{
  "id": "2005.04393",
  "version": "v1",
  "published": "2020-05-09T08:34:40.000Z",
  "updated": "2020-05-09T08:34:40.000Z",
  "title": "Combinatorics for calculating expectations of stochastic differential equations",
  "authors": [
    "Jun Ohkubo"
  ],
  "comment": "16 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "Combinatorial discussion is proposed and applied for calculating expectations of stochastic differential equations. Starting from the duality theory of stochastic processes, some modifications of interpretation and usages of time-ordering operators naturally lead to combinatorial discussions. As a demonstration, the first and second moments for the Ornstein-Uhlenbeck process are re-derived from the combinatorial discussion. Furthermore, two numerical methods for practical applications are proposed. One is based on a conventional exponential expansion and the Pade approximation. Another uses a resolvent of a time-evolution operator, and the Aitken series acceleration method is also employed. These two proposals recover the correct results approximately.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-09T08:34:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential equations",
      "calculating expectations",
      "combinatorial discussion",
      "combinatorics",
      "aitken series acceleration method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}