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Notebook[{

Cell[CellGroupData[{
Cell["AnnuitySolve Examples", "Title"],

Cell[CellGroupData[{

Cell["Start up", "Section"],

Cell[BoxData[
    \(Off[General::spell1]\)], "Input"],

Cell[BoxData[
    \(<< AnnuitySolve`\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 1: Getting started. One-factor model", "Section"],

Cell[CellGroupData[{

Cell["Setup", "Subsection"],

Cell[BoxData[{
    \(\(pde\  = \ \ k \((m - x)\) D[A[x, t], \ x]\  + \ 
          x\ s^2/2\ D[A[x, t], \ {x, \ 2}]\  - \n\tD[A[x, t], \ t]\  - \ 
          x\ A[x, t]\  + \ \[Nu]\ x\ s^2/2\ D[A[x, t], \ x]^2/
              A[x, t];\)\n\), "\n", 
    \(\(Npde[
          k_, \ \[Nu]_]\  = \ \ pde\  + 1 /. \ {m\  -> \ 1/20, \ 
            s\  -> \ 1/5};\)\), "\n", 
    \(\(Nbondpde[k_, \ \[Nu]_]\  = \ 
        pde\  /. \ {m\  -> \ 1/20, \ s\  -> \ 1/5};\)\n\), "\n", 
    \(\(x0\  = \ 1/20;\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solve", "Subsection"],

Cell[CellGroupData[{

Cell["First part: Getting started", "Subsubsection"],

Cell["Solve linear pde with k = 1, 3rd-order polynomial", "Text"],

Cell[BoxData[
    \(\(Asoln\  = \ 
        NAnnuitySolve[Npde[1, \ 0], \ 
          A, \ {t, \ 0, \ 1000}, \ {x, \ x0}, \ \ 3];\)\)], "Input"],

Cell["Plot the solution", "Text"],

Cell[BoxData[
    \(\(Plot3D[
        Asoln[x, t], \ {x, \ 0, \  .2}, \ {t, \ 0, \ 1000}];\)\)], "Input"],

Cell["How well does the solution fit the pde?", "Text"],

Cell[BoxData[{
    \(\(resid\  = \ ResidualFromPDE[Npde[1, \ 0], \ \ Asoln];\)\), "\n", 
    \(\(Plot3D[
        Abs[resid[x, t]], \ {x, \ 0, \  .2}, \ {t, \ 0, \ 
          1000}];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Second part: More complicated", "Subsubsection"],

Cell["\<\
Loop over values of k. (Series solution with low k converges slowly.)\
\>", "Text"],

Cell[BoxData[
    \(\(Atab\  = \ 
        Table[\n\t
          NAnnuitySolve[Npde[10^\((\(-i\))\), \ 0], \ 
            A, \ {t, \ 0, \ 1000}, {x, x0}, \ \ order], \n\t{i, \ 0, \ 
            3}, \ {order, \ 1, \ 5}];\)\)], "Input"],

Cell["Get annuity series from bond prices", "Text"],

Cell[BoxData[{
    \(\(Asoln\  = \ 
        NAnnuitySolve[Npde[1/1000, \ 0], \ 
          A, \ \ {t, \ 0, \ 1000}, \ {x, \ x0}, 3];\)\), "\n", 
    \(\(resid\  = \ ResidualFromPDE[Npde[1/1000, 0], \ Asoln];\)\), "\n", 
    \(\(Bsoln\  = \ 
        NAnnuitySolve[Nbondpde[1/1000, 0], \ 
          A, \ \ {t, \ 0, \ 1000}, {x, x0}, \ 1, \n\t
          FunctionalForm\  -> \ Exp];\)\), "\n", 
    \(\(Bcoeffs\  = \ 
        NBondToAnnuitySeries[Bsoln[x, t], {t, \ 0, \ 1000}, \ {x, x0}, \ 
          5];\)\)}], "Input"],

Cell["Use Bcoeffs to fix series solution ", "Text"],

Cell[BoxData[{
    \(\(Asoln1\  = \ 
        NAnnuitySolve[Npde[1/1000, 0], \ A, \ {t, \ 0, \ 1000}, \ {x, x0}, 
          3, \n\tPolynomialOrderDifferential\  -> \ 2, \ 
          DifferentialLoadings\  -> \ Bcoeffs];\)\), "\n", 
    \(\(resid1\  = \ 
        ResidualFromPDE[Npde[1/1000, 0], \ Asoln1];\)\)}], "Input"],

Cell["Compare the two solutions", "Text"],

Cell[BoxData[{
    \(\(Plot3D[
        Asoln[x, t], \ {x, \ 0, \  .2}, \ {t, \ 0, \ 1000}];\)\), "\n", 
    \(\(Plot3D[
        Asoln1[x, t], \ {x, \ 0, \  .2}, \ {t, \ 0, \ 1000}];\)\), "\n", 
    \(\(Plot3D[
        Abs[resid[x, t]], \ {x, \ 0, \  .2}, \ {t, \ 0, \ 1000}];\)\), "\n", 
    \(\(Plot3D[
        Abs[resid1[x, t]], \ {x, \ 0, \  .2}, \ {t, \ 0, \ 
          1000}];\)\)}], "Input"]
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}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 2: Two-factor model", "Section"],

Cell[CellGroupData[{

Cell["Setup", "Subsection"],

Cell[BoxData[{
    \(\(pde2\  = \ \ 1\  - \ \((x1\  + \ x2)\)*
            A[x1, \ x2, \ t]\  - \ \(\(Derivative[0, \ 0, \ 1]\)[A]\)[x1, \ 
            x2, \ t]\  + \n\ \ \((k2*\((m2\  - \ x2)\)\  - \ 
                sX2*x2*\[Lambda]2)\)*\(\(Derivative[0, \ 1, \ 0]\)[A]\)[x1, \ 
              x2, \ t]\  + \n\ \ \((k1*\((m1\  - \ x1)\)\  - \ 
                sX1*x1*\[Lambda]1)\)*\(\(Derivative[1, \ 0, \ 0]\)[A]\)[x1, \ 
              x2, \ t]\  - \n\ \ \((\[Nu]*\((\((sX2^2*
                          x2*\(\(Derivative[0, \ 1, \ 0]\)[A]\)[x1, \ x2, \ 
                              t]^2)\)/
                      A[x1, \ x2, \ t]^2\  + \n\ \ \((sX1^2*
                          x1*\(\(Derivative[1, \ 0, \ 0]\)[A]\)[x1, \ x2, \ 
                              t]^2)\)/A[x1, \ x2, \ t]^2)\))\)/
            2\  + \n\ \ \((sX2^2*
                  x2*\(\(Derivative[0, \ 2, \ 0]\)[A]\)[x1, \ x2, \ 
                    t]\  + \n\ \ \ \ \ sX1^2*
                  x1*\(\(Derivative[2, \ 0, \ 0]\)[A]\)[x1, \ x2, \ t])\)/
            2\ ;\)\n\), "\n", 
    \(\(nrule2\  = \ {k1\  -> \ 97/125, \ m1\  -> \ 3209/100000, \ 
          sX1\  -> \ 82/625, \n\ \ \[Lambda]1\  -> \ \(-593\)/5000, \ 
          k2\  -> \ 8991/10000000, \n\ \ m2\  -> \ 1061/50000, \ 
          sX2\  -> \ 2657/50000, \n\ \ \[Lambda]2\  -> \ \(-4151\)/100000, \ 
          k3\  -> \ 3/500, \ 
          m3\  -> \ 1/100, \n\ \ sX3\  -> \ 
            9/50, \ \[Lambda]3\  -> \ \(-13\)/100, \ \[Nu]\  -> \ 
            0}\ ;\)\n\), "\n", 
    \(\(npde2\  = \ pde2\  /. \ nrule2;\)\), "\n", 
    \(\(xargs\  = \ 
        Sequence\  @@ \ \((Transpose[{{x1, \ x2}, \ {m1, \ m2}}]\  /. \ 
              nrule2)\);\)\)}], "Input"]
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Cell[CellGroupData[{

Cell["Solve", "Subsection"],

Cell[BoxData[
    \(\(Asoln2\  = \ 
        NAnnuitySolve[npde2, \ A, \ {t, \ 0, \ 10000}, \ xargs, \ 
          3];\)\)], "Input"],

Cell["Get annuity series from bond prices", "Text"],

Cell[BoxData[{
    \(\(Bsoln2\  = \ 
        NAnnuitySolve[npde2\  - \ 1, \ A, {t, \ 0, \ 1000}, \ xargs, \ 1, \ 
          FunctionalForm\  -> \ Exp];\)\), "\n", 
    \(\(Bcoeffs2\  = \ 
        NBondToAnnuitySeries[Bsoln2[x1, \ x2, \ t], \ {t, \ 0, \ 1000}, \ 
          xargs, \ 5];\)\)}], "Input"],

Cell["Use Bcoeffs to fix series solution", "Text"],

Cell[BoxData[
    \(\(Asoln3\  = \ 
        NAnnuitySolve[npde2, \ A, {t, \ 0, \ 1000}, xargs, \ 3, \ 
          PolynomialOrderDifferential\  -> \ 2, \n\t
          DifferentialLoadings\  -> \ Bcoeffs2];\)\)], "Input"],

Cell["Compare with direct bond series at t = 1000", "Text"],

Cell[BoxData[{
    \(\(bondseries\  = \ 
        MakePolynomial[{x1, x2} - {m1, m2}\  /. \ nrule2, \ t, \n\t5]\  /. \ 
          Bcoeffs2;\)\), "\n", 
    \(\(Plot3D[
        Asoln3[x1, \ x2, \ 1000]\  - \ bondseries\  /. \ 
          t\  -> \ 1000, \n\t{x1, \ 0, \  .2}, \ {x2, \ 
          0, \  .2}];\)\)}], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 3: Symbolic solution to Vasicek bond prices.", "Section"],

Cell[BoxData[
    \(\(pde\  = \ 
        k \((m - x)\) D[B[x, t], \ x]\  + \ 
          s^2/2\ D[B[x, t], \ {x, \ 2}]\  - \n\tD[B[x, t], \ t]\  - \ 
          x\ B[x, t];\)\)], "Input"],

Cell[BoxData[
    \(Bsoln\  = \ 
      AnnuitySolve[pde, \ B, \ t, \ {x, \ m}, \ \ 1, \ 
        FunctionalForm\  -> \ Exp]\)], "Input"],

Cell[BoxData[
    \(\(ResidualFromPDE[pde, \ Bsoln]\)[x, t] // Simplify\)], "Input"],

Cell[BoxData[
    \(BondToAnnuitySeries[Bsoln[x, t] /. s -> 0, \ t, \ {x, \ m}, \ 
      2]\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Example 4: Symbolic solution for 1st 3 terms for annuity with deterministic \
state variable.\
\>", "Section"],

Cell["Compare this to the last part of the previous example.", "Text"],

Cell[BoxData[
    \(\(pde\  = \ 
        1\  + \ k \((m - x)\) D[A[x, t], \ x]\  - \ D[A[x, t], \ t]\  - \ 
          x\ A[x, t];\)\)], "Input"],

Cell[BoxData[
    \(Asoln\  = \ 
      Simplify\  /@ \ \(AnnuitySolve[pde, \ A, \ t, {x, \ m}, \ 2]\)[x, 
          t]\)], "Input"]
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