(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 54190, 1621]*) (*NotebookOutlinePosition[ 55051, 1651]*) (* CellTagsIndexPosition[ 55007, 1647]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Reflect.nb"], "Title", Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center], Cell[TextData["Wilfrid S. Kendall"], "Subtitle", Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center], Cell[TextData[ "Statistics, University of Warwick, \nCoventry CV4 7AL, UK."], "Subsubtitle", Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center], Cell[CellGroupData[{ Cell[TextData["Contact information:"], "SmallText", Editable->False, Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0.501961, 0, 0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData["Email: w.s.kendall@warwick.ac.uk"], "Name", Editable->False, Evaluatable->False, CellHorizontalScrolling->False, FontFamily->"Times New Roman", FontSize->10, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1], Background->GrayLevel[1]], Cell[TextData["URL: http://www.warwick.ac.uk/statsdept/Staff/WSK"], "Name", Editable->False, Evaluatable->False, CellHorizontalScrolling->False, FontFamily->"Times New Roman", FontSize->10, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Acknowledgements:"], "SmallText", Editable->False, Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0.501961, 0, 0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[ "The research reported here was supported by EPSRC grants GR/71677 \ (Stochastic calculus in AXIOM using modules of stochastic differentials) and \ GR/L56831 (Perfect simulation in stochastic geometry), and a joint \ EPSRC/BBSRC research grant (Multi-strain species modelling and control via \ differential algebra reductions). This Mathematica notebook was constructed \ on a visit to MSRI Berkeley CA during its 1997-1998 program Stochastic \ Analysis. Finally, it is a pleasure to expres my gratitude to my friends \ Suzanne Scotchmer and Joseph Farrell for the generous hospitality they showed \ to me during my visit to MSRI."], "SmallText", Editable->False, Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }, Closed]], Cell[TextData[ "Purpose: this is a Mathematica package (working under Mathematica 3.0) which \ demonstrates a pretty fact about coupled pairs of Brownian motions reflecting \ off a half-plane, and coupled by reflection. The result is easy to prove \ directly once known: however it is indeed the case that it was discovered \ using Itovsn3 (in fact in its REDUCE implementation)."], "SmallText", Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontSlant->"Italic", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell[TextData["References:"], "SmallText", Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[ "This calculation was a significant step in the following paper:"], "SmallText", Editable->False, Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[ "K.Burdzy and WSK: \"Efficient Markovian Couplings: Examples and \ Counterexamples\", in preparation (1998)."], "SmallText", Editable->False, Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontSlant->"Italic", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[TextData["It also contributed usefully to"], "SmallText", Editable->False, Evaluatable->False], Cell[TextData[ "R. Banuelos and K. Burdzy: \"On the `hot spots' conjecture of J. Rauch\", \ Preprint, Department of Mathematics, University of Washington (1997),\nK. \ Burdzy and W. Werner: \"A counterexample to the ``hot spots'' conjecture\", \ Preprint, Department of Mathematics, University of Washington (1998)."], "SmallText", Editable->False, Evaluatable->False, FontFamily->"Times New Roman", FontSize->10, FontSlant->"Italic", FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "IMPORTANT: this notebook assumes nothing has been previously defined. Quit \ Mathematica and reload if this is not the case!\n\nFirst we must load the \ Itovsn3 package."], "Text", Evaluatable->False], Cell[BoxData[ \(Needs["\"]\)], "Input", PageWidth->Infinity] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Set up basic semimartingales including local times"], "Section", Evaluatable->False], Cell[TextData[ "Now we set up Itovsn3 with time t as the basic semimartingale and add a \ two-dimensional Brownian motion:"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(ItoReset[t, dt]\), \(BrownBasis[{A, B}, {A0, B0}]\)}], "Input", PageWidth->Infinity], Cell[BoxData[ InterpretationBox[GridBox[{ {"\<\"Itovsn3 resetting ...\"\>"}, {"\<\"Itovsn3 initialized\"\>"}, { InterpretationBox[ \("with time semimartingale "\[InvisibleSpace]t\), SequenceForm[ "with time semimartingale ", t], Editable->False]}, { InterpretationBox[ \("and time differential "\[InvisibleSpace]dt\), SequenceForm[ "and time differential ", dt], Editable->False]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {"Itovsn3 resetting ...", "Itovsn3 initialized", SequenceForm[ "with time semimartingale ", t], SequenceForm[ "and time differential ", dt]}]]], "Output"] }, Open ]], Cell[TextData[ "Next step is to introduce the two local times, one for each of the two \ coupled two-dimensional reflecting Brownian motions."], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(Introduce[L, dL]\), \(\(AddQuadVar[dL\^2, 0]; \)\), \(\(AddQuadVar[dL\ dA, 0]; \)\), \(\(AddQuadVar[dL\ dB, 0]; \)\), \(\(AddDrift[dL, dL]; \)\), \(\(AddFixed[0, L, L0]; \)\), \(Introduce[H, dH]\), \(\(AddQuadVar[dH\^2, 0]; \)\), \(\(AddQuadVar[dH\ dA, 0]; \)\), \(\(AddQuadVar[dH\ dB, 0]; \)\), \(\(AddQuadVar[dH\ dL, 0]; \)\), \(\(AddDrift[dH, dH]; \)\), \(\(AddFixed[0, H, H0]; \)\)}], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \({dL, dB, dA, dt}\)], "Output"], Cell[BoxData[ \({dH, dL, dB, dA, dt}\)], "Output"] }, Open ]], Cell[TextData[ "We now redefine the drifts for (A,B) to allow for reflection (more \ economical than introducing further semimartingales!). We assume the \ half-plane of reflection is the region where A is negative."], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(AddDrift[dA, dL]\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(dL\/dt\)], "Output"] }, Open ]], Cell[TextData[ "Notice that the output is mathematically formal nonsense! Note also that the \ local time dL has the property A*dL == 0, which we will need to use in \ simplifications at the end!"], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[ "We inspect the status of the stochastic differential multiplication and \ drift tables:"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(ItoStatus[]\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \("---------------"\)], "Print"], Cell[BoxData[ \("Summary of current structure of stochastic differentials"\)], "Print"], Cell[BoxData[ \("- - - - - - - -"\)], "Print"], Cell[BoxData[ \("Current second-order structure of semimartingale differentials:"\)], "Print"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "dH", "dL", "dB", "dA", "dt"}, {"dH", "0", "0", "0", "0", "0"}, {"dL", "0", "0", "0", "0", "0"}, {"dB", "0", "0", "dt", "0", "0"}, {"dA", "0", "0", "0", "dt", "0"}, {"dt", "0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableHeadings -> {{dH, dL, dB, dA, dt}, {dH, dL, dB, dA, dt}}]&)]], "Print"], Cell[BoxData[ \("- - - - - - - -"\)], "Print"], Cell[BoxData[ \("Current first-order structure of semimartingale differentials:"\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {"dH", "dL", "dB", "dA", "dt"}, {"dH", "dL", "0", "dL", "dt"} }, RowSpacings->3, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {dH, dL, 0, dL, dt}, TableDirections -> Row, TableHeadings -> {{dH, dL, dB, dA, dt}, {"Drifts:"}}]]], "Print"], Cell[BoxData[ \("- - - - - - - -"\)], "Print"], Cell[BoxData[ \("Current initial values:"\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {"H", "L", "B", "A", "t"}, {"H0", "L0", "B0", "A0", "0"} }, RowSpacings->3, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {H0, L0, B0, A0, 0}, TableDirections -> Row, TableHeadings -> {{H, L, B, A, t}, {"Initially:"}}]]], "Print"], Cell[BoxData[ \("---------------"\)], "Print"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "Set up reflection coupling using elementary geometry"], "Section", Evaluatable->False], Cell[TextData[ "The next step involves some simple geometry. We construct a unit vector UU \ pointing from (A,B) to (X,Y), and a perpendicular unit vector VV."], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(AA = {A, B}; XX = {X, Y}; \), \(\(UU = XX - AA; \)\), \(UU = UU\/\@\(UU . UU\)\), \(VV = {{0, \(-1\)}, {1, 0}} . UU\), \(0 == UU . VV\)}], "Input", PageWidth->Infinity], Cell[BoxData[ \({\(\(-A\) + X\)\/\@\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\), \(\(-B\) + Y\)\/\@\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\)}\)], "Output"], Cell[BoxData[ \({\(- \(\(\(-B\) + Y \)\/\@\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\)\)\), \(\(-A\) + X\)\/\@\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\)}\)], "Output"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "We use the unit vector UU to build a vector of stochastic differentials \ reflected in the UU direction based on {dA-dL,dB}: notice we take out the \ local time term from the dA differential to make it into a standard \ Brownian differential!"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(dAA = {dA - dL, dB}\), \(dXX = \((IdentityMatrix[2] - 2\ Transpose[{UU}] . {UU})\) . dAA\)}], "Input", PageWidth->Infinity], Cell[BoxData[ \({dA - dL, dB}\)], "Output"], Cell[BoxData[ \({\(- \(\(2\ dB\ \((\(-A\) + X)\)\ \((\(-B\) + Y)\)\)\/\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\)\)\) + \((dA - dL)\)\ \((1 - \(2\ \((\(-A\) + X)\)\^2\)\/\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\))\), \(-\(\(2\ \((dA - dL)\)\ \((\(-A\) + X)\)\ \((\(-B\) + Y)\)\)\/\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\)\)\) + dB\ \((1 - \(2\ \((\(-B\) + Y)\)\^2\)\/\(\((\(-A\) + X)\)\^2 + \((\(-B\) + Y)\)\^2\))\)}\)], "Output"] }, Open ]], Cell[TextData[ "We now introduce the reflection coupling, by using the geometry to construct \ the coupled process (X,Y). Note the local time term added in the dX \ sde."], "Text", Evaluatable->False], Cell[BoxData[{ \(Itosde[X, dX == dXX . {1, 0} + dH, X0]\), \(Itosde[Y, dY == dXX . {0, 1}, Y0]\)}], "Input", PageWidth->Infinity], Cell[TextData[ "Again the local time term dH has the property X*dH == 0. We summarize \ the local time relationships:"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(localtimes = {A\ dL \[Rule] 0, X\ dH \[Rule] 0}\)], "Input", PageWidth->Infinity], Cell[BoxData[ \({A\ dL \[Rule] 0, dH\ X \[Rule] 0}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Computation of hinge location"], "Section", Evaluatable->False], Cell[TextData[ "The next step is to compute the location of the hinge (in the terminology of \ Burdzy and Kendall, 1998), the vertical intercept Z of the perpendicular \ bisector between {A,B} and {X,Y}. 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"], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[TextData["Click here for an amusing exercise: "], "Subsubsection", Evaluatable->False], Cell[TextData["Now figure out a computation-free approach. "], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[TextData["OPEN THIS CELL-GROUP FOR A HINT"], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[ "\t... it all depends on looking at the problem from the right viewpoint, or \ origin ..."], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "Stochastic dynamics of distance between hinge and bisector"], "Section", Evaluatable->False], Cell[TextData[ "Just to finish off, we examine the distance rho between {0,Z} and the \ perpendicular bisector point. First we obtain the expression for rho from \ our previous solution of simultaneous linear equations:"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(rho = Simplify[rho /. solution]\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(\(\((A + X)\)\ \@\(\((A - X)\)\^2 + \((B - Y)\)\^2\)\)\/\(2\ \((\(-B\) + Y)\)\)\)], "Output"] }, Open ]], Cell[TextData[ "Now we compute its stochastic differential sd and its drift sd1. It is \ immediately clear that the drift is zero when the local time terms vanish, as \ the coefficient of dt is zero: "], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(sd = ItoD[rho]; \)\), \(\(sd1 = Expand[Together[Drift[sd]]]; \)\), \(0 == Coefficient[sd1, dt]\)}], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "So consider the coefficient of dL. Here we know A must be zero, as we \ interpret dL as the differential of the local time of A at zero:"], "Text",\ Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(0 == Together[Coefficient[sd1, dL] - \(X - A\)\/\(4\ rho\)] /. A \[Rule] 0\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "Now consider the differential of dH. Similarly, here we know X must be \ zero, as we interpret dH as the differential of the local time of X at \ zero:"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(0 == Together[Coefficient[sd1, dH] - \(A - X\)\/\(4\ rho\)] /. X \[Rule] 0\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "These last two calculations give us simpler forms for the drift of rho \ when either A or X is zero. We deduce the drift of rho is given in \ general by"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(driftdrho = \(\((dL - dH)\)\ \((X - A)\)\)\/\(4\ rho\)\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(\(\((\(-dH\) + dL)\)\ \((\(-A\) + X)\)\ \((\(-B\) + Y)\)\)\/\(2\ \((A + X)\)\ \@\(\((A - X)\)\^2 + \((B - Y)\)\^2\)\)\)], "Output"] }, Open ]], Cell[TextData[ "Here is a check. We know either dH, dL are both zero (off the boundary!)"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(0 == Together[sd1 - driftdrho] /. {dL \[Rule] 0, dH \[Rule] 0}\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData["or X, dL are both zero (AA is on the axis X=0)"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(0 == Together[sd1 - driftdrho] /. {dL \[Rule] 0, X \[Rule] 0}\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData["or dH, A are both zero (XX is on the axis X=0)"], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[BoxData[ \(0 == Together[sd1 - driftdrho] /. {A \[Rule] 0, dH \[Rule] 0}\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "Now consider the second-order aspect of sd = ItoD[rho]. For some reason it \ is computationally burdensome to compute ItoExpand[sd^2] so we ease the \ task by noting that dA and dB generate the filtration."], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(sd2a = Together[ItoExpand[sd\ dA]\/dt]\), \(sd2b = Together[ItoExpand[sd\ dB]\/dt]\)}], "Input", PageWidth->Infinity], Cell[BoxData[ \(\(\(-B\) + Y\)\/\@\(\((A - X)\)\^2 + \((B - Y)\)\^2\)\)], "Output"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(sd2b\)\" is similar to \ existing symbol \"\!\(sd2a\)\"."\)], "Message"], Cell[BoxData[ \(\(A - X\)\/\@\(\((A - X)\)\^2 + \((B - Y)\)\^2\)\)], "Output"] }, Open ]], Cell[TextData[ "We deduce that rho behaves as a Brownian motion when XX and AA are \ away from the boundary: we have already computed the behaviour on the \ boundary!"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(1 == Together[sd2a\^2 + sd2b\^2]\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[ "So rho behaves as a standard Brownian motion with a singular drift imposed \ whenever one of the coupled Brownian motions hits the boundary.\n\nAs a \ further exercise, use Itovsn3 to compute the characteristics of the \ semimartingale given by the angle made by the bisector with the boundary."], "Text", Evaluatable->False] }, Closed]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 800}, {0, 544}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{771, 478}, WindowMargins->{{2, Automatic}, {Automatic, 2}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (*********************************************************************** Cached data follows. 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