Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Population Systems: Stochastic versusDeterministic
Xuerong Mao
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are
number of cancer cells,number of HIV infected individuals,numbers of interacting species,share price in a company,price of gold, oil or electricity.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are
number of cancer cells,number of HIV infected individuals,numbers of interacting species,share price in a company,price of gold, oil or electricity.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are
number of cancer cells,number of HIV infected individuals,numbers of interacting species,share price in a company,price of gold, oil or electricity.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are
number of cancer cells,number of HIV infected individuals,numbers of interacting species,share price in a company,price of gold, oil or electricity.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are
number of cancer cells,number of HIV infected individuals,numbers of interacting species,share price in a company,price of gold, oil or electricity.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Now suppose that at time t the underlying quantity is x(t). Letus consider a small subsequent time interval dt , during whichx(t) changes to x(t) + dx(t). (We use the notation d · for thesmall change in any quantity over this time interval when weintend to consider it as an infinitesimal change.) By definition,the intrinsic growth rate at t is dx(t)/x(t). How might we modelthis rate?
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
If, given x(t) at time t , the rate of change is deterministic, sayR = R(x(t), t), then
dx(t)x(t)
= R(x(t), t)dt .
This gives the ordinary differential equation (ODE)
dx(t)dt
= R(x(t), t)x(t).
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
However the rate of change is in general not deterministic as itis often subjective to many factors and uncertainties e.g.system uncertainty, environmental disturbances. To model theuncertainty, we may decompose
dx(t)x(t)
= deterministic change + random change.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
The deterministic change may be modeled by
Rdt = R(x(t), t)dt
where R = r(x(t), t) is the average rate of change given x(t) attime t . So
dx(t)x(t)
= R(x(t), t)dt + random change.
How may we model the random change?
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
In general, the random change is affected by many factorsindependently. By the well-known central limit theorem thischange can be represented by a normal distribution with meanzero and and variance V 2dt , namely
random change = N(0,V 2dt) = V N(0,dt),
where V = V (x(t), t) is the standard deviation of the rate ofchange given x(t) at time t , and N(0,dt) is a normal distributionwith mean zero and and variance dt . Hence
dx(t)x(t)
= R(x(t), t)dt + V (x(t), t)N(0,dt).
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
A convenient way to model N(0,dt) as a process is to use theBrownian motion B(t) (t ≥ 0) which has the followingproperties:
B(0) = 0,dB(t) = B(t + dt)− B(t) is independent of B(t),dB(t) follows N(0,dt).
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
The stochastic model can therefore be written as
dx(t)x(t)
= R(x(t), t)dt + V (x(t), t)dB(t),
or
dx(t) = R(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)
which is a stochastic differential equation (SDE).
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Exponential growth model
If both R and V are constants, say
R(x(t), t) = µ, V (x(t), t) = σ,
then the SDE becomes
dx(t) = µx(t)dt + σx(t)dB(t).
This is a linear SDE. It is known as the geometric Brownianmotion in finance and the exponential growth model in thepopulation systems.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Mean-reverting process
IfR(x(t), t) =
α(µ− x(t))
x(t), V (x(t), t) = σ,
then the SDE becomes
dx(t) = α(µ− x(t))dt + σx(t)dB(t).
This is known as the mean-reverting process.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Logistic model
IfR(x(t), t) = b + ax(t), V (x(t), t) = σx(t),
then the SDE becomes
dx(t) = x(t)(
[b + ax(t)]dt + σx(t)dB(t)).
This is the well-known Logistic model in population.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Square root process
IfR(x(t), t) = µ, V (x(t), t) =
σ√x(t)
,
then the SDE becomes the well-known square root process
dx(t) = µx(t)dt + σ√
x(t)dB(t).
This is used widely in engineering and finance.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Mean-reverting square root process
IfR(x(t), t) =
α(µ− x(t))
x(t), V (x(t), t) =
σ√x(t)
,
then the SDE becomes
dx(t) = α(µ− x(t))dt + σ√
x(t)dB(t).
This is the mean-reverting square root process used widely infinance and population.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Linear SDE modelsNon-linear SDE models
Theta process
IfR(x(t), t) = µ, V (x(t), t) = σ(x(t))θ−1,
then the SDE becomes
dx(t) = µx(t)dt + σ(x(t))θdB(t),
which is known as the theta process.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Consider the Lotka-Volterra model for a system with ninteracting species, namely
x(t) = diag(x1(t), · · · , xn(t))[b + Ax(t)], (3.1)
where
x = (x1, · · · , xn)T , b = (b1, · · · ,bn)T , A = (aij)n×n.
We need some condition on A, e.g., A + AT is negative-definite,for the system not to explode to infinity at a finite time (namely,no explosion).
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
We first consider the corresponding SDE model where thevariance dependent on the state, say
dx(t) = x(t)(
[b + Ax(t)]dt + σx(t)dB(t)), (3.2)
where σ = (σij)n×n and B(t) is a scalar Brownian motion.
How is this SDE different from its corresponding ODE?
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Theorem 1
Assume that
σii > 0 for 1 ≤ i ≤ n whilst σij ≥ 0 for i 6= j .
Then for any system parameters b ∈ Rn and A ∈ Rn×n, and anygiven initial data x(0) = x0 ∈ Rn, there is a unique solution x(t)to equation (3.2) on t ≥ 0 and the solution will remain in Rn
+
with probability 1, namely x(t) ∈ Rn+ for all t ≥ 0 almost surely.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Significant Difference between ODE (3.1) and SDE(3.2)
ODE (3.1): The solution may explode to infinity at a finitetime if, e.g., A + AT is positive definite.SDE (3.2): With probability one, the solution will no longerexplode to infinity in a finite time, even in the case whenA + AT is positive definite, as long as σii > 0 for 1 ≤ i ≤ nwhilst σij ≥ 0 for i 6= j . Moreover, the stochastic populationsystem is persistent.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Significant Difference between ODE (3.1) and SDE(3.2)
ODE (3.1): The solution may explode to infinity at a finitetime if, e.g., A + AT is positive definite.SDE (3.2): With probability one, the solution will no longerexplode to infinity in a finite time, even in the case whenA + AT is positive definite, as long as σii > 0 for 1 ≤ i ≤ nwhilst σij ≥ 0 for i 6= j . Moreover, the stochastic populationsystem is persistent.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Example
dy(t)dt
= y(t)[1 + y(t)], t ≥ 0, x(0) = x0 > 0
has the solution
y(t) =1
−1 + e−t (1 + x0)/x0(0 ≤ t < T ) ,
which explodes to infinity at the finite time
T = log(
1 + x0
x0
).
However, the SDE
dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)]
will never explode as long as σ 6= 0.Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
0 2 4 6 8 10
01
23
45
t
x(t)
or
y(t)
x(t)
y(t)
0 2 4 6 8 100
.51
.01
.5
t
x(t)
or
y(t)
x(t)
y(t)
Left: σ = 3; right: σ = 6
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Key Point:When a > 0 and ε = 0 the solution explodes at the finite timet = T ; whilst conversely, no matter how small ε > 0, thesolution will not explode in a finite time. In other words,
noise may suppress explosion.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Consider the SDE model where the noise is independent of thestate, namely
dx(t) = x(t)(
[b + Ax(t)]dt + σdB(t)), (3.3)
where σ = (σij) ∈ Rn×n and B(t) is an n-dimensional Brownianmotion.
How is this SDE different from others?
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
For this SDE model, we require, e.g. A + AT isnegative-definite, in order to have no explosion.If λ+max(Q) < 0, where Q = (qij)n×n is defined by
qij = bi + bj −n∑
k=1
σikσjk ,
then limt→∞ x(t) = 0 with probability 1, namely the noisemakes the population extinct.If both σ and −A are non-singular M-matrices while∑n
k=1 σ2ik < 2bi for 1 ≤ i ≤ n, the the population system is
persistent. In this case
limt→∞
Ex(t) = (−A)−1(ξ1, · · · , ξn)T ,
where ξi = bi − 0.5∑n
k=1 σ2ik .
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
For this SDE model, we require, e.g. A + AT isnegative-definite, in order to have no explosion.If λ+max(Q) < 0, where Q = (qij)n×n is defined by
qij = bi + bj −n∑
k=1
σikσjk ,
then limt→∞ x(t) = 0 with probability 1, namely the noisemakes the population extinct.If both σ and −A are non-singular M-matrices while∑n
k=1 σ2ik < 2bi for 1 ≤ i ≤ n, the the population system is
persistent. In this case
limt→∞
Ex(t) = (−A)−1(ξ1, · · · , ξn)T ,
where ξi = bi − 0.5∑n
k=1 σ2ik .
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
For this SDE model, we require, e.g. A + AT isnegative-definite, in order to have no explosion.If λ+max(Q) < 0, where Q = (qij)n×n is defined by
qij = bi + bj −n∑
k=1
σikσjk ,
then limt→∞ x(t) = 0 with probability 1, namely the noisemakes the population extinct.If both σ and −A are non-singular M-matrices while∑n
k=1 σ2ik < 2bi for 1 ≤ i ≤ n, the the population system is
persistent. In this case
limt→∞
Ex(t) = (−A)−1(ξ1, · · · , ξn)T ,
where ξi = bi − 0.5∑n
k=1 σ2ik .
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Example
The SDE
dx(t) = x(t)[(1− x(t))dt + σx(t)dw(t)]
will become extinct if σ >√
2 but will be persistent if σ <√
2.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
t
x(t)
0 2 4 6 8 100.5
1.0
1.5
2.0
2.5
t
x(t)
Left: σ = 2; right: σ = 0.5
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Consider the linear scalar ordinary differential equation
y(t) = a + by(t) on t ≥ 0 (3.4)
with initial value y(0) = y0 > 0, where a,b > 0. This equationhas its explicit solution
y(t) =(
y0 +ab
)ebt − a
b.
Hencelim
t→∞
1t
log(y(t)) = b,
that is, the solution tends to infinity exponentially.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
On the other hand, considr its corresponding SDE
dx(t) = [a + bx(t)]dt + σx(t)dB(t) on t ≥ 0 , (3.5)
where σ > 0 and B(t) is a scalar Brownian motion. Given initialvalue x(0) = x0 > 0, this SDE has its explicit solution
x(t) = exp[(b−1
2σ2)t+σB(t)
](x0+a
∫ t
0exp
[(b−1
2σ2)s+σB(s)
]ds).
If σ2 > 2b then the solution of equation (3.5) obeys
lim supt→∞
log(x(t))
log t≤ σ2
σ2 − 2ba.s. (3.6)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
This shows that for any ε > 0, there is a positive randomvariable Tε such that, with probability one,
x(t) ≤ tε+σ2/(σ2−2b) ∀t ≥ Tε.
In other words, the solution will grow at most polynomially withorder ε+ σ2/(σ2 − 2b). In particular, by increasing the noiseintensity σ, we can make the order as close to 1 as we like.Comparing this polynomial growth with the exponential growthof the solution (3.4), we see the important fact that the noisesuppresses the exponential growth.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Generally, consider a nonlinear system described by ann-dimensional ordinary differential equation
y(t) = f (y(t), t),
whose coefficient obeys
〈y , f (y , t)〉 ≤ α + β|y |2, ∀(y , t) ∈ Rn × R+.
Clearly, the solution of this equation may grow exponentially.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
However, its stochastically linearly perturbed system
dx(t) = f (x(t), t)dt +m∑
i=1
Aix(t)dBi(t) (3.7)
may grow at most polynomially with probability one, where Bi(t)are m indepdenent Brownian motions and Ai ∈ Rn×n.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
In fact, if there are two positive constants γ and δ such that
m∑i=1
|Aix |2 ≤ γ|x |2,m∑
i=1
|xT Aix |2 ≥ δ|x |4 (3.8)
for all x ∈ Rn, andδ > β + 1
2γ, (3.9)
then, for any initial value x(0) = x0 ∈ Rn, the solution of thestochastically controlled system (3.7) obeys
lim supt→∞
log(|x(t)|)log t
≤ δ
2δ − 2β − γa.s. (3.10)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Consider the n-dimensional linear ordinary differential equation
dy(t)dt
= q + Qy(t), t ≥ 0 (3.11)
with initial value y(0) = y0 ∈ Rn, where q ∈ Rn and Q ∈ Rn×n.Assume that
λmax(Q + QT ) < 0. (3.12)
Equation (3.12) can be solved explicitly, which implies easilythat
lim supt→∞
|y(t)| ≤ |q|. (3.13)
That is, under condition (3.12), the solution of equation (3.11) isasymptotically bounded and the bound is independent of theinitial value.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Generally, consider an n-dimensional ordinary differentialequation
dy(t)dt
= f (y(t), t), t ≥ 0. (3.14)
Assume that f is sufficiently smooth and, in particular, it obeys
− 2〈x , f (x , t)〉 ≤ c1 + c2|x |2, ∀(x , t) ∈ Rn × R (3.15)
for some non-negative numbers c1 and c2. Our aim here isshow that the solution of its corresponding SDE
dx(t) = f (x(t), t)dt + g(x(t))dB(t) (3.16)
may grow exponentially with probability one.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
First, let the dimension of the state space n be an even numberand choose the dimension of the Brownian motion m to be 2.Let g : Rn → Rn×2 by
g(x) = (ξ,Ax),
where ξ = (ξ1, · · · , ξn)T ∈ Rn and, of course, ξ 6= 0, while
A =
0 σ−σ 0
. . .0 σ−σ 0
with σ > 0. So the SDE (3.16) becomes
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
dx(t) = f (x(t), t)dt + ξdB1 + σ
x2(t)−x1(t)
...xn(t)−xn−1(t)
dB2(t). (3.17)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
If we have
|ξ|2 > c1 and σ2 = c1 + c2 + |ξ|2 (3.18)
then the solution of equation (3.17) obeys
lim inft→∞
1t
log(|x(t)|) ≥|ξ|2 − c2
14|ξ|2
a.s. (3.19)
Alternatively, if we choose
|ξ|2 > c1 and σ2 = 3|ξ|2 + c2 − c1 (3.20)
thenlim inft→∞
1t
log(|x(t)|) ≥ 12(|ξ|2 − c1) a.s. (3.21)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
We next consider the dimension of the state space n be an oddnumber but n ≥ 3. Choose the dimension of the Brownianmotion m to be 3 and design g : Rn → Rn×3 by
g(x) = (ξ,A2x ,A3x),
where ξ = (ξ1, · · · , ξn)T ∈ Rn and, of course, ξ 6= 0, while
A2 =
0 σ−σ 0
. . .0 σ−σ 0
0
, A3 =
00 σ−σ 0
. . .0 σ−σ 0
with σ > 0.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
So the stochastically perturbed system (3.16) becomes
dx(t) = f (x(t), t)dt+ξdB1+σ
x2(t)−x1(t)
...xn−1(t)−xn−2(t)
0
dB2+σ
0x2(t)−x3(t)
...xn(t)−xn−1(t)
dB3.
(3.22)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
If we choose ξ and σ as (3.18) then the solution of equation(3.22) obeys (3.19), while if we choose ξ and σ as (3.20) thenthe solution of equation (3.22) obeys (3.21).
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Outline
1 Stochastic Modelling
2 Well-known ModelsLinear SDE modelsNon-linear SDE models
3 Stochastic vs DeterministicLotka-Volterra model
Variance dependent on x(t)Variance independent of x(t)
Noise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
4 Summary
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Following Bailey (The Mathematical Theory of InfectiousDiseases, Second Edition, Griffin, London, 1975), let X be theproportion of the human population that is infective and Y bethe proportion of infective (female) mosquitoes. Thedeterministic epidemic equations are then
dX/dt = abmY (t)[1− X (t)]− rX (t),dY/dt = aX (t)[1− Y (t)]− µY (t), (3.23)
where m is the number of female mosquitoes per human host,a the number of bites per unit time on man by a singlemosquito, b the proportion of infected bites on man thatproduce an infection, r the per capita rate of recovery inhumans and µ the per capita mortality rate for mosquitoes.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
The deterministic steady state endemic levels, whendX/dt = dY/dt = 0, are
X∞ =a2bm − µra(abm + r)
and
Y∞ =a2bm − µrabm(a + µ)
Ifa2bm ≤ µr
the endemic level is zero, and initial outbreaks will collapse.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
If a = b = m = r = 1 and µ = 0.9, then the equations are
dX/dt = Y (t)[1− X (t)]− X (t),dY/dt = X (t)[1− Y (t)]− 0.9Y (t). (3.24)
The endemic state (X∞ = 0.05,Y∞ = 0.0526) is a stable state.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
Consider the corresponding SDE model
dX (t) = (Y (t)[1− X (t)]− X (t))dt + 0.4Y (t)dB1(t),dY (t) = (X (t)[1− Y (t)]− 0.9Y (t))dt + 0.4X (t)dB2(t).
The following simulation shows that the system divergesmarkedly from the low endemic level (initially (0.05,0.05)),generating repeated and increasing peaks of infection.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
0 10 20 30 40 50
0.02
0.06
0.10
t
X(t)
0 10 20 30 40 50
0.02
0.06
0.10
t
Y(t)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
On the other hand, consider another SDE model
dX (t) = (Y (t)[1− X (t)]− X (t))dt+0.4X (t)dB1(t) + 0.4Y (t)dB2(t),
dY (t) = (X (t)[1− Y (t)]− 0.9Y (t))dt+0.4X (t)dB1(t) + 0.4Y (t)dB2(t).
The following simulation shows that Individual outbreaks do notconverge on the endemic level, but fall to zero, in the presenceof sufficient symmetric stochasticity.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Lotka-Volterra modelNoise suppresses exponential growthNoise expresses exponential growthAn example by R. Wallace
0 10 20 30 40 50
0.00
0.02
0.04
t
X(t)
0 10 20 30 40 50
0.00
0.02
0.04
0.06
t
Y(t)
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.Noise may suppress or express the exponential growth.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.Noise may suppress or express the exponential growth.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.Noise may suppress or express the exponential growth.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.Noise may suppress or express the exponential growth.
Xuerong Mao SDE Models
Stochastic ModellingWell-known Models
Stochastic vs DeterministicSummary
Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.Noise may suppress or express the exponential growth.
Xuerong Mao SDE Models