@(@\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\W}[1]{ \; #1 \; }@)@
This is dismod_at-20221105 documentation: Here is a link to its
current documentation
.
Age Grid
We are given a grid of @(@
M
@)@ age values
@(@
\{ a_i \; | \; i = 0, \ldots , M-1 \}
@)@.
Time Grid
We are also given a grid of @(@
N
@)@ time values
@(@
\{ t_j \; | \; j = 0, \ldots , N-1 \}
@)@.
Function Grid
In addition,
we are given a grid of function values to be interpolated
@[@
\left\{
\left. v_{i,j} = f( a_i , t_j ) \; \right| \;
i = 0, \ldots , M-1 , j = 0 , \ldots , N-1
\right\} \; .
@]@
Interpolant
The corresponding interpolating function at age @(@
\alpha
@)@ and time
@(@
s
@)@, @(@
f( \alpha , s )
@)@ is defined as follows:
Bilinear
Consider the case
where there is an index pair @(@
i < M-1
@)@, @(@
j < N-1
@)@ such that
@(@
a_i \leq \alpha \leq a_{i+1}
@)@ and
@(@
t_j \leq s \leq t_{j+1}
@)@.
In this case the function is defined by
@[@
f( \alpha , s )
=
\frac{a_{i+1} - \alpha}{a_{i+1} - a_i}
\frac{t_{j+1} - s}{t_{j+1} - t_j} v_{i,j}
+
\frac{\alpha - a_i}{a_{i+1} - a_i}
\frac{t_{j+1} - s}{t_{j+1} - t_j} v_{i+1,j}
+
\frac{a_{i+1} - s}{a_{i+1} - a_i}
\frac{s - t_j}{t_{j+1} - t_j} v_{i+1,j}
+
\frac{\alpha - a_i}{a_{i+1} - a_i}
\frac{s - t_j}{t_{j+1} - t_j} v_{i+1,j+1}
@]@
Below Minimum Age
Consider the case where @(@
\alpha < a_0
@)@ and
there is an index @(@
j < N-1
@)@ such that
@(@
t_j \leq s \leq t_{j+1}
@)@.
In this case the function is defined by
@[@
f( \alpha , s )
=
\frac{t_{j+1} - s}{t_{j+1} - t_j} v_{0,j}
+
\frac{s - t_j}{t_{j+1} - t_j} v_{0,j+1}
@]@
Above Maximum Age
Consider the case where @(@
a_{M-1} < \alpha
@)@ and
there is an index @(@
j < N-1
@)@ such that
@(@
t_j \leq s \leq t_{j+1}
@)@.
In this case the function is defined by
@[@
f( \alpha , s )
=
\frac{t_{j+1} - s}{t_{j+1} - t_j} v_{M-1,j}
+
\frac{s - t_j}{t_{j+1} - t_j} v_{M-1,j+1}
@]@
Below Minimum Time
Consider the case where @(@
s < t_0
@)@ and
there is an index @(@
i < M-1
@)@ such that
@(@
a_i \leq \alpha \leq a_{i+1}
@)@.
In this case the function is defined by
@[@
f( \alpha , s )
=
\frac{a_{i+1} - \alpha}{a_{i+1} - a_i} v_{i,0}
+
\frac{\alpha - a_i}{a_{i+1} - a_i} v_{i+1,0}
@]@
Above Maximum Time
Consider the case where @(@
t_{N-1} < s
@)@ and
there is an index @(@
i < M-1
@)@ such that
@(@
a_i \leq \alpha \leq a_{i+1}
@)@.
In this case the function is defined by
@[@
f( \alpha , s )
=
\frac{a_{i+1} - \alpha}{a_{i+1} - a_i} v_{i,N-1}
+
\frac{\alpha - a_i}{a_{i+1} - a_i} v_{i+1,N-1}
@]@
If @(@
\alpha < a_0
@)@ and @(@
s < t_0
@)@,
@(@
f( \alpha , s ) = v_{0,0}
@)@.
If @(@
\alpha < a_0
@)@ and @(@
t_{N-1} < s
@)@,
@(@
f( \alpha , s ) = v_{0,N-1}
@)@.
If @(@
a_{M-1} < \alpha
@)@ and @(@
t_{N-1} < s
@)@,
@(@
f( \alpha , s ) = v_{M-1,N-1}
@)@.
If @(@
a_{M-1} < \alpha
@)@ and @(@
s < t_0
@)@,
@(@
f( \alpha , s ) = v_{M-1,0}
@)@.
Plotting
Functions of age and time are usually plotted with
age on the vertical axis and time on the horizontal axis.
This is opposite the usual convention where
the first variable is plotted on the horizontal axis
and the second variable on the vertical axis.
Input File: omh/model/bilinear.omh
