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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Levy-Konstruktion (wie in d
er Aufgabenstellung beschrieben):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 478 "wiener:= proc(N,omega,a,b)\nglobal _seed:\nlocal u1,
 u2, g1, W, delta, delta_W, f, g, X, Y,  A, B, C, k, n:\n_seed:=omega:
\nu1:=rand()/1e12;\nu2:=rand()/1e12;\ng1:=evalf(sqrt(-2*ln(u1))*cos(2*
Pi*u2));\nW[0]:=0;\nW[1]:=g1;\ndelta:=1/2:\nwhile delta >= 2^(-N) do\n
\nfor k from 1 to 1/delta-1 by 2 do\n  u1:=rand()/1e12;\n  u2:=rand()/
1e12;\n  g1:=evalf(sqrt(-2*ln(u1))*cos(2*Pi*u2));\n  W[k*delta]:=(W[(k
-1)*delta]+W[(k+1)*delta])/2 +sqrt(delta/2.0)*g1:\nod:\n\ndelta:=delta
/2:\nod:\n\ndelta:=2^(-N):\n\n" }{TEXT -1 20 "Euler-Approximation:" }
{MPLTEXT 1 0 152 "\nf:=x->a*x:\ng:=x->b*x:\nX[0]:=1:\nfor n from 0 to \+
2^N-1 do\n  delta_W[n]:=W[(n+1)*delta]-W[n*delta]:\n  X[n+1]:=X[n]+f(X
[n])*delta+g(X[n])*delta_W[n]:\nod:\n\n" }{TEXT -1 15 "Heun-Verfahren:
" }{MPLTEXT 1 0 183 "\nY[0]:=1:\nfor n from 0 to 2^N-1 do\n  Y[n+1]:=Y
[n]+1/2*(f(Y[n])+f(Y[n]+f(Y[n])*delta+g(Y[n])*delta_W[n]))*delta+1/2*(
f(Y[n])+f(Y[n]+f(Y[n])*delta+g(Y[n])*delta_W[n]))*delta_W[n]:\nod:\n\n
" }{TEXT -1 44 "Graphische Darstellung:\nEuler-Approximation:" }
{MPLTEXT 1 0 177 "\nA:=PLOT(CURVES([seq([evalf(i*delta),X[i]],i=0..2^N
)]),COLOR(HUE,0.9),THICKNESS(1),TITLE(\"Exakte L\366sung (blau) und zw
ei Approximationen f\374r N =\".N),FONT(HELVETICA,OBLIQUE,10)):\n" }
{TEXT -1 14 "Exakte L\366sung:" }{MPLTEXT 1 0 150 "\nB:=PLOT(CURVES([s
eq([evalf(j*delta),X[0]*exp((a-b^2/2)*(j*delta)+b*W[j*delta])],j=0..2^
N)]),COLOR(HUE,0.7),THICKNESS(1),FONT(HELVETICA,OBLIQUE,10)):\n" }
{TEXT -1 19 "Heun-Approximation:" }{MPLTEXT 1 0 133 "\nC:=PLOT(CURVES(
[seq([evalf(i*delta),Y[i]],i=0..2^N)]),COLOR(HUE,0.1),THICKNESS(1),FON
T(HELVETICA,OBLIQUE,10)):\ndisplay(A,B,C);\nend:\n" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 46 "Graphische Darstellung f\374r N = 2, 4, 6, 8, 10:
" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fo
r j from 1 to 7 do  \n  wiener(j,1111,1.5,2):\nod;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Offensichtl
ich konvergiert die HEUN-Approximation nicht." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 1 1 1803 }