{
  "id": "2405.07519",
  "version": "v1",
  "published": "2024-05-13T07:34:55.000Z",
  "updated": "2024-05-13T07:34:55.000Z",
  "title": "Stability equivalence for stochastic differential equations, stochastic differential delay equations and their corresponding Euler-Maruyama methods in $G$-framework",
  "authors": [
    "Wen Lu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we investigate the stability equivalence problem for stochastic differential delay equations, the auxiliary stochastic differential equations and their corresponding Euler-Maruyama (EM) methods under $G$-framework. More precisely, for $p\\geq 2$, we prove the equivalence of practical exponential stability in $p$-th moment sense among stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs), the auxiliary stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs), and their corresponding Euler-Maruyama methods, provided the delay or the step size is small enough. Thus, we can carry out careful simulations to examine the practical exponential stability of the underlying $G$-SDDE or $G$-SDE under some reasonable assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-13T07:34:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential delay equations",
      "corresponding euler-maruyama methods",
      "stability equivalence",
      "differential delay equations driven",
      "stochastic differential equations driven"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}