sde

KO | EN

gitlite — search

#hacktoberfest #react #llm #go #android #deep-learning #docker #macos #windows #kubernetes #vue #awesome-list #ios #open-source

(★ 21)

Numerical experiments with stochastic differential equations

sde Latest Version Download

Download Latest Version (.zip)

// repository documentation

Was this content helpful?

★ 0 (0 ratings)

# sde

[![Build Status](https://travis-ci.org/ocramz/sde.png)](https://travis-ci.org/ocramz/sde)

Numerical experiments with stochastic differential equations in Haskell

![stochastic-volatility-1](https://rawgit.com/ocramz/sde/master/doc/stoch_volatility1.png)

In the figure above, a sample path (left) and histogram of the variable `y_t` as formulated in the following stochastic volatility model [1]:

y_t = a exp(x_t / 2) v_t           (1)

x_t = b x_{t-1} + sigma u_t

where

u_t ~ N(0, 1)
    
    v_t ~ S_alpha(1, 0, 0).

`S_alpha(1, 0, 0)` denotes the Levy-stable distribution with parameters `(alpha, 1)`, where `0 < alpha <= 2`. The special cases `alpha=1` and `alpha=2` correspond to the Cauchy-Lorentz and Gaussian distributions, respectively. Values of the `alpha` parameter smaller than 2 result in sudden large deviations typical of "heavy tailed" distributions, which can be used to model shocks or phase changes in underlying phenomena.

## Implementation details and usage notes

The library relies on `mwc-probability` for its primitive sampling functionality, and on the `StateT` monad transformer.

newtype Transition m a = Trans { runTrans :: Gen (PrimState m) -> StateT a m a }

The SDE integration process can be seen as an interleaved sequence of random sampling and state transformation. Formally, we are sampling independent _increments_ (a Wiener process) of the state variable. The `transition` function, shown below, declares one such stochastic transion:

transition :: Monad m => Prob m a -> (b -> a -> b) -> Transition m b
    transition msf f = Trans $ \g -> do
      x <- get
      w <- lift $ sample msf g
      let z = f x w
      put z
      return z

For example, the stochastic volatility model shown in the beginning can be implemented as follows :

data SV1 = SV1 {sv1x :: Double, sv1y :: Double} deriving (Eq, Show)

stochVolatility1 :: PrimMonad m => Double -> Double -> Double -> Double -> Transition m SV1
    stochVolatility1 a b sig alpha = transition randf f where
      randf = (,) <$> normal 0 1
                  <*> alphaStable100 alpha
      f (SV1 x _) (ut, vt) = let xt = b * x + sig * ut
                                 yt = a * exp (xt / 2) * vt
                             in SV1 xt yt

This formulation lets the library user focus exclusively on the mathematical details of the model she wishes to simulate.

Once the stochastic model is written in terms of the `Transition` type, we may simulate a sample path with a runner function such as the following:

samplePathSDE :: Monad m => Int -> Transition m s -> s -> Gen (PrimState m) -> m [s]
    samplePathSDE n sde x0 g = evalStateT (replicateM n (runTrans sde g)) x0

The random generator `g :: Gen (PrimState m)` can be instantiated either with `create` (which by default will reuse the seed therefore producing the same random sequence at every call), or within the bracket `withSystemRandom . asIOGen`, which taps from the operating system's random generator. Please refer to the documentation of `mwc-probability` for details.

## Notes

Currently, only the Euler-Maruyama integration method has been implemented (the `sampleSDE` function shown above); this works for some models but is inaccurate and unstable for others; therefore one of the next development goals will be to include higher-order integration schemes such as those shown in Wilkie [2].

[1] Li, K. and Oestergaard, J. - Inference for a stochastic volatility model using MCMC with ABC-SMC approximation

[2] Wilkie, J. - Numerical methods for stochastic differential equations (https://arxiv.org/abs/quant-ph/0407039)

sde

sde Latest Version Download

Today's picks

Was this content helpful?

Recent Feedback

sde

sde Latest Version Download

Today's picks

Was this content helpful?

Recent Feedback

Download README