FeynmanKac (JMSL Numerical Library)

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com.imsl.math
Class FeynmanKac

java.lang.Object
  com.imsl.math.FeynmanKac

All Implemented Interfaces:: Serializable, Cloneable

public class FeynmanKac
extends Object
implements Serializable, Cloneable
extends Object
implements Serializable, Cloneable

Solves the generalized Feynman-Kac PDE.

Class FeynmanKac solves the generalized Feynman-Kac PDE on a rectangular grid with boundary conditions using a finite element Galerkin method. The generalized Feynman-Kac differential equation has the form

$f_t +mu(x,t) f_x +frac{sigma^2(x,t)}{2}f_{xx}-kappa(x,t)f=phi(f,x,t),$

where the initial data satisfies

. The derivatives are $f_t=frac{partial{f}}{partial{t}}$ , etc. Method computeCoefficients uses a finite element Galerkin method over the rectangle

$[x_{min},x_{max}] times [bar{T},T]$

to compute the approximate solution. The interval $[x_{min},x_{max}]$ is decomposed with a grid

$(x_{min}=)x_1 lt x_2 lt ldots lt x_m(=x_{max}),.$

On each subinterval the solution is represented by

$f(x,t) = f_ib_0(z)+f_{i+1}b_0(1-z)+h_i f_i' b_1(z)-h_i f_{i+1}' b_1(1-z)+ h_i^2f_i''b_2(z)+h_i^2f_{i+1}''b_2(1-z).$

The values

$f_i,f_i',f_i'',f_{i+1},f_{i+1}',f_{i+1}''$

are time-dependent coefficients associated with each interval. The basis functions

are given for

$x in [x_i, x_{i+1}], , h_i=x_{i+1}-x_i, , z=(x-x_i)/h_i, z in [0,1],$

$b_2(z)=frac{1}{2}(-z^5+3z^4-3z^3+z^2)=frac{1}{2}(1-z)^3z^2.$

By adding the piece-wise definitions the unknown solution function may be arranged as the series

$f(x,t) = sum_{i=1}^{3m}y_ibeta_i(x), ; x in [x_{min},x_{max}],$

where the time-dependent coefficients are defined by re-labeling:

$y_{3i-2} = f_i,, y_{3i-1} = f_i',, y_{3i} = f_i'',, i=1,ldots,m.$

The Galerkin principle is then applied. Using the provided initial and boundary conditions leads to an Index 1 differential-algebraic equation for the coefficients

This system is integrated using the variable order, variable step algorithm DDASLX noted in Hanson, R. and Krogh, F. (2008), Solving Constrained Differential-Algebraic Systems Using Projections. Solution values and their time derivatives at a grid preceding time T, expressed in units of time remaining, are given back by methods getSplineCoefficients and getSplineCoefficientsPrime, respectively. For further details of deriving and solving the system see Hanson, R. (2008), Integrating Feynman-Kac Equations Using Hermite Quintic Finite Elements. To evaluate f or its partials $f_x, f_{xx}, f_{xxx}, f_t, f_{tx}, f_{txx}, f_{txxx}$ at any time point in the grid, use method getSplineValue.

One useful application of the FeynmanKac class is financial analytics. This is illustrated in Example 2, which solves a diffusion model for call options which, in the special case reduces to the Black-Scholes (BS) model. Another useful application for the FeynmanKac class is the calculation of the Greeks, i.e. various derivatives of Feynman-Kac solutions applicable to, e.g., the pricing of options and related financial derivatives. In Example 5, the FeynmanKac class is used to calculate eleven of the Greeks for the same diffusion model introduced in Example 2 in the special BS case. These Greeks are also calculated using the BS closed form Greek equations (see http://en.wikipedia.org/wiki/The_Greeks). The Feynman-Kac and BS solutions are output and compared. Example 5 illustrates that the FeynmanKac class can be used to explore the Greeks for a much wider class of financial models than can BS.

See Also:: Example 1, Example 2, Example 3, Example 4, Example 5, Serialized Form

Nested Class Summary
`static interface`	`FeynmanKac.Boundaries` Public interface for user supplied boundary coefficients and terminal condition the PDE must satisfy.
`static class`	`FeynmanKac.BoundaryInconsistentException` The boundary conditions are inconsistent.
`static class`	`FeynmanKac.ConstraintsInconsistentException` The constraints are inconsistent.
`static class`	`FeynmanKac.CorrectorConvergenceException` Corrector failed to converge.
`static class`	`FeynmanKac.ErrorTestException` Error test failure detected.
`static interface`	`FeynmanKac.ForcingTerm` Public interface for non-zero forcing term in the Feynman-Kac equation.
`static class`	`FeynmanKac.InitialConstraintsException` The constraints at the initial point are inconsistent.
`static interface`	`FeynmanKac.InitialData` Public interface for adjustment of initial data or as an opportunity for output during the integration steps.
`static class`	`FeynmanKac.IterationMatrixSingularException` Iteration matrix is singular.
`static interface`	`FeynmanKac.PdeCoefficients` Public interface for user supplied PDE coefficients in the Feynman-Kac PDE.
`static class`	`FeynmanKac.TcurrentTstopInconsistentException` The end value for the integration in time, tout, is not consistent with the current time value, t.
`static class`	`FeynmanKac.TEqualsToutException` The current integration point in time and the end point are equal.
`static class`	`FeynmanKac.TimeIntervalTooSmallException` Distance between starting time point and end point for the integration is too small.
`static class`	`FeynmanKac.ToleranceTooSmallException` Tolerance is too small.
`static class`	`FeynmanKac.TooManyIterationsException` Too many iterations required by the DAE solver.

Field Summary
`static int`	`METHOD_OF_PETZOLD` Used by method `setStepControlMethod` to indicate that the step control algorithm of the original Petzold code is used in the integration.
`static int`	`METHOD_OF_SOEDERLIND` Used by method `setStepControlMethod` to indicate that the step control method by Soederlind is used in the integration.

Constructor Summary
`FeynmanKac(FeynmanKac.PdeCoefficients pdeCoeffs)` Constructs a PDE solver to solve the Feynman-Kac PDE.

Method Summary
`void`	`computeCoefficients(int numLeftBounds, int numRightBounds, FeynmanKac.Boundaries pdeBounds, double[] xGrid, double[] tGrid)` Determines the coefficients of the Hermite quintic splines that represent an approximate solution for the Feynman-Kac PDE.
`double[]`	`getAbsoluteErrorTolerances()` Returns absolute error tolerances.
`int`	`getGaussLegendreDegree()` Returns the number of quadrature points used in the Gauss-Legendre quadrature formula.
`double`	`getInitialStepsize()` Returns the starting step size for the integration.
`int`	`getMaximumBDFOrder()` Returns the maximum order of the BDF formulas.
`double`	`getMaximumStepsize()` Returns the maximum internal step size used by the integrator.
`int`	`getMaxSteps()` Returns the maximum number of internal steps allowed.
`double[]`	`getRelativeErrorTolerances()` Returns relative error tolerances.
`double[][]`	`getSplineCoefficients()` Returns the coefficients of the Hermite quintic splines that represent an approximate solution of the Feynman-Kac PDE.
`double[][]`	`getSplineCoefficientsPrime()` Returns the first derivatives of the Hermite quintic spline coefficients that represent an approximate solution of the Feynman-Kac PDE.
`double[]`	`getSplineValue(double[] evaluationPoints, double[] coefficients, int ideriv)` Evaluates for time value 0 or a time value in tGrid the derivative of the Hermite quintic spline interpolant at evaluation points within the range of `xGrid`.
`int`	`getStepControlMethod()` Returns the step control method used in the integration of the Feynman-Kac PDE.
`double`	`getTimeBarrier()` Returns the barrier set for integration in the time direction.
`void`	`setAbsoluteErrorTolerances(double atol)` Sets the absolute error tolerances.
`void`	`setAbsoluteErrorTolerances(double[] atol)` Sets the absolute error tolerances.
`void`	`setForcingTerm(FeynmanKac.ForcingTerm forceTerm)` Sets the user-supplied method that computes approximations to the forcing term and its derivative used in the FeynmanKac PDE.
`void`	`setGaussLegendreDegree(int degree)` Sets the number of quadrature points used in the Gauss-Legendre quadrature formula.
`void`	`setInitialData(FeynmanKac.InitialData initData)` Sets the user-supplied method for adjustment of initial data or as an opportunity for output during the integration steps.
`void`	`setInitialStepsize(double initStepsize)` Sets the starting stepsize for the integration.
`void`	`setMaximumBDFOrder(int maxBDFOrder)` Sets the maximum order of the BDF formulas.
`void`	`setMaximumStepsize(double maximumStepsize)` Sets the maximum internal step size used by the integrator.
`void`	`setMaxSteps(int maxSteps)` Sets the maximum number of internal steps allowed.
`void`	`setRelativeErrorTolerances(double rtol)` Sets the relative error tolerances.
`void`	`setRelativeErrorTolerances(double[] rtol)` Sets the relative error tolerances.
`void`	`setStepControlMethod(int stepControlMethod)` Sets the step control method used in the integration of the Feynman-Kac PDE.
`void`	`setTimeBarrier(double timeBarrier)` Sets a barrier for the integration in the time direction.
`void`	`setTimeDependence(boolean[] timeFlag)` Sets the time dependence of the coefficients, boundary conditions and function in the Feynman Kac equation.

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Field Detail

METHOD_OF_PETZOLD

public static final int METHOD_OF_PETZOLD

Used by method setStepControlMethod to indicate that the step control algorithm of the original Petzold code is used in the integration.

See Also:: Constant Field Values

METHOD_OF_SOEDERLIND

public static final int METHOD_OF_SOEDERLIND

Used by method setStepControlMethod to indicate that the step control method by Soederlind is used in the integration.

See Also:: Constant Field Values

Constructor Detail

FeynmanKac

public FeynmanKac(FeynmanKac.PdeCoefficients pdeCoeffs)

Constructs a PDE solver to solve the Feynman-Kac PDE.

Parameters:: pdeCoeffs - Implementation of interface PdeCoefficients that computes the values of the Feynman-Kac coefficients at a given point (t,x).

Method Detail

computeCoefficients

public void computeCoefficients(int numLeftBounds,
                                int numRightBounds,
                                FeynmanKac.Boundaries pdeBounds,
                                double[] xGrid,
                                double[] tGrid)
                         throws FeynmanKac.ToleranceTooSmallException,
                                FeynmanKac.TooManyIterationsException,
                                FeynmanKac.ErrorTestException,
                                FeynmanKac.CorrectorConvergenceException,
                                FeynmanKac.IterationMatrixSingularException,
                                FeynmanKac.TimeIntervalTooSmallException,
                                FeynmanKac.TcurrentTstopInconsistentException,
                                FeynmanKac.TEqualsToutException,
                                FeynmanKac.InitialConstraintsException,
                                FeynmanKac.ConstraintsInconsistentException,
                                SingularMatrixException,
                                FeynmanKac.BoundaryInconsistentException

Determines the coefficients of the Hermite quintic splines that represent an approximate solution for the Feynman-Kac PDE.

Parameters:: numLeftBounds - an int, the number of left boundary conditions. It is required that
1 numLeftBounds 3.; numRightBounds - an int, the number of right boundary conditions. It is required that
1 numRightBounds 3.; pdeBounds - Implementation of interface Boundaries that computes the boundary coefficients and terminal condition for given (t,x).; xGrid - a double array containing the breakpoints for the Hermite quintic splines used in the x discretization. The length of xGrid must be at least 2, xGrid.length >= 2, and the elements in xGrid must be in strictly increasing order.; tGrid - a double array containing the set of time points (in time-remaining units) where an approximate solution is returned. The elements in array tGrid must be positive and in strictly increasing order.
Throws:: FeynmanKac.ToleranceTooSmallException - is thrown if the absolute or relative error tolerances used in the integrator are too small.; FeynmanKac.TooManyIterationsException - is thrown if the integrator needs too many iteration steps.; FeynmanKac.ErrorTestException - is thrown if the error test used in the integrator failed repeatedly.; FeynmanKac.CorrectorConvergenceException - is thrown if the corrector failed to converge repeatedly.; FeynmanKac.IterationMatrixSingularException - thrown if one of the iteration matrices used in the integrator is singular.; FeynmanKac.TimeIntervalTooSmallException - is thrown if the distance between an intermediate starting and end point for the integration is too small.; FeynmanKac.TcurrentTstopInconsistentException - is thrown if during the integration the current integration time and given stepsize is inconsistent with the endpoint of the integration.; FeynmanKac.TEqualsToutException - is thrown if during the integration process the actual integration time and the end time of the integration are identical.; FeynmanKac.InitialConstraintsException - is thrown if at the initial integration point some of the constraints are inconsistent.; FeynmanKac.ConstraintsInconsistentException - is thrown if during the integration process the constraints for the actual time point and given stepsize are inconsistent.; SingularMatrixException - is thrown if one of the matrices used outside the integrator is singular.; FeynmanKac.BoundaryInconsistentException - is thrown if the boundary conditions are inconsistent.

getAbsoluteErrorTolerances

public double[] getAbsoluteErrorTolerances()

Returns absolute error tolerances.

Returns:: a double array of length 3*xGrid.length containing absolute error tolerances for the solutions.

getGaussLegendreDegree

public int getGaussLegendreDegree()

Returns the number of quadrature points used in the Gauss-Legendre quadrature formula.

Returns:: an int, the degree of the polynomial used in the Gauss-Legendre quadrature.

getInitialStepsize

public double getInitialStepsize()

Returns the starting step size for the integration.

Returns:: a double, the starting step size used in the integrator.

getMaximumBDFOrder

public int getMaximumBDFOrder()

Returns the maximum order of the BDF formulas.

Returns:: an int, the maximum order of the backward differentiation formulas used in the integrator.

getMaximumStepsize

public double getMaximumStepsize()

Returns the maximum internal step size used by the integrator.

Returns:: a double, the maximum internal step size.

getMaxSteps

public int getMaxSteps()

Returns the maximum number of internal steps allowed.

Returns:: an int specifying the maximum number of internal steps allowed between each output point of the integration.

getRelativeErrorTolerances

public double[] getRelativeErrorTolerances()

Returns relative error tolerances.

Returns:: a double array of length 3*xGrid.length containing relative error tolerances for the solutions.

getSplineCoefficients

public double[][] getSplineCoefficients()

Returns the coefficients of the Hermite quintic splines that represent an approximate solution of the Feynman-Kac PDE.

Returns:

a double array of dimension (tGrid.length+1) by (3*xGrid.length) containing the coefficients of the Hermite quintic spline representation of the approximate solution for the Feynman-Kac PDE at time points 0, tGrid[0],...,tGrid[tGrid.length-1]. Setting ntGrid = tGrid.length and nxGrid = xGrid.length the approximate solution is given by

$f(x,t) = sum_{j=0}^{3*text{nxGrid}-1}y_{ij}beta_j(x) ; mbox{for} ; t=text{tGrid}[i-1], i=1,ldots,text{ntGrid}.$

The representation for the initial data at t=0 is

$p(x) = sum_{j=0}^{3*text{nxGrid}-1} y_{0j}beta_j(x),.$

The (ntGrid+1) by (3*nxGrid) matrix

$(y_{ij})_{i=0,ldots,text{ntGrid}}^{j=0,ldots,3*text{nxGrid}-1}$

is stored row-wise in the returned array.

getSplineCoefficientsPrime

public double[][] getSplineCoefficientsPrime()

Returns the first derivatives of the Hermite quintic spline coefficients that represent an approximate solution of the Feynman-Kac PDE.

Returns:

a double array of dimension (tGrid.length+1) by (3*xGrid.length) containing the first derivatives (in time) of the coefficients of the Hermite quintic spline representation of the approximate solution for the Feynman-Kac PDE at time points 0, tGrid[0],...,tGrid[tGrid.length-1]. The approximate solution itself is given by

$f_t(x,bar{t}) = sum_{j=0}^{3*text{nxGrid}-1}y_{ij}' beta_j(x) quad mbox{for} ; bar{t} = text{tGrid}[i-1], i=1,ldots,text{ntGrid},$

and

$f_t(x,bar{t})=sum_{j=0}^{3*text{nxGrid}-1}y_{0j}^primebeta_j(x) quad mbox{for} ; bar{t} = 0 , .$

The (ntGrid+1) by (3*nxGrid) matrix

$(y_{ij}')_{i=0,ldots,text{ntGrid}}^{j=0,ldots,3*text{nxGrid}-1}$

is stored row-wise in the returned array.

getSplineValue

public double[] getSplineValue(double[] evaluationPoints,
                               double[] coefficients,
                               int ideriv)

Evaluates for time value 0 or a time value in tGrid the derivative of the Hermite quintic spline interpolant at evaluation points within the range of xGrid.

Parameters:: evaluationPoints - a double array containing the points in x-direction at which the Hermite quintic spline representing the approximate solution to the Feynman-Kac PDE or one of its derivatives is to be evaluated. It is required that all elements in array evaluationPoints are greater than or equal to xGrid[0] and less than or equal to xGrid[xGrid.length-1].; coefficients - a double array of length 3*xGrid.length containing the coefficients of the Hermite quintic spline representing the approximate solution f or to the Feynman-Kac PDE. These coefficients are the rows of the arrays splineCoeffs and splineCoeffsPrime returned by methods getSplineCoefficients and getSplineCoefficientsPrime. If the user wants to compute approximate solutions or $f_x,f_{xx},f_{xxx}$ to the Feynman-Kac PDE at time point 0, one must assign row splineCoeffs[0] to array coefficients. If the user wants to compute these approximate solutions for time points t=tGrid[i], i=0,...,tGrid.length-1, one must assign row splineCoeffs[i+1] to array coefficients. The same reasoning applies to the computation of approximate solutions and $f_{tx},f_{txx},f_{txxx}$ and assignment of rows of array getSplineCoefficientsPrime to array coefficients.; ideriv - an int specifying the derivative to be computed. It must be 0, 1, 2 or 3.
Returns:: a double array containing the derivative of order ideriv of the Hermite quintic spline representing the approximate solution f or to the Feynman Kac PDE at evaluationPoints. If ideriv=0, then the spline values are returned. If ideriv=1, then the first derivative is returned, etc.

getStepControlMethod

public int getStepControlMethod()

Returns the step control method used in the integration of the Feynman-Kac PDE.

Returns:

an int scalar specifying which step control method to be used.

Return Value	Description
`METHOD_OF_SOEDERLIND`	Method of Soederlind
`METHOD_OF_PETZOLD`	Method from the original Petzold code DASSL

getTimeBarrier

public double getTimeBarrier()

Returns the barrier set for integration in the time direction.

Returns:: a double, the time point beyond which the integrator should not integrate.

setAbsoluteErrorTolerances

public void setAbsoluteErrorTolerances(double atol)

Sets the absolute error tolerances.

Parameters:: atol - a double scalar specifying the absolute error tolerances for the row-wise solutions returned by method getSplineCoefficients. The tolerance value atol is applied to all 3*xGrid.length solution components. atol must be greater than or equal to zero. Also, not all entries in atol and rtol are allowed to be equal to 0 simultaneously. Default value is 1.0e-5 for each solution component.

setAbsoluteErrorTolerances

public void setAbsoluteErrorTolerances(double[] atol)

Sets the absolute error tolerances.

Parameters:: atol - a double array of length 3*xGrid.length specifying the absolute error tolerances for the row-wise solutions returned by method getSplineCoefficients. All entries in atol must be greater than or equal to zero. Also, not all entries in atol and rtol are allowed to be equal to 0 simultaneously. Default value is 1.0e-5 for each solution component.

setForcingTerm

public void setForcingTerm(FeynmanKac.ForcingTerm forceTerm)

Sets the user-supplied method that computes approximations to the forcing term

and its derivative

used in the FeynmanKac PDE.

Parameters:: forceTerm - a ForcingTerm interface specifying the user defined function used for computation of the forcing term and its derivative . If this member function is not called it is assumed that is identically zero.

setGaussLegendreDegree

public void setGaussLegendreDegree(int degree)

Sets the number of quadrature points used in the Gauss-Legendre quadrature formula.

Parameters:: degree - an int, the degree of the polynomial used in the Gauss-Legendre quadrature. It is required that degree is greater than or equal to 6. The default value is 6.

setInitialData

public void setInitialData(FeynmanKac.InitialData initData)

Sets the user-supplied method for adjustment of initial data or as an opportunity for output during the integration steps.

Parameters:: initData - an InitialData object specifying the user-defined function for adjustment of initial data or the object can be used as an opportunity for output during the integration steps. If this member function is not called, no adjustment of initial data or output during the integration steps is done.

setInitialStepsize

public void setInitialStepsize(double initStepsize)

Sets the starting stepsize for the integration.

Parameters:: initStepsize - a double, the starting stepsize used by the integrator. Must be less than zero since the integration is internally done from t=0 to t=tGrid[tGrid.length-1] in a negative direction. The default is initStepsize = -1.1102230246252e-16.

setMaximumBDFOrder

public void setMaximumBDFOrder(int maxBDFOrder)

Sets the maximum order of the BDF formulas.

Parameters:: maxBDFOrder - an int specifying the maximum order of the backward differentiation formulas used in the integrator. It is required that maxBDFOrder is greater than zero and smaller than 6. The default is maxBDFOrder = 5.

setMaximumStepsize

public void setMaximumStepsize(double maximumStepsize)

Sets the maximum internal step size used by the integrator.

Parameters:: maximumStepsize - a positive scalar of type double, the maximum internal step size. Default value is Double.MAX_VALUE, the largest possible machine number.

setMaxSteps

public void setMaxSteps(int maxSteps)

Sets the maximum number of internal steps allowed.

Parameters:: maxSteps - an int specifying the maximum number of internal steps allowed between each output point of the integration. maxSteps must be positive. The default value is 500000.

setRelativeErrorTolerances

public void setRelativeErrorTolerances(double rtol)

Sets the relative error tolerances.

Parameters:: rtol - a double scalar specifying the relative error tolerances for the solution. The tolerance value rtol is applied to all 3*xGrid.length solution components. rtol must be greater than or equal to zero. Also, not all entries in atol and rtol are allowed to be equal to 0 simultaneously. Default value is 1.0e-5 for each solution component.

setRelativeErrorTolerances

public void setRelativeErrorTolerances(double[] rtol)

Sets the relative error tolerances.

Parameters:: rtol - a double array of length 3*xGrid.length specifying the relative error tolerances for the solution. All entries in rtol must be greater than or equal to zero. Also, not all entries in atol and rtol are allowed to be equal to 0 simultaneously. Default value is 1.0e-5 for each solution component.

setStepControlMethod

public void setStepControlMethod(int stepControlMethod)

Sets the step control method used in the integration of the Feynman-Kac PDE.

Parameters:

stepControlMethod - an int scalar specifying the step control method to be used in the integration. If this member function is not called stepControlMethod is set to METHOD_OF_SOEDERLIND by default.

`stepControlMethod`	Description
`METHOD_OF_SOEDERLIND`	Use method of Soederlind.
`METHOD_OF_PETZOLD`	Use method from the original Petzold code DASSL.

setTimeBarrier

public void setTimeBarrier(double timeBarrier)

Sets a barrier for the integration in the time direction.

Parameters:: timeBarrier - a double, controls whether the integrator should integrate in the time direction beyond a special point, timeBarrier, and then interpolate to get the Hermite quintic spline coefficients and its derivatives at the points in tGrid. It is required that timeBarrier be greater than or equal to tGrid[tGrid.length-1]. The default is timeBarrier = tGrid[tGrid.length-1].

setTimeDependence

public void setTimeDependence(boolean[] timeFlag)

Sets the time dependence of the coefficients, boundary conditions and function

in the Feynman Kac equation.

Parameters:

timeFlag - a boolean vector of length 7 indicating time dependencies in the Feynman-Kac PDE.

Index	Time dependency of
`0`
`1`
`2`
`3`
`4`	Left boundary conditions
`5`	Right boundary conditions
`6`

timeFlag[i] = true indicates that the associated value is time-dependent, timeFlag[i] = false indicates that the associated value is not time-dependent. By default, timeFlag[i] = false for

Overview

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JMSL^TM Numerical Library 6.0

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com.imsl.math Class FeynmanKac

METHOD_OF_PETZOLD

METHOD_OF_SOEDERLIND

FeynmanKac

computeCoefficients

getAbsoluteErrorTolerances

getGaussLegendreDegree

getInitialStepsize

getMaximumBDFOrder

getMaximumStepsize

getMaxSteps

getRelativeErrorTolerances

getSplineCoefficients

getSplineCoefficientsPrime

getSplineValue

getStepControlMethod

getTimeBarrier

setAbsoluteErrorTolerances

setAbsoluteErrorTolerances

setForcingTerm

setGaussLegendreDegree

setInitialData

setInitialStepsize

setMaximumBDFOrder

setMaximumStepsize

setMaxSteps

setRelativeErrorTolerances

setRelativeErrorTolerances

setStepControlMethod

setTimeBarrier

setTimeDependence

com.imsl.math
Class FeynmanKac