Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their...
Transcript of Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their...
![Page 1: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/1.jpg)
Stochastic models and theirdeterministic analogues
Phil Pollett
Department of Mathematics and MASCOS
University of Queensland
AUSTRALIAN RESEARCH COUNCILCentre of Excellence for Mathematicsand Statistics of Complex Systems
MASCOS APWSPM06, February 2006 - Page 1
![Page 2: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/2.jpg)
A precipitation reaction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.3513
14
15
16
17
18
19
20A + B ⇀↽ C : a = b = 0, c = 20, k1 = 4, k2 = 5, V = 103
Time
Con
centr
atio
nof
C
Na+ + Cl− ⇋ NaCl
MASCOS APWSPM06, February 2006 - Page 2
![Page 3: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/3.jpg)
A precipitation reaction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.3513
14
15
16
17
18
19
20A + B ⇀↽ C : a = b = 0, c = 20, k1 = 4, k2 = 5, V = 103
Time
Con
centr
atio
nof
C
dX
dt= k1(c −X)2 − k2X
Xt =x1(c − x2)− x2(c − x1)e
−λt
c − x2 − (c − x1)e−λt
λ = k1(x2 − x1) x2 = c2/x1 x1 = 15.5861
Na+ + Cl− ⇋ NaCl
MASCOS APWSPM06, February 2006 - Page 3
![Page 4: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/4.jpg)
A precipitation reaction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.3513
14
15
16
17
18
19
20A + B ⇀↽ C : a = b = 0, c = 20, k1 = 4, k2 = 5, V = 103
Time
Con
centr
atio
nof
C
dX
dt= k1(c −X)2 − k2X
Xt =x1(c − x2)− x2(c − x1)e
−λt
c − x2 − (c − x1)e−λt
λ = k1(x2 − x1) x2 = c2/x1 x1 = 15.5861
Na+ + Cl− ⇋ NaCl
MASCOS APWSPM06, February 2006 - Page 4
![Page 5: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/5.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
Davidson, J. (1938) On the growth of the sheep populationin Tasmania, Trans. Roy. Soc. Sth. Austral. 62, 342–346.
MASCOS APWSPM06, February 2006 - Page 5
![Page 6: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/6.jpg)
A deterministic model
dn
dt= nf(n).
The net growth rate per individual is a function of thepopulation size n.
We want f(n) to be positive for small n and negative forlarge n.
MASCOS APWSPM06, February 2006 - Page 6
![Page 7: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/7.jpg)
A deterministic model
dn
dt= nf(n).
The net growth rate per individual is a function of thepopulation size n.
We want f(n) to be positive for small n and negative forlarge n. Simply set f(n) = r − sn to give
dn
dt= n(r − sn).
MASCOS APWSPM06, February 2006 - Page 6
![Page 8: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/8.jpg)
A deterministic model
dn
dt= nf(n).
The net growth rate per individual is a function of thepopulation size n.
We want f(n) to be positive for small n and negative forlarge n. Simply set f(n) = r − sn to give
dn
dt= n(r − sn).
This is the classical Verhulst∗ model (or logistic model):
∗Verhulst, P.F. (1838) Notice sur la loi que la population suit dans son accroisement,
Corr. Math. et Phys. X, 113–121.
MASCOS APWSPM06, February 2006 - Page 6
![Page 9: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/9.jpg)
The Verhulst model
Pierre Francois Verhulst (1804–1849, Brussels, Belgium)
MASCOS APWSPM06, February 2006 - Page 7
![Page 10: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/10.jpg)
The Verhulst model
MASCOS APWSPM06, February 2006 - Page 8
![Page 11: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/11.jpg)
The Verhulst model
MASCOS APWSPM06, February 2006 - Page 9
![Page 12: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/12.jpg)
The Verhulst model
An alternative formulation has r being the growth rate withunlimited resources and K being the “natural” population size(the carrying capacity). We put f(n) = r(1 − n/K) giving
dn
dt= rn(1 − n/K),
which is the original model with s = r/K.
MASCOS APWSPM06, February 2006 - Page 10
![Page 13: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/13.jpg)
The Verhulst model
An alternative formulation has r being the growth rate withunlimited resources and K being the “natural” population size(the carrying capacity). We put f(n) = r(1 − n/K) giving
dn
dt= rn(1 − n/K),
which is the original model with s = r/K.
Integration gives
nt =K
1 +(
K−n0n0
)
e−rt(t ≥ 0).
MASCOS APWSPM06, February 2006 - Page 10
![Page 14: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/14.jpg)
Verhulst-Pearl model
This formulation is due to Raymond Pearl:
Pearl, R. and Reed, L. (1920) On the rate of growth of population of the
United States since 1790 and its mathematical representation, Proc. Nat.
Academy Sci. 6, 275–288.
Pearl, R. (1925) The biology of population growth, Alfred A. Knopf, New
York.
Pearl, R. (1927) The growth of populations, Quart. Rev. Biol. 2, 532–548.
MASCOS APWSPM06, February 2006 - Page 11
![Page 15: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/15.jpg)
Verhulst-Pearl model
Raymond Pearl (1879–1940, Farmington, N.H., USA)
MASCOS APWSPM06, February 2006 - Page 12
![Page 16: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/16.jpg)
Pearl was a “social drinker”
Pearl was widely known for his lust for life and his love of food,drink, music and parties. He was a key member of theSaturday Night Club. Prohibition made no dent in Pearl’sdrinking habits (which were legendary).
MASCOS APWSPM06, February 2006 - Page 13
![Page 17: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/17.jpg)
Pearl was a “social drinker”
Pearl was widely known for his lust for life and his love of food,drink, music and parties. He was a key member of theSaturday Night Club. Prohibition made no dent in Pearl’sdrinking habits (which were legendary).
In 1926, his book, Alcohol and Longevity, demonstrated thatdrinking alcohol in moderation is associated with greaterlongevity than either abstaining or drinking heavily.
Pearl, R. (1926) Alcohol and Longevity , Alfred A. Knopf, New York.
MASCOS APWSPM06, February 2006 - Page 13
![Page 18: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/18.jpg)
Verhulst-Pearl model
0 200 400 600 800 1000 1200 1400 1600 1800 20000
500
1000
1500
2000
2500
3000Trajectories of the logistic model: K = 1670, r = 0.007
t
nt
nt =K
1 +(
K−n0
n0
)
e−rt
MASCOS APWSPM06, February 2006 - Page 14
![Page 19: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/19.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
nt = 1670/(1 + e240.81−0.13125 t )
Davidson, J. (1938) On the growth of the sheep populationin Tasmania, Trans. Roy. Soc. Sth. Austral. 62, 342–346.
MASCOS APWSPM06, February 2006 - Page 15
![Page 20: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/20.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 1940−400
−300
−200
−100
0
100
200
300
400
500
600
700Growth of Tasmanian sheep population from 1818 to 1936
Year
Sta
ndar
dize
d nu
mbe
r of
she
ep (
thou
sand
s)
(With the deterministic trajectory subtracted)
MASCOS APWSPM06, February 2006 - Page 16
![Page 21: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/21.jpg)
A stochastic model
We really need to account for the variation observed.
A recent approach to stochastic modelling in AppliedMathematics can be summarised as follows:
“I feel guilty – I should add some noise”
(promulgated by stochastic modelling “experts” and courses inFinancial Mathematics that require no background instochastic processes).
MASCOS APWSPM06, February 2006 - Page 17
![Page 22: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/22.jpg)
A stochastic model
We really need to account for the variation observed.
A recent approach to stochastic modelling in AppliedMathematics can be summarised as follows:
“I feel guilty – I should add some noise”
(promulgated by stochastic modelling “experts” and courses inFinancial Mathematics that require no background instochastic processes)∗.
∗Zen Maxim (for survival in a modern university): Before you criticize someone, you
should walk a mile in their shoes. That way, when you criticize them, you’re a mile
away and you have their shoes.
MASCOS APWSPM06, February 2006 - Page 17
![Page 23: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/23.jpg)
Adding noise
In our case,
nt =K
1 +(
K−n0n0
)
e−rt+ something random
or perhapsdn
dt= rn
(
1 − n
K
)
+ σ × noise.
MASCOS APWSPM06, February 2006 - Page 18
![Page 24: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/24.jpg)
Noise?
The usual model for “noise” is white noise (or pure Gaussiannoise).
Imagine a random process (ξt, t ≥ 0) with ξt ∼ N(0, 1) for all tand ξt1 , . . . , ξtn independent for all finite sequences of timest1, . . . , tn.
MASCOS APWSPM06, February 2006 - Page 19
![Page 25: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/25.jpg)
White noise
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3
−2
−1
0
1
2
3
t
ξ t
White noise on [0,1] sampled 1000 times
MASCOS APWSPM06, February 2006 - Page 20
![Page 26: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/26.jpg)
Brownian motion
The white noise process (ξt, t ≥ 0) is formally defined as thederivative of standard Brownian motion (Bt, t ≥ 0).
Brownian motion (or Wiener process) can be constructed byway of a random walk. A particle starts at 0 and takes smallsteps of size +∆ or −∆ with equal probability p = 1/2 aftersuccessive time steps of size h. If ∆ ∼
√h, as h → 0, then the
limit process is standard Brownian motion.
MASCOS APWSPM06, February 2006 - Page 21
![Page 27: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/27.jpg)
Symmetric random walk: ∆ =√
h
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Random walk simulation: h = 2.5e-005, ∆ = 0.005
t
Xt
MASCOS APWSPM06, February 2006 - Page 22
![Page 28: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/28.jpg)
Brownian motion
The white noise process (ξt, t ≥ 0) is formally defined as thederivative of standard Brownian motion (Bt, t ≥ 0).
Brownian motion (or Wiener process) can be constructed byway of a random walk. A particle starts at 0 and takes smallsteps of size +∆ or −∆ with equal probability p = 1/2 aftersuccessive time steps of size h. If ∆ ∼
√h, as h → 0, then the
limit process is standard Brownian motion.
MASCOS APWSPM06, February 2006 - Page 23
![Page 29: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/29.jpg)
Brownian motion
The white noise process (ξt, t ≥ 0) is formally defined as thederivative of standard Brownian motion (Bt, t ≥ 0).
Brownian motion (or Wiener process) can be constructed byway of a random walk. A particle starts at 0 and takes smallsteps of size +∆ or −∆ with equal probability p = 1/2 aftersuccessive time steps of size h. If ∆ ∼
√h, as h → 0, then the
limit process is standard Brownian motion.
This construction permits us to write dBt = ξt
√dt, with the
interpretation that a change in Bt in time dt is a Gaussianrandom variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt, dBs) = 0 (s 6= t).
MASCOS APWSPM06, February 2006 - Page 23
![Page 30: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/30.jpg)
Brownian motion
The white noise process (ξt, t ≥ 0) is formally defined as thederivative of standard Brownian motion (Bt, t ≥ 0).
Brownian motion (or Wiener process) can be constructed byway of a random walk. A particle starts at 0 and takes smallsteps of size +∆ or −∆ with equal probability p = 1/2 aftersuccessive time steps of size h. If ∆ ∼
√h, as h → 0, then the
limit process is standard Brownian motion.
This construction permits us to write dBt = ξt
√dt, with the
interpretation that a change in Bt in time dt is a Gaussianrandom variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt, dBs) = 0 (s 6= t).
The correct interpretation is by way of the Itô integral:
Bt =∫ t
0 dBs =∫ t
0 ξs ds.
MASCOS APWSPM06, February 2006 - Page 23
![Page 31: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/31.jpg)
Brownian motion
General Brownian motion (Wt, t ≥ 0), with drift µ and varianceσ2, can be constructed in the same way but with ∆ ∼ σ
√h and
p = 12
(
1 + (µ/σ)√
h)
, and we may write
dWt = µ dt + σ dBt,
with the interpretation that a change in Wt in time dt is aGaussian random variable with E(dWt) = µdt, Var(dWt) = σ2dtand Cov(dWt, dWs) = 0.
MASCOS APWSPM06, February 2006 - Page 24
![Page 32: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/32.jpg)
Brownian motion
General Brownian motion (Wt, t ≥ 0), with drift µ and varianceσ2, can be constructed in the same way but with ∆ ∼ σ
√h and
p = 12
(
1 + (µ/σ)√
h)
, and we may write
dWt = µ dt + σ dBt,
with the interpretation that a change in Wt in time dt is aGaussian random variable with E(dWt) = µdt, Var(dWt) = σ2dtand Cov(dWt, dWs) = 0. This stochastic differential equation(SDE) can be integrated to give Wt = µt + σBt.
MASCOS APWSPM06, February 2006 - Page 24
![Page 33: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/33.jpg)
Brownian motion
General Brownian motion (Wt, t ≥ 0), with drift µ and varianceσ2, can be constructed in the same way but with ∆ ∼ σ
√h and
p = 12
(
1 + (µ/σ)√
h)
, and we may write
dWt = µ dt + σ dBt,
with the interpretation that a change in Wt in time dt is aGaussian random variable with E(dWt) = µdt, Var(dWt) = σ2dtand Cov(dWt, dWs) = 0. This stochastic differential equation(SDE) can be integrated to give Wt = µt + σBt.
It does not require an enormous leap of faith for us now towrite down, and properly interpret, the SDE
dnt = rnt (1 − nt/K) dt + σdBt
as a model for growth of our sheep population.
MASCOS APWSPM06, February 2006 - Page 24
![Page 34: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/34.jpg)
Adding noise
The idea (indeed the very idea of an SDE) can be traced backto Paul Langevin’s 1908 paper “On the theory of BrownianMotion”:
Langevin, P. (1908) Sur la théorie du mouvement brownien, Comptes
Rendus 146, 530–533.
He derived a “dynamic theory” of Brownian Motion three yearsafter Einstein’s ground breaking paper on Brownian Motion:
Einstein, A. (1905) On the movement of small particles suspended in
stationary liquids required by the molecular-kinetic theory of heat, Ann.
Phys. 17, 549–560 [English translation by Anna Beck in The Collected
Papers of Albert Einstein, Princeton University Press, Princeton, USA,
1989, Vol. 2, pp. 123–134.]
MASCOS APWSPM06, February 2006 - Page 25
![Page 35: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/35.jpg)
Langevin
Langevin introduced a “stochastic force” (his phrase“complementary force”–complimenting the viscous drag µ)pushing the Brownian particle around in velocity space(Einstein worked in configuration space).
In modern terminology, Langevin described the Brownianparticle’s velocity as an Ornstein-Uhlenbeck (OU) processand its position as the time integral of its velocity, whileEinstein described its position as a Wiener process.
The Langevin equation (for a particle of unit mass) is
dvt = −µvt dt + σdBt.
This is Newton’s law (−µv = Force = mv̇) plus noise. The(strong) solution to this SDE is the OU process.
MASCOS APWSPM06, February 2006 - Page 26
![Page 36: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/36.jpg)
Langevin
Langevin introduced a “stochastic force” (his phrase“complementary force”–complimenting the viscous drag µ)pushing the Brownian particle around in velocity space(Einstein worked in configuration space).
In modern terminology, Langevin described the Brownianparticle’s velocity as an Ornstein-Uhlenbeck (OU) processand its position as the time integral of its velocity, whileEinstein described its position as a Wiener process.
The Langevin equation (for a particle of unit mass) is
dvt = −µvt dt + σdBt.
This is Newton’s law (−µv = Force = mv̇) plus noise. The(strong) solution to this SDE is the OU process. Warning:∫ t
0 vs ds 6= Bt; this functional is not even Markovian.
MASCOS APWSPM06, February 2006 - Page 27
![Page 37: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/37.jpg)
Langevin
Einstein said of Langevin “...It seems to me certain thathe would have developedthe special theory of rela-tivity if that had not beendone elsewhere, for he hadclearly recognized the es-sential points.”
Paul Langevin (1872-1946, Paris, France)
MASCOS APWSPM06, February 2006 - Page 28
![Page 38: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/38.jpg)
Langevin was a dark horse
In 1910 he had an affair with Marie Curie (Polish physicist).
MASCOS APWSPM06, February 2006 - Page 29
![Page 39: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/39.jpg)
Langevin was a dark horse
In 1910 he had an affair with Marie Curie (Polish physicist).
MASCOS APWSPM06, February 2006 - Page 29
![Page 40: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/40.jpg)
Langevin was a dark horse
In 1910 he had an affair with Marie Curie (Polish physicist).
The person on the right is not Langevin, but Langevin’s PhDsupervisor Pierre Curie.
MASCOS APWSPM06, February 2006 - Page 29
![Page 41: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/41.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
MASCOS APWSPM06, February 2006 - Page 30
![Page 42: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/42.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
Differentiation (Itô calculus!) gives dyt = eµtdvt + µeµtvtdt.
MASCOS APWSPM06, February 2006 - Page 30
![Page 43: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/43.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
Differentiation (Itô calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt,
MASCOS APWSPM06, February 2006 - Page 30
![Page 44: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/44.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
Differentiation (Itô calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt,
and hence that dyt = σeµtdBt.
MASCOS APWSPM06, February 2006 - Page 30
![Page 45: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/45.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
Differentiation (Itô calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt,
and hence that dyt = σeµtdBt. Integration gives
yt = y0 +∫ t
0 σeµsdBs,
MASCOS APWSPM06, February 2006 - Page 30
![Page 46: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/46.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
Differentiation (Itô calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt,
and hence that dyt = σeµtdBt. Integration gives
yt = y0 +∫ t
0 σeµsdBs,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0 σe−µ(t−s)dBs.
MASCOS APWSPM06, February 2006 - Page 30
![Page 47: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/47.jpg)
Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt, consider the process yt = vteµt.
Differentiation (Itô calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt,
and hence that dyt = σeµtdBt. Integration gives
yt = y0 +∫ t
0 σeµsdBs,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0 σe−µ(t−s)dBs.
We can deduce much from this. For example, vt is a Gaussianprocess with E(vt) = v0e
−µt and Var(vt) = σ2
2µ(1 − e−2µt), and
Cov(vt, vt+s) = Var(vt)e−µ|s|.
MASCOS APWSPM06, February 2006 - Page 30
![Page 48: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/48.jpg)
Where were we?
We had just added noise to our logistic model:
dnt = rnt
(
1 − nt
K
)
dt + σ dBt. (1)
MASCOS APWSPM06, February 2006 - Page 31
![Page 49: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/49.jpg)
Where were we?
We had just added noise to our logistic model:
dnt = rnt
(
1 − nt
K
)
dt + σ dBt. (1)
So, what the hell is wrong with (1)?
MASCOS APWSPM06, February 2006 - Page 31
![Page 50: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/50.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
nt = 1670/(1 + e240.81−0.13125 t )
MASCOS APWSPM06, February 2006 - Page 32
![Page 51: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/51.jpg)
Solution to SDE (Run 1)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(
1 − nt
K
)
dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 33
![Page 52: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/52.jpg)
Solution to SDE (Run 2)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(
1 − nt
K
)
dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 34
![Page 53: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/53.jpg)
Solution to SDE (Run 3)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(
1 − nt
K
)
dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 35
![Page 54: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/54.jpg)
Solution to SDE (Run 4)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(
1 − nt
K
)
dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 36
![Page 55: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/55.jpg)
Solution to SDE (Run 5)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(
1 − nt
K
)
dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 37
![Page 56: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/56.jpg)
Solution to SDE
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Mean path of SDE solution with ± 2 standard deviations (1000 runs)
dnt = rnt
(
1 − nt
K
)
dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 38
![Page 57: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/57.jpg)
Solution to SDE
1820 1840 1860 1880 1900 1920 1940
−1000
−500
0
500
1000
1500
t
Sta
ndar
diz
ednum
ber
Mean path of SDE solution with ± 2 standard deviations (1000 runs)
(With the solution to the deterministic model subtracted)
MASCOS APWSPM06, February 2006 - Page 39
![Page 58: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/58.jpg)
Logistic model with noise
So, what is wrong with the model?
dnt = rnt
(
1 − nt
K
)
dt + σ dBt.
MASCOS APWSPM06, February 2006 - Page 40
![Page 59: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/59.jpg)
Logistic model with noise
So, what is wrong with the model?
dnt = rnt
(
1 − nt
K
)
dt + σ dBt.
For a start:
MASCOS APWSPM06, February 2006 - Page 40
![Page 60: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/60.jpg)
Logistic model with noise
So, what is wrong with the model?
dnt = rnt
(
1 − nt
K
)
dt + σ dBt.
For a start:
• 0 is reflecting;
MASCOS APWSPM06, February 2006 - Page 40
![Page 61: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/61.jpg)
Logistic model with noise
So, what is wrong with the model?
dnt = rnt
(
1 − nt
K
)
dt + σ dBt.
For a start:
• 0 is reflecting;• The mean path of the SDE solution does not follow a
logistic curve;
MASCOS APWSPM06, February 2006 - Page 40
![Page 62: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/62.jpg)
Logistic model with noise
So, what is wrong with the model?
dnt = rnt
(
1 − nt
K
)
dt + σ dBt.
For a start:
• 0 is reflecting;• The mean path of the SDE solution does not follow a
logistic curve;• The variance in the solution is large for the non-
equilibrium phase–is this okay?
MASCOS APWSPM06, February 2006 - Page 40
![Page 63: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/63.jpg)
Logistic model with noise
So, what is wrong with the model?
dnt = rnt
(
1 − nt
K
)
dt + σ dBt.
For a start:
• 0 is reflecting;• The mean path of the SDE solution does not follow a
logistic curve;• The variance in the solution is large for the non-
equilibrium phase–is this okay?
. . . not to mention the fact that nt is a continuous variable, yetpopulation size is an integer-valued process!
MASCOS APWSPM06, February 2006 - Page 40
![Page 64: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/64.jpg)
The variance!
Since the variance is not uniform over time, we should at leasthave
dnt = rnt
(
1 − nt
K
)
dt + σ(nt) dBt,if not
dnt = rnt
(
1 − nt
K
)
dt + σ(nt, t) dBt.
MASCOS APWSPM06, February 2006 - Page 41
![Page 65: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/65.jpg)
A different approach
Let’s start from scratch specifying a stochastic model withvariation being an inherent property: a Markovian model .
MASCOS APWSPM06, February 2006 - Page 42
![Page 66: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/66.jpg)
A different approach
Let’s start from scratch specifying a stochastic model withvariation being an inherent property: a Markovian model .
We will suppose that nt (integer-valued!) evolves as abirth-death process with rates
qn,n+1 = λn(
1 − nN
)
and qn,n−1 = µn,
where λ and µ (both positive) are per-capita birth and deathrates (for λ when the population is small). Here N is thepopulation ceiling (nt now takes values in S = {0, 1, . . . , N}).
I will call this model the stochastic logistic (SL) model , thoughit has many names, having been rediscovered several timessince Feller proposed it in 1939.
MASCOS APWSPM06, February 2006 - Page 42
![Page 67: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/67.jpg)
A different approach
Let’s start from scratch specifying a stochastic model withvariation being an inherent property: a Markovian model .
We will suppose that nt (integer-valued!) evolves as abirth-death process with rates
qn,n+1 = λn(
1 − nN
)
and qn,n−1 = µn,
where λ and µ (both positive) are per-capita birth and deathrates (for λ when the population is small). Here N is thepopulation ceiling (nt now takes values in S = {0, 1, . . . , N}).
I will call this model the stochastic logistic (SL) model , thoughit has many names, having been rediscovered several timessince Feller proposed it in 1939.
It shares an important property with the deterministic logisticmodel: that of density dependence.MASCOS APWSPM06, February 2006 - Page 42
![Page 68: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/68.jpg)
Density dependence
The Verhulst-Pearl model dndt
= rn(
1 − nK
)
can be written
1
N
dn
dt= r
n
N
(
1 − N
K
n
N
)
.
The rate of change of nt depends on nt only through nt
N.
So, letting xt = nt/N be the “population density”, we get
dx
dt= rx
(
1 − x
E
)
, where E = K/N.
This is a convenient space scaling. We could have setxt = nt/A, where A is habitat area, and then
dx
dt= rx
(
1 − x
DE
)
, where D = N/A.
MASCOS APWSPM06, February 2006 - Page 43
![Page 69: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/69.jpg)
Markovian models
Let (nt, t ≥ 0) be a continuous-time Markov chain takingvalues in S ⊆ Zk with transition rates Q = (qnm, n, m ∈ S). Weidentify a quantity N , usually related to the size of the systembeing modelled.
Definition (Kurtz∗) The model is density dependent if there isa subset E of Rk and a continuous function f : Zk × E → R,such that
qn,n+l = Nfl
(
nN
)
, l 6= 0 (l ∈ Zk).
(So, the idea is the same: the rate of change of nt depends onnt only through the “density” nt/N .)
∗Kurtz, T. (1970) Solutions of ordinary differential equations as limits of pure jump
Markov processes, J. of Appl. Probab. 7, 49–58.
MASCOS APWSPM06, February 2006 - Page 44
![Page 70: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/70.jpg)
Markovian models
Let (nt, t ≥ 0) be a continuous-time Markov chain takingvalues in S ⊆ Zk with transition rates Q = (qnm, n, m ∈ S). Weidentify a quantity N , usually related to the size of the systembeing modelled.
Definition (Kurtz∗) The model is density dependent if there isa subset E of Rk and a continuous function f : Zk × E → R,such that
qn,n+l = Nfl
(
nN
)
, l 6= 0 (l ∈ Zk).
(So, the idea is the same: the rate of change of nt depends onnt only through the “density” nt/N .)
∗Kurtz, T. (1970) Solutions of ordinary differential equations as limits of pure jump
Markov processes, J. of Appl. Probab. 7, 49–58.
MASCOS APWSPM06, February 2006 - Page 45
![Page 71: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/71.jpg)
Tom Kurtz
Thomas Kurtz (taken in 2003)
MASCOS APWSPM06, February 2006 - Page 46
![Page 72: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/72.jpg)
Density dependence
Consider the forward equations for pn(t) := Pr(nt = n). Letqn =
∑
m6=n qnm. Then,
p ′n(t) = −qnpn(t) +
∑
m6=n pm(t)qmn,
MASCOS APWSPM06, February 2006 - Page 47
![Page 73: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/73.jpg)
Density dependence
Consider the forward equations for pn(t) := Pr(nt = n). Letqn =
∑
m6=n qnm. Then,
p ′n(t) = −qnpn(t) +
∑
m6=n pm(t)qmn,
and so (formally) E(nt) =∑
n npn(t) satisfies
ddt
E(nt) = −∑
n qnnpn(t) +∑
m pm(t)∑
n6=m nqmn.
MASCOS APWSPM06, February 2006 - Page 47
![Page 74: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/74.jpg)
Density dependence
Consider the forward equations for pn(t) := Pr(nt = n). Letqn =
∑
m6=n qnm. Then,
p ′n(t) = −qnpn(t) +
∑
m6=n pm(t)qmn,
and so (formally) E(nt) =∑
n npn(t) satisfies
ddt
E(nt) = −∑
n qnnpn(t) +∑
m pm(t)∑
n6=m nqmn.
So if qn,n+l = Nfl(n/N), then
ddt
E(nt) = −∑
n
∑
l 6=0 Nfl(n/N)npn(t)
+∑
m pm(t)∑
l 6=0(m + l)Nfl(m/N)
=∑
m pm(t)N∑
l 6=0 lfl(m/N) = NE
(
∑
l 6=0 lfl(nt/N))
.
MASCOS APWSPM06, February 2006 - Page 47
![Page 75: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/75.jpg)
Density dependence
For an arbitrary density dependent model, define F : E → Rby F (x) =
∑
l 6=0 lfl (x). Then,
ddt
E(nt) = N E
(
F(nt
N
))
,
or, setting Xt = nt/N (the density process),
ddt
E(Xt) = E (F (Xt)) .
MASCOS APWSPM06, February 2006 - Page 48
![Page 76: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/76.jpg)
Density dependence
For an arbitrary density dependent model, define F : E → Rby F (x) =
∑
l 6=0 lfl (x). Then,
ddt
E(nt) = N E
(
F(nt
N
))
,
or, setting Xt = nt/N (the density process),
ddt
E(Xt) = E (F (Xt)) .
Warning: I’m not saying that ddt
E(Xt) = F (E(Xt)).
MASCOS APWSPM06, February 2006 - Page 48
![Page 77: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/77.jpg)
Density dependence
For an arbitrary density dependent model, define F : E → Rby F (x) =
∑
l 6=0 lfl (x). Then,
ddt
E(nt) = N E
(
F(nt
N
))
,
or, setting Xt = nt/N (the density process),
ddt
E(Xt) = E (F (Xt)) .
Warning: I’m not saying that ddt
E(Xt) = F (E(Xt)).
(But, I am hoping for something like that to be true!)
MASCOS APWSPM06, February 2006 - Page 48
![Page 78: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/78.jpg)
Density dependence
For the SL model we have S = {0, 1, . . . , N} and
qn,n+1 = λn(
1 − nN
)
and qn,n−1 = µn.
Therefore, f+1(x) = λx (1 − x) and f−1(x) = µx, x ∈ E := [0, 1],and so F (x) = λx (1 − ρ − x), x ∈ E, where ρ = µ/λ.
MASCOS APWSPM06, February 2006 - Page 49
![Page 79: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/79.jpg)
Density dependence
For the SL model we have S = {0, 1, . . . , N} and
qn,n+1 = λn(
1 − nN
)
and qn,n−1 = µn.
Therefore, f+1(x) = λx (1 − x) and f−1(x) = µx, x ∈ E := [0, 1],and so F (x) = λx (1 − ρ − x), x ∈ E, where ρ = µ/λ.
Now compare F (x) with the right-hand side of theVerhulst-Pearl model for the density process:
dxdt
= rx(
1 − xE
)
, where E = K/N . (2)
If K ∼ βN for N large, so that K/N → β, then we may identifyβ with 1 − ρ and r with λβ, and discover that (2) can berewritten as dx/dt = F (x).
MASCOS APWSPM06, February 2006 - Page 49
![Page 80: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/80.jpg)
Recall that ...
Recall that (nt, t ≥ 0) is a continuous-time Markov chaintaking values in S ⊆ Zk with transition ratesQ = (qnm, n, m ∈ S), and we have identified a quantity N ,usually related to the size of the system being modelled.
The model is assumed to be density dependent : there is asubset E of Rk and a continuous function f : Zk × E → R,such that
qn,n+l = Nfl
(
nN
)
, l 6= 0 (l ∈ Zk).
We set F (x) =∑
l 6=0 lfl (x), x ∈ E.
We now formally define the density process (X(N)
t ) byX(N)
t = nt/N , t ≥ 0. We hope that (X (N)
t ) becomes moredeterministic as N gets large.
MASCOS APWSPM06, February 2006 - Page 50
![Page 81: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/81.jpg)
Recall that ...
Recall that (nt, t ≥ 0) is a continuous-time Markov chaintaking values in S ⊆ Zk with transition ratesQ = (qnm, n, m ∈ S), and we have identified a quantity N ,usually related to the size of the system being modelled.
The model is assumed to be density dependent : there is asubset E of Rk and a continuous function f : Zk × E → R,such that
qn,n+l = Nfl
(
nN
)
, l 6= 0 (l ∈ Zk).
We set F (x) =∑
l 6=0 lfl (x), x ∈ E.
We now formally define the density process (X(N)
t ) byX(N)
t = nt/N , t ≥ 0. We hope that (X (N)
t ) becomes moredeterministic as N gets large. To simplify the statement ofresults, I’m going to assume that the state space S is finite.
MASCOS APWSPM06, February 2006 - Page 50
![Page 82: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/82.jpg)
A law of large numbers
The following functional law of large numbers establishesconvergence of the family (X(N)
t ) to the unique trajectory of anappropriate approximating deterministic model.
Theorem (Kurtz∗) Suppose F is Lipschitz on E (that is,|F (x) − F (y)| < ME |x − y|). If limN→∞ X(N)
0 = x0, then thedensity process (X(N)
t ) converges uniformly in probability on[0, t] to (xt), the unique (deterministic) trajectory satisfying
dds
xs = F (xs), xs ∈ E, s ∈ [0, t].
∗Kurtz, T. (1970) Solutions of ordinary differential equations as limits of pure jump
Markov processes, J. of Appl. Probab. 7, 49–58.
MASCOS APWSPM06, February 2006 - Page 51
![Page 83: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/83.jpg)
A law of large numbers
The following functional law of large numbers establishesconvergence of the family (X(N)
t ) to the unique trajectory of anappropriate approximating deterministic model.
Theorem (Kurtz∗) Suppose F is Lipschitz on E (that is,|F (x) − F (y)| < ME |x − y|). If limN→∞ X(N)
0 = x0, then thedensity process (X(N)
t ) converges uniformly in probability on[0, t] to (xt), the unique (deterministic) trajectory satisfying
dds
xs = F (xs), xs ∈ E, s ∈ [0, t].
∗Kurtz, T. (1970) Solutions of ordinary differential equations as limits of pure jump
Markov processes, J. of Appl. Probab. 7, 49–58.
(If S is an infinite set, we have the additional conditionssupx∈E
∑
l 6=0 |l|fl(x) < ∞ and limd→∞∑
|l|>d |l|fl(x) = 0, x ∈ E.)
MASCOS APWSPM06, February 2006 - Page 51
![Page 84: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/84.jpg)
A law of large numbers
Convergence uniformly in probability on [0, t] means that forevery ǫ > 0,
limN→∞ Pr(
sups≤t
∣
∣X(N)
t − xt
∣
∣ > ǫ)
= 0.
MASCOS APWSPM06, February 2006 - Page 52
![Page 85: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/85.jpg)
A law of large numbers
Convergence uniformly in probability on [0, t] means that forevery ǫ > 0,
limN→∞ Pr(
sups≤t
∣
∣X(N)
t − xt
∣
∣ > ǫ)
= 0.
The conditions of the theorem hold for the SL model: sinceF (x) = λx(1 − ρ − x), we have, for all x, y ∈ E = [0, 1], that
|F (x) − F (y)| = λ|x − y||1 − ρ − (x + y)| ≤ (1 + ρ)λ|x − y|.
So, provided X(N)
0 → x0 as N → ∞, the population density(X(N)
t ) converges (uniformly in probability on finite timeintervals) to the solution (xt) of the deterministic model
dx
dt= λx(1 − ρ − x) (xt ∈ E).
MASCOS APWSPM06, February 2006 - Page 52
![Page 86: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/86.jpg)
Simulation of the SL model
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Simulation of SL Model (N =10000, λ =0.78593, µ =0.65468, K =1670)
t
nt
(Solution to the deterministic model is in green)
MASCOS APWSPM06, February 2006 - Page 53
![Page 87: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/87.jpg)
A central limit law
In a later paper Kurtz∗ proved a functional central limit lawwhich establishes that, for large N , the fluctuations about thedeterministic trajectory follow a Gaussian diffusion, providedthat some mild extra conditions are satisfied.
He considered the family of processes {(Z(N)
t )} defined by
Z(N)s =
√N
(
X(N)s − xs
)
, 0 ≤ s ≤ t.
∗Kurtz, T. (1971) Limit theorems for sequences of jump Markov processes approximat-
ing ordinary differential processes. J. Appl. Probab. 8, 344–356.
MASCOS APWSPM06, February 2006 - Page 54
![Page 88: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/88.jpg)
The SL model (N = 20)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =20, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 55
![Page 89: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/89.jpg)
The SL model (N = 50)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =50, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 56
![Page 90: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/90.jpg)
The SL model (N = 100)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =100, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 57
![Page 91: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/91.jpg)
The SL model (N = 200)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =200, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 58
![Page 92: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/92.jpg)
The SL model (N = 500)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =500, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 59
![Page 93: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/93.jpg)
The SL model (N = 1 000)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =1000, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 60
![Page 94: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/94.jpg)
The SL model (N = 10 000)
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Simulation of SL Model (N =10000, λ =0.1625, µ =0.0325)
t
Z(N
)t
=√
N(X
(N)
t−
xt)
MASCOS APWSPM06, February 2006 - Page 61
![Page 95: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/95.jpg)
A central limit law
Theorem Suppose that F is Lipschitz and has uniformlycontinuous first derivative on E, and that the k × k matrixG(x), defined for x ∈ E by Gij(x) =
∑
l 6=0 liljfl(x), is uniformlycontinuous on E.
Let (xt) be the unique deterministic trajectory starting at x0
and suppose that limN→∞
√N
(
X(N)
0 − x0
)
= z.
Then, {(Z(N)
t )} converges weakly in D[0, t] (the space ofright-continuous, left-hand limits functions on [0, t]) to aGaussian diffusion (Zt) with initial value Z0 = z and with meanand covariance given by µs := E(Zs) = Msz, whereMs = exp(
∫ s
0 Bu du) and Bs = ∇F (xs), and
Vs := Cov(Zs) = Ms
(∫ s
0 M−1u G(xu)(M−1
u )T du)
MTs .
MASCOS APWSPM06, February 2006 - Page 62
![Page 96: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/96.jpg)
A central limit law
The functional central limit theorem tells us that, for large N ,the scaled density process Z(N)
t can be approximated overfinite time intervals by the Gaussian diffusion (Zt).
In particular, for all t > 0, X(N)
t has an approximate normaldistribution with Cov(X(N)
t ) ≃ Vt/N .
We would usually take x0 = X(N)
0 , thus giving E(X(N)
t ) ≃ xt.
MASCOS APWSPM06, February 2006 - Page 63
![Page 97: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/97.jpg)
A central limit law
For the SL model we have F (x) = λx(1 − ρ − x), and thesolution to dx/dt = F (x) is
x(t) = (1−ρ)x0
x0+(1−ρ−x0)e−λ(1−ρ)t .
We also have F ′(x) = λ(1 − ρ − 2x) and
G(x) =∑
l l2fl(x) = λx(1 + ρ − x) = F (x) + 2µx,
giving
Mt = exp(
∫ t
0 F ′(xs) ds)
= (1−ρ)2e−λ(1−ρ)t
(x0+(1−ρ−x0)e−λ(1−ρ)t)2.
We can evaluate
Vt := Var(Zt) = M2t
(
∫ t
0 G(xs)/M2s ds
)
numerically, or ...
MASCOS APWSPM06, February 2006 - Page 64
![Page 98: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/98.jpg)
Or ....
Vt = x0
(
ρx30 + x2
0(1 + 5ρ)(1 − ρ − x0)e−λ(1−ρ)t
+ 2x0(1 + 2ρ)(1 − ρ − x0)2(λ(1 − ρ)t)e−2λ(1−ρ)t
−(
(1 − ρ − x0)[3ρx20 + (2 + ρ)(1 − ρ)x0 − ((1 + 2ρ))(1 − ρ)2]
+ ρ(1 − ρ)3)
e−2λ(1−ρ)t
− (1+ρ)(1−ρ−x0)3e−3λ(1−ρ)t
)/(
x0 +(1−ρ−x0)e−λ(1−ρ)t
)4.
MASCOS APWSPM06, February 2006 - Page 65
![Page 99: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/99.jpg)
The SL model
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Simulation of SL Model (N =10000, λ =0.78593, µ =0.65468, K =1670)
t
nt
(Deterministic trajectory plus or minus two standard deviations in green)
MASCOS APWSPM06, February 2006 - Page 66
![Page 100: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/100.jpg)
The OU approximation
If the initial point x0 of the deterministic trajectory is chosen tobe an equilibrium point of the deterministic model, we can befar more precise about the approximating diffusion.
Corollary If xeq satisfies F (xeq) = 0, then, under theconditions of the theorem, the family {(Z(N)
t )}, defined by
Z(N)s =
√N(X(N)
s − xeq), 0 ≤ s ≤ t,
converges weakly in D[0, t] to an OU process (Zt) with initialvalue Z0 = z, local drift matrix B = ∇F (xeq) and localcovariance matrix G(xeq). In particular, Zs is normallydistributed with mean and covariance given byµs := E(Zs) = eBsz and
Vs := Cov(Zs) =∫ s
0 eBuG(xeq)eBT u du .
MASCOS APWSPM06, February 2006 - Page 67
![Page 101: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/101.jpg)
The OU approximation
Note that
Vs =∫ s
0 eBuG(xeq)eBT u du = V∞ − eBsV∞eBT s,
where V∞, the stationary covariance matrix, satisfies
BV∞ + V∞BT + G(xeq) = 0.
MASCOS APWSPM06, February 2006 - Page 68
![Page 102: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/102.jpg)
The OU approximation
Note that
Vs =∫ s
0 eBuG(xeq)eBT u du = V∞ − eBsV∞eBT s,
where V∞, the stationary covariance matrix, satisfies
BV∞ + V∞BT + G(xeq) = 0.
We conclude that, for N large, X(N)
t has an approximateGaussian distribution with Cov(X(N)
t ) ≃ Vt/N .
MASCOS APWSPM06, February 2006 - Page 68
![Page 103: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/103.jpg)
The OU approximation
Note that
Vs =∫ s
0 eBuG(xeq)eBT u du = V∞ − eBsV∞eBT s,
where V∞, the stationary covariance matrix, satisfies
BV∞ + V∞BT + G(xeq) = 0.
We conclude that, for N large, X(N)
t has an approximateGaussian distribution with Cov(X(N)
t ) ≃ Vt/N .
For the SL model, we get Var(X(N)
t ) ≃ ρ(1 − e−2λ(1−ρ)t)/N .
MASCOS APWSPM06, February 2006 - Page 68
![Page 104: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/104.jpg)
The OU approximation
Note that
Vs =∫ s
0 eBuG(xeq)eBT u du = V∞ − eBsV∞eBT s,
where V∞, the stationary covariance matrix, satisfies
BV∞ + V∞BT + G(xeq) = 0.
We conclude that, for N large, X(N)
t has an approximateGaussian distribution with Cov(X(N)
t ) ≃ Vt/N .
For the SL model, we get Var(X(N)
t ) ≃ ρ(1 − e−2λ(1−ρ)t)/N .
This brings us “full circle” to the approximating SDE
dnt = −α(nt − K) dt +√
2Nαρ dBt, where α = λ(1 − ρ).
MASCOS APWSPM06, February 2006 - Page 68
![Page 105: Stochastic models and their deterministic analogues · 2006-02-15 · Stochastic models and their deterministic analogues Phil Pollett ... Pearl was a “social drinker ... stochastic](https://reader034.fdocuments.us/reader034/viewer/2022050219/5f6529fc31c986747a270e68/html5/thumbnails/105.jpg)
The SL model
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Simulation of SL Model (N =10000, λ =0.78593, µ =0.65468, K =1670)
t
nt
(Deterministic equilibrium plus or minus two standard deviations is in black)
MASCOS APWSPM06, February 2006 - Page 69