{
  "id": "1304.0217",
  "version": "v3",
  "published": "2013-03-31T15:38:03.000Z",
  "updated": "2014-10-27T17:00:44.000Z",
  "title": "Causal interpretation of stochastic differential equations",
  "authors": [
    "Alexander Sokol",
    "Niels Richard Hansen"
  ],
  "journal": "Electronic Journal of Probability, Vol. 19 (100) (2014)",
  "doi": "10.1214/EJP.v19-2891",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a L\\'evy process, the postintervention distribution is identifiable from the generator of the SDE.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-08T10:41:45.000Z",
      "abstract": "We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a L\\'evy process, the postintervention distribution is identifiable from the semigroup of the SDE. Also for the case of L\\'evy driving noise, we relate our results to the notion of weak conditional local independence (WCLI) by proving that if a coordinate $X^i$ is locally unaffected by an intervention in another coordinate $X^j$, then $X^i$ is WCLI of $X^j$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-27T17:00:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential equations",
      "causal interpretation",
      "weak conditional local independence",
      "postintervention structural equation models",
      "postintervention sde"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.0217S"
    }
  }
}