{
  "config": {
    "gamma": 1.0,
    "dt": 0.01,
    "steps": 60000,
    "burn": 30000
  },
  "bound_sweep": [
    {
      "lambda": 0.1,
      "bound": 1.0,
      "max_abs": 0.9999993702580099,
      "ratio": 0.9999993702580099
    },
    {
      "lambda": 0.7,
      "bound": 1.0,
      "max_abs": 0.9567823718178728,
      "ratio": 0.9567823718178728
    },
    {
      "lambda": 2.0,
      "bound": 1.0,
      "max_abs": 0.7876499928340427,
      "ratio": 0.7876499928340427
    },
    {
      "lambda": 8.0,
      "bound": 1.0,
      "max_abs": 0.4914642969231532,
      "ratio": 0.4914642969231532
    }
  ],
  "variance_sweep": [
    {
      "lambda": 0.1,
      "emp_var": 0.22184790725174827,
      "pred_var": 0.20833333333333334,
      "rel_err": 0.06486995480839175
    },
    {
      "lambda": 0.3,
      "emp_var": 0.12992748191808579,
      "pred_var": 0.15625,
      "rel_err": -0.16846411572425102
    },
    {
      "lambda": 0.7,
      "emp_var": 0.10295221051962604,
      "pred_var": 0.10416666666666667,
      "rel_err": -0.01165877901159007
    },
    {
      "lambda": 1.5,
      "emp_var": 0.06407518340680937,
      "pred_var": 0.0625,
      "rel_err": 0.025202934508949948
    },
    {
      "lambda": 3.0,
      "emp_var": 0.03290677451371123,
      "pred_var": 0.03571428571428571,
      "rel_err": -0.07861031361608539
    },
    {
      "lambda": 8.0,
      "emp_var": 0.014326501207616789,
      "pred_var": 0.014705882352941176,
      "rel_err": -0.02579791788205832
    }
  ],
  "covariance_check": {
    "lags": [
      0.0,
      0.1,
      0.25,
      0.5,
      1.0,
      2.0,
      4.0
    ],
    "empirical": [
      0.37658421035616224,
      0.3736366455986064,
      0.3605711860031419,
      0.3246079799616098,
      0.23884382486567812,
      0.11436068995098818,
      0.016533536598099724
    ],
    "corrected": [
      0.4166666666666667,
      0.41397307276201817,
      0.401701048522139,
      0.36724752063153887,
      0.27961901426918007,
      0.13402013948545816,
      0.02285836534140096
    ],
    "supplied": [
      -0.0,
      0.03695748191370174,
      0.07720072224238693,
      0.11452641241794159,
      0.12633591378107903,
      0.07763043813686953,
      0.015226849137761721
    ],
    "rmse_corrected": 0.03552170395906621,
    "rmse_supplied": 0.23712879574183154
  },
  "kurtosis_sweep": [
    {
      "K": 1,
      "excess_kurtosis": -1.3340502460584176,
      "variance": 0.4004448444372525,
      "pred_var": 0.4166666666666667
    },
    {
      "K": 2,
      "excess_kurtosis": -0.5310430581684686,
      "variance": 0.3941007196072442,
      "pred_var": 0.4166666666666666
    },
    {
      "K": 4,
      "excess_kurtosis": -0.65313176508716,
      "variance": 0.39936926351644425,
      "pred_var": 0.4166666666666667
    },
    {
      "K": 8,
      "excess_kurtosis": 0.058334978098325774,
      "variance": 0.4070997988505528,
      "pred_var": 0.4166666666666666
    },
    {
      "K": 16,
      "excess_kurtosis": -0.22682709084103436,
      "variance": 0.508631491649338,
      "pred_var": 0.4166666666666667
    }
  ],
  "quadratic_optimizer": {
    "none": [
      0.0,
      0.0
    ],
    "telegraph": [
      0.0346325191211659,
      0.028322584889397096
    ],
    "ou": [
      0.03496700678582951,
      0.025026321910527633
    ]
  },
  "note": "The covariance formula in the prompt is tested literally; it is zero at tau=0 and is not a valid covariance. corrected is derived from the same SDE."
}