The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method

<table class="fixed-width table-group" id="tab1"><tr><td><table class="table"><colgroup><col style="width:4.73em"/><col style="width:4.66em"/></colgroup><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td class="align_left">step-size </td><td class="align_center">error (<svg height="6.1673pt" id="M320" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 5.63229 6.1673" width="5.63229pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M401 397C401 420 368 448 302 448C245 448 169 416 122 377C62 327 23 254 23 169C23 45 83 -12 181 -12C252 -12 323 29 374 85L358 107C305 62 257 43 210 43C147 43 110 98 110 189V214L313 208L321 256L115 250C132 342 190 405 253 405C291 405 323 389 346 360C356 348 364 348 377 357C392 367 401 384 401 397Z" id="g113-227"></path></g></svg>)</td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td class="align_left">0.0050</td><td class="align_center">0.1318</td></tr><tr><td class="align_left">0.0025</td><td class="align_center">0.0673</td></tr><tr><td class="align_left">0.00125</td><td class="align_center">0.0347</td></tr><tr><td class="align_left">0.00062</td><td class="align_center">0.0177</td></tr><tr><td class="align_left">0.00031</td><td class="align_center">0.0088</td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>The error results for the Milstein scheme in 2<svg height="9.49473pt" id="M319" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 7.34169 9.49473" width="7.34169pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z" id="g113-101"></path></g></svg> case.</div>

Abstract and Applied Analysis

tab1

Table 1

Table 1: The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method 