Hidden Geometry of Bidirectional Grid-Constrained Stochastic Processes

<div>Itô diffusions with &amp; without BGC. The zero drift in (a) is constrained in (b) the more it deviates from the origin, causing the hidden reflective upper barrier <svg height="12.5501pt" id="M25" style="vertical-align:-3.313951pt" version="1.1" viewbox="-0.0498162 -9.23615 17.9826 12.5501" width="17.9826pt" 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="M644 408C714 434 749 483 749 555C749 596 732 631 697 660C663 689 615 704 553 704L426 604H424C387 657 322 695 246 695C194 695 150 681 114 653C79 626 61 588 61 540C61 508 78 479 93 463L128 433C160 408 175 390 175 350C175 311 136 281 100 269L109 247C154 258 192 278 225 307S275 368 275 401C275 521 160 504 160 597C160 634 186 664 226 664C293 664 340 574 340 431C340 296 306 211 237 174H227C191 174 157 161 126 134C95 108 69 78 48 44L68 30C87 63 124 79 178 79C198 79 218 71 227 63C238 56 263 34 275 25C322 -11 368 -24 452 -24C540 -24 616 -1 679 46S774 150 774 217C774 297 727 373 644 406V408ZM632 531C632 463 598 385 455 379C462 408 465 436 465 465C465 501 459 537 446 573C468 595 484 609 495 616S521 627 539 627C568 627 591 617 607 596C624 575 632 554 632 531ZM282 163C330 193 366 220 389 244C412 269 431 301 446 342C459 346 473 348 486 348C531 348 570 334 603 305C637 277 654 242 654 199C654 96 609 25 524 25C495 25 457 43 409 80C400 88 370 113 354 124C339 136 310 152 282 160V163Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.686,3.132)"><path d="M723 650H473V615C541 611 564 591 570 556C575 526 580 488 580 417V310C580 143 528 53 409 53C301 53 247 140 247 290V508C247 602 257 608 330 615V650H29V615C103 608 113 602 113 508V272C113 155 140 81 196 36C244 -1 307 -15 373 -15C444 -15 512 10 558 58C614 118 630 202 630 308V417C630 488 634 525 639 556C644 591 669 612 723 615V650Z"></path></g></svg> and hidden reflective lower barrier <svg height="12.4177pt" id="M26" style="vertical-align:-3.18155pt" version="1.1" viewbox="-0.0498162 -9.23615 16.3567 12.4177" width="16.3567pt" 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="M644 408C714 434 749 483 749 555C749 596 732 631 697 660C663 689 615 704 553 704L426 604H424C387 657 322 695 246 695C194 695 150 681 114 653C79 626 61 588 61 540C61 508 78 479 93 463L128 433C160 408 175 390 175 350C175 311 136 281 100 269L109 247C154 258 192 278 225 307S275 368 275 401C275 521 160 504 160 597C160 634 186 664 226 664C293 664 340 574 340 431C340 296 306 211 237 174H227C191 174 157 161 126 134C95 108 69 78 48 44L68 30C87 63 124 79 178 79C198 79 218 71 227 63C238 56 263 34 275 25C322 -11 368 -24 452 -24C540 -24 616 -1 679 46S774 150 774 217C774 297 727 373 644 406V408ZM632 531C632 463 598 385 455 379C462 408 465 436 465 465C465 501 459 537 446 573C468 595 484 609 495 616S521 627 539 627C568 627 591 617 607 596C624 575 632 554 632 531ZM282 163C330 193 366 220 389 244C412 269 431 301 446 342C459 346 473 348 486 348C531 348 570 334 603 305C637 277 654 242 654 199C654 96 609 25 524 25C495 25 457 43 409 80C400 88 370 113 354 124C339 136 310 152 282 160V163Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.686,3.132)"><path d="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g></svg> to emerge, together with horizontal bands to form due to the discretization effect of BGC.</div>

Journal of Probability and Statistics

fig1

Figure 1

Figure 1: Hidden Geometry of Bidirectional Grid-Constrained Stochastic Processes 