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McKean–
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Interacting-particle Euler scheme for $dX_t = θ(\bar X_t - X_t) dt + σ `mean_reverting_mckean_vlasov` ). The empirical mean is preserved; the empirical variance approaches the diffusion-only equilibrium.
.. note :: Companion executed notebook: `14_mckean_vlasov.ipynb <../../examples/notebooks/14_mckean_vlasov.ipynb> `_
14 — McKean–
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.. code-block :: python
import numpy as np
import matplotlib.pyplot as plt
from optimizr import _core as opt
plt . rcParams [ 'figure.figsize' ] = ( 7 , 4 )
plt . rcParams [ 'figure.dpi' ] = 110
.. code-block :: python
init = np . linspace ( - 2.0 , 2.0 , 200 ) . tolist ()
init_mean = float ( np . mean ( init ))
res = opt . mean_reverting_mckean_vlasov (
initial = init , theta = 1.0 , sigma = 0.1 ,
n_steps = 1000 , t_horizon = 1.0 , seed = 42 ,
)
n_t = res [ 'n_steps' ]; n_p = res [ 'n_particles' ]
X = np . array ( res [ 'paths_flat' ]) . reshape ( n_t , n_p )
tg = np . array ( res [ 'time_grid' ])
print ( 'initial mean =' , init_mean )
print ( 'final mean =' , float ( X [ - 1 ] . mean ()))
print ( 'final std =' , float ( X [ - 1 ] . std ()))
.. code-block :: python
fig , ax = plt . subplots ()
ax . plot ( tg , X [:, :: 20 ], color = 'tab:blue' , alpha = 0.2 , lw = 0.6 )
ax . plot ( tg , X . mean ( axis = 1 ), color = 'red' , lw = 2 , label = 'empirical mean' )
ax . axhline ( init_mean , color = 'k' , ls = ':' , label = 'initial mean' )
ax . set_xlabel ( 't' ); ax . set_ylabel ( 'X^i_t' ); ax . legend (); ax . grid ( alpha = 0.3 )
ax . set_title ( 'Mean-reverting McKean– )
fig . tight_layout (); plt . show ()
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.. AUTO-PLOT-BEGIN
.. image :: ../_static/auto/algorithms__mckean_vlasov/block_03_fig_01.png
:align: center
:width: 80%
.. AUTO-PLOT-END
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.. image :: ../_static/v2/mckean_vlasov/plot_01.png
:align: center
:width: 80%
.. code-block :: python
fig , ax = plt . subplots ()
ax . hist ( X [ 0 ], bins = 30 , alpha = 0.5 , label = 't = 0' , density = True )
ax . hist ( X [ - 1 ], bins = 30 , alpha = 0.5 , label = 't = T' , density = True )
ax . set_xlabel ( 'x' ); ax . set_ylabel ( 'empirical density' ); ax . legend (); ax . grid ( alpha = 0.3 )
ax . set_title ( 'Marginal density at t = 0 and t = T' )
fig . tight_layout (); plt . show ()
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.. AUTO-PLOT-BEGIN
.. image :: ../_static/auto/algorithms__mckean_vlasov/block_04_fig_01.png
:align: center
:width: 80%
.. AUTO-PLOT-END
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.. image :: ../_static/v2/mckean_vlasov/plot_02.png
:align: center
:width: 80%
**Verified:** empirical mean stays within `0.05` of the initial mean.
API
---
.. code-block :: rust
pub fn simulate_mckean_vlasov < B > ( initial : & [ f64 ], drift : B , cfg : & McKeanVlasovConfig ) -> Result < McKeanVlasovResult >
where B : Fn ( f64 , & [ f64 ]) -> f64 ;
pub struct McKeanVlasovConfig { pub n_particles : usize , pub n_steps : usize , pub t_horizon : f64 , pub sigma : f64 , pub seed : u64 }
pub struct McKeanVlasovResult { pub paths : Array2 < f64 > , pub time_grid : Array1 < f64 > }
