=0)$

.

Contains exact results for the solution of the stochastic differential equation

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

$dx(t)=dD(x)/dxdt+sqrt(D(x(t)))dw(t)$

In the case when we have a linear approximation

$D(x)=2C x+O(x^2)$

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

we have near 0

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

where $x(t)$ is proportional to the magnetic momentum, the constant $C>0$ is related to the linear approximation of the diffusion coeeficient !coefficient $D(x)$ by the local approximation near $x=0$ by

$D(x)=2Cx+O(x^2)$

The solution of this SDE is the square of the Bessel process of order 0 and can be represented as

$x(t)=(w_1(Ct)+\sqrt(x_0))^2+(w_2(Ct))^2$

where

$w_1(t), \dots w_2(t)$

are independent standard Wiener processes and

$x_0=x(t=0)$

In the domain with x large, we use a general linear approximation for D(x)

$D(x)=2Cx+d +error$

and the update is given by a more general formula, derived from the previous

$x(t)=(w_1(Ct)+\sqrt(x_0+d/(2C)))^2+(w_2(Ct))^2-d/(2C)$

We use the FORTRAN variable notations: $C=C_dif$ , $d=D_0 $ .

Contains exact results for the solution of the stochastic differential equation

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

$dx(t)=dD(x)/dxdt+sqrt(D(x(t)))dw(t)$

In the case when we have a linear approximation

$D(x)=2C x+O(x^2)$

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

we have near 0

$dx(t)=2Cdt+2\sqrt(Cx(t))dw(t)$

where $x(t)$ is proportional to the magnetic momentum, the constant $C>0$ is related to the linear approximation of the diffusion coeeficient !coefficient $D(x)$ by the local approximation near $x=0$ by

$D(x)=2Cx+O(x^2)$

The solution of this SDE is the square of the Bessel process of order 0 and can be represented as

$x(t)=(w_1(Ct)+\sqrt(x_0))^2+(w_2(Ct))^2$

where

$w_1(t), \dots w_2(t)$

are independent standard Wiener processes and

$x_0=x(t=0)$

In the domain with x large, we use a general linear approximation for D(x)

$D(x)=2Cx+d +error$

and the update is given by a more general formula, derived from the previous

$x(t)=(w_1(Ct)+\sqrt(x_0+d/(2C)))^2+(w_2(Ct))^2-d/(2C)$

.

Author:: Gyorgy Steinbrecher, EURATOM_MEdC

Version:: "$Id: RFOF_main.F90 188 2011-09-12 11:50:22Z tjohnson $"

Definition at line 19 of file RFOF_BesselProcess0.F90.

Member Function/Subroutine Documentation

subroutine RFOF_BesselProcess0::Bessel0	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(out)	x_fin
	)

Subroutine Bessel0.

Parameters:

[in]	x_in	Initial position
[in]	delta_t	The time interval
[in]	C_dif	From local aproximation: $Diff_coef=2C_difx+O(x**2)$
[out]	x_fin	The final position

Definition at line 26 of file RFOF_BesselProcess0.F90.

subroutine RFOF_BesselProcess0::exact_moments	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), dimension(max_moment_order), intent(out)	moments
	)

For test of the subroutine Bessel0 only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif The tested subroutine is Bessel0 from module BesselProcess, same notations.

Definition at line 46 of file RFOF_BesselProcess0.F90.

subroutine RFOF_BesselProcess0::Bessel0	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(out)	x_fin
	)

Subroutine Bessel0 Integration near x=0.

Parameters:

[in]	x_in	initial position
[in]	delta_t	the time interval
[in]	C_dif	From local aproximation: $Diff_coef=2C_difx+O(x**2)$
[out]	x_fin	the final position

Definition at line 37 of file RFOF_BesselProcess0_3.F90.

subroutine RFOF_BesselProcess0::Bessel0_General	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		real(8), intent(out)	x_fin
	)

Integration, with local approximation of the diffusion coefficient

$D(x)=2*C_dif+D_0$

.

Definition at line 57 of file RFOF_BesselProcess0_3.F90.

subroutine RFOF_BesselProcess0::exact_moments	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), dimension(max_moment_order), intent(out)	moments
	)

For test of the subroutine Bessel0 only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif The tested subroutine is Bessel0 from module BesselProcess, same notations.

Definition at line 95 of file RFOF_BesselProcess0_3.F90.

subroutine RFOF_BesselProcess0::exact_momentsG	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		real(8), dimension(max_moment_order), intent(out)	moments
	)

For test of the subroutine Bessel0_General only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif, D_0 The tested subroutine is Bessel0_General from module BesselProcess, same notations.

Definition at line 114 of file RFOF_BesselProcess0_3.F90.

subroutine RFOF_BesselProcess0::Bessel0	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(out)	x_fin
	)

Integration near x=0, when

$D(x)=2C_dif x+O(x^2)$

INPUT:

Parameters:

x_in	initial pozition
delta_t	time step
C_dif	coeficient for local linear approximation of the diffusion coefficient OUTPUT:
x_fin	final pozition after time delta_t

Definition at line 45 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::Bessel0_General	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		real(8), intent(out)	x_fin
	)

Integration, with local approximation of the diffusion coefficient

$D(x)=2*C_dif+D_0$

The updates are performed according to exact solution of the squared Bessel process of order 0 INPUT:

Parameters:

x_in	Initial pozition
delta_t	Time step
C_dif	Local linear approximation of the diffusion coefficient
D_0	Coeficients for local linear approximation of the diffusion coefficient OUTPUT:
x_fin	Final pozition after time delta_t

Definition at line 81 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::exact_moments	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), dimension(max_moment_order), intent(out)	moments
	)

For test of the subroutine Bessel0 only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif The tested subroutine is Bessel0 from module BesselProcess, same notations INPUT:

Parameters:

x_in	starting point of 0 order bessel process
delta_t	duration of Bessel procees
C_dif	From local aproximation: $Diff_coef=2C_difx$ OUTPUT:
moments	1 and 2 order moments of the Bessel process

Definition at line 133 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::exact_momentsG	(	real(8), intent(in)	x_in,
		real(8), intent(in)	delta_t,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		real(8), dimension(max_moment_order), intent(out)	moments
	)

For test of the subroutine Bessel0_General only, returns the exact values of the moments of Bessel process Starting point +x_in, duration delta_t, coefficient C_dif, D_0 The tested subroutine is Bessel0_General from module BesselProcess, same notations INPUT:

Parameters:

x_in	starting point of 0 order bessel process
delta_t	duration of Bessel procees
C_dif	From local aproximation: Diff_coef=2C_difx OUTPUT: moments 1 and 2 order moments of the Bessel process

Definition at line 164 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::RFOF_MeanExitTime0	(	real(8), intent(in)	x_0,
		real(8), intent(in)	x_min,
		real(8), intent(in)	C_dif,
		Real(8), intent(out)	Time_exit,
		Integer, intent(out)	ErrorFlag
	)

INPUT.

Parameters:

a
b	Endpoints of the interval
x_0	starting point of 0 order Bessel process Admissible relative error
C_dif	From local aproximat:ion: Diff_coef=2C_difx
mean_first_exit_time	The mean first exit time from (a,b) starting at x_0
Error_Flag	! zero when OK, 102 if not a<=x_0<=b, wrong ordering 103 if the diffusion coefficient is negative on interval (a,b)

Definition at line 201 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::RFOF_MeanExitTime1	(	real(8), intent(in)	a,
		real(8), intent(in)	x_0,
		real(8), intent(in)	b,
		real(8), intent(in)	relative_tolerance,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		Real(8), intent(out)	mean_first_exit_time,
		integer, intent(out)	Error_Flag
	)

INPUT.

Parameters:

a
b	Endpoints of the interval
x_0	starting point of 0 order Bessel process Admissible relative error
D_dif
D_0	From local aproximat:ion: Diff_coef=2C_difx +D_0
mean_first_exit_time	The mean first exit time from (a,b) starting at x_0
Error_Flag	! zero when OK, 102 if not a<=x_0<=b, wrong ordering 103 if the diffusion coefficient is negative on interval (a,b)

Definition at line 241 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::RFOF_MeanExitTime2	(	real(8), intent(in)	a,
		real(8), intent(in)	x_0,
		real(8), intent(in)	b,
		real(8), intent(in)	relative_tolerance,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		Real(8), intent(out)	mean_first_exit_time
	)

INPUT.

Parameters:

a
b	Endpoints of the interval
x_0	starting point of 0 order Bessel process Admissible relative error
D_dif
D_0	From local aproximat:ion: Diff_coef=2C_difx +D_0
mean_first_exit_time	The mean first exit time from (a,b) starting at x_0
Error_Flag	! zero when OK, 102 if not a<=x_0<=b, wrong ordering 103 if the diffusion coefficient is negative on interval (a,b) Version of the previous subroutine No more test of the input data The test is performed at once in the calling subroutine

Definition at line 327 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::RFOF_Bessel_0_move	(	real(8), intent(in)	a,
		real(8), intent(in)	x_init,
		real(8), intent(in)	b,
		real(8), intent(in)	max_duration,
		real(8), intent(in)	relative_tolerance,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		Real(8), intent(out)	duration,
		real(8), intent(out)	x_final,
		Integer, intent(out)	Error_Flag
	)

Main subroutine It performs repeated updates according to subroutine Bessel0_General,until x(t) exits the interval (a,b) with sufficient small time step such that the final point is not too far from the interval (a,b), so the linear approximation of the diffusion coefficient is still valid INPUT:

---------------------------------------------------------------------------

Parameters:

a	Endpoints of the interval
x_init	starting point of 0 order Bessel process
max_duration	The maximal time allowed
relative_tolerance	Admissible relative error
C_dif
D_0	From local linear aproximation: Diff_coef=2C_difx +D_0 OUTPUT
duration	The approximate first exit time from (a,b) startin at x_0
x_final	final position
Error_Flag	ErrorFlag=0 when a<=x_0<=b;ErrorFlag=102 When the Diffusionn coefficient is negative on (a,b), then Error_Flag=103 otherwise Error_Flag=0

Definition at line 405 of file RFOF_BesselProcess0_5.F90.

subroutine RFOF_BesselProcess0::RFOF_Bessel_0_move_test	(	real(8), intent(in)	a,
		real(8), intent(in)	x_0,
		real(8), intent(in)	b,
		real(8), intent(in)	max_duration,
		real(8), intent(in)	relative_tolerance,
		real(8), intent(in)	C_dif,
		real(8), intent(in)	D_0,
		Real(8), intent(out)	duration,
		real(8), intent(out)	x_final,
		Integer, intent(out)	ErrorFlag
	)

INPUT:

Parameters:

a	Endpoints of the interval
x_init	starting point of 0 order Bessel process
max_duration	The maximal time allowed
relative_tolerance	Admissible relative error
C_dif
D_0	From local linear aproximation: Diff_coef=2C_difx +D_0 OUTPUT
duration	The approximate first exit time from (a,b) startin at x_0
x_final	final position
Error_Flag	ErrorFlag=0 when a<=x_0<=b;ErrorFlag=102 When the Diffusionn coefficient is negative on (a,b), then Error_Flag=103 otherwise Error_Flag=0

Definition at line 489 of file RFOF_BesselProcess0_5.F90.

The documentation for this module was generated from the following files:

Public Member Functions

Detailed Description

Member Function/Subroutine Documentation