Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Stochastic Differential Equations inApplications
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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,share price in a company,price of gold, oil or electricity.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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,share price in a company,price of gold, oil or electricity.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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,share price in a company,price of gold, oil or electricity.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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,share price in a company,price of gold, oil or electricity.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
The deterministic change may be modeled by
R̄dt = R̄(x(t), t)dt
where R̄ = r̄(x(t), t) is the average rate of change given x(t) attime t . So
dx(t)x(t)
= R̄(x(t), t)dt + random change.
How may we model the random change?
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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)
= R̄(x(t), t)dt + V (x(t), t)N(0,dt).
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
The stochastic model can therefore be written as
dx(t)x(t)
= R̄(x(t), t)dt + V (x(t), t)dB(t),
or
dx(t) = R̄(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)
which is a stochastic differential equation (SDE).
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Exponential growth model
If both R̄ and V are constants, say
R̄(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 theory.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Exponential growth model
IfR̄(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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Logistic model
IfR̄(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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Square root process
IfR̄(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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Mean-reverting square root process
IfR̄(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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Linear SDE modelsNon-linear SDE models
Theta process
IfR̄(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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
In the classical theory of population dynamics, it is assumedthat the grow rate is constant µ. Thus
dx(t)x(t)
= µdt ,
which is often written as the familiar ordinary differentialequation (ODE)
dx(t)dt
= µx(t).
This linear ODE can be solved exactly to give exponentialgrowth (or decay) in the population, i.e.
x(t) = x(0)eµt ,
where x(0) is the initial population at time t = 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
We observe:
If µ > 0, x(t)→∞ exponentially, i.e. the population willgrow exponentially fast.If µ < 0, x(t)→ 0 exponentially, that is the population willbecome extinct.If µ = 0, x(t) = x(0) for all t , namely the population isstationary.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
We observe:
If µ > 0, x(t)→∞ exponentially, i.e. the population willgrow exponentially fast.If µ < 0, x(t)→ 0 exponentially, that is the population willbecome extinct.If µ = 0, x(t) = x(0) for all t , namely the population isstationary.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
We observe:
If µ > 0, x(t)→∞ exponentially, i.e. the population willgrow exponentially fast.If µ < 0, x(t)→ 0 exponentially, that is the population willbecome extinct.If µ = 0, x(t) = x(0) for all t , namely the population isstationary.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
However, if we take the uncertainty into account as explainedbefore, we may have
dx(t)x(t)
= rdt + σdB(t).
This is often written as the linear SDE
dx(t) = µx(t)dt + σx(t)dB(t).
It has the explicit solution
x(t) = x(0) exp[(µ− 0.5σ2)t + σB(t)
].
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Recall the properties of the Brownian motion
lim supt→∞
B(t)√2t log log t
= 1 a.s.
andlim inft→∞
B(t)√2t log log t
= −1 a.s.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If µ > 0.5σ2, x(t)→∞ exponentially with probability 1, i.e.the population will grow exponentially fast.If µ < 0.5σ2, x(t)→ 0 exponentially with probability 1, thatis the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct. (Comment on stochastic stabilisation.)
If µ = 0.5σ2, lim supt→∞ x(t) =∞ while lim inft→∞ x(t) = 0with probability 1.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If µ > 0.5σ2, x(t)→∞ exponentially with probability 1, i.e.the population will grow exponentially fast.If µ < 0.5σ2, x(t)→ 0 exponentially with probability 1, thatis the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct. (Comment on stochastic stabilisation.)
If µ = 0.5σ2, lim supt→∞ x(t) =∞ while lim inft→∞ x(t) = 0with probability 1.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If µ > 0.5σ2, x(t)→∞ exponentially with probability 1, i.e.the population will grow exponentially fast.If µ < 0.5σ2, x(t)→ 0 exponentially with probability 1, thatis the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct. (Comment on stochastic stabilisation.)
If µ = 0.5σ2, lim supt→∞ x(t) =∞ while lim inft→∞ x(t) = 0with probability 1.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If µ > 0.5σ2, x(t)→∞ exponentially with probability 1, i.e.the population will grow exponentially fast.If µ < 0.5σ2, x(t)→ 0 exponentially with probability 1, thatis the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct. (Comment on stochastic stabilisation.)
If µ = 0.5σ2, lim supt→∞ x(t) =∞ while lim inft→∞ x(t) = 0with probability 1.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If µ > 0.5σ2, x(t)→∞ exponentially with probability 1, i.e.the population will grow exponentially fast.If µ < 0.5σ2, x(t)→ 0 exponentially with probability 1, thatis the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct. (Comment on stochastic stabilisation.)
If µ = 0.5σ2, lim supt→∞ x(t) =∞ while lim inft→∞ x(t) = 0with probability 1.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
ExampleThe solution of the ODE
dx(t)dt
= x(t)
obeyslim
t→∞x(t) =∞.
However, The solution of the SDE
dx(t) = x(t)dt + 2x(t)dB(t)
obeyslim
t→∞x(t) = 0 a.s.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
0 5 10 15
010
0020
0030
0040
00
t
X(t)
or x
(t)
true solnEM soln
0 5 10 15
050
150
250
350
t
X(t)
or x
(t)
true solnEM soln
0 5 10 15
050
0010
000
1500
0
t
X(t)
or x
(t)
true solnEM soln
0 5 10 15
010
2030
4050
t
X(t)
or x
(t)
true solnEM soln
x
x
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
The logistic model for single-species population dynamics isgiven by the ODE
dx(t)dt
= x(t)[b + ax(t)]. (3.1)
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If a < 0 and b > 0, the equation has the global solution
x(t) =b
−a + e−bt (b + ax0)/x0(t ≥ 0) ,
which is not only positive and bounded but also
limt→∞
x(t) =b|a|.
If a > 0, whilst retaining b > 0, then the equation has onlythe local solution
x(t) =b
−a + e−bt (b + ax0)/x0(0 ≤ t < T ) ,
which explodes to infinity at the finite time
T = −1b
log(
ax0
b + ax0
).
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If a < 0 and b > 0, the equation has the global solution
x(t) =b
−a + e−bt (b + ax0)/x0(t ≥ 0) ,
which is not only positive and bounded but also
limt→∞
x(t) =b|a|.
If a > 0, whilst retaining b > 0, then the equation has onlythe local solution
x(t) =b
−a + e−bt (b + ax0)/x0(0 ≤ t < T ) ,
which explodes to infinity at the finite time
T = −1b
log(
ax0
b + ax0
).
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Once again, the growth rate b here is not a constant but astochastic process. Therefore, bdt should be replaced by
bdt + N(0, v2dt) = bdt + vN(0,dt) = bdt + vdB(t),
where v2 is the variance of the noise intensity. Hence the ODEevolves to an SDE
dx(t) = x(t)(
[b + ax(t)]dt + vdB(t)). (3.2)
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
The variance may or may not depend on the state x(t). We firstconsider the latter, say
v = σx(t).
Then the SDE (3.2) becomes
dx(t) = x(t)(
[b + ax(t)]dt + σx(t)dB(t)). (3.3)
How is this SDE different from its corresponding ODE?
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Significant Difference between ODE (3.1) and SDE(3.3)
ODE (3.1): The solution explodes to infinity at a finite timeif a > 0 and b > 0.SDE (3.3): With probability one, the solution will no longerexplode in a finite time, even in the case when a > 0 andb > 0, as long as σ 6= 0. Moreover, the stochasticpopulation system is persistent.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Significant Difference between ODE (3.1) and SDE(3.3)
ODE (3.1): The solution explodes to infinity at a finite timeif a > 0 and b > 0.SDE (3.3): With probability one, the solution will no longerexplode in a finite time, even in the case when a > 0 andb > 0, as long as σ 6= 0. Moreover, the stochasticpopulation system is persistent.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Example
dx(t)dt
= x(t)[1 + x(t)], t ≥ 0, x(0) = x0 > 0
has the solution
x(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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
0�
2�
4�
6�
8�
10�0
20
40
60
80
0�
2�
4�
6�
8�
10�0
50
100
150
200(a)�
(b)�
x
x
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Note on the graphs:In graph (a) the solid curve shows a stochastic trajectorygenerated by the Euler scheme for time step ∆t = 10−7 andσ = 0.25 for a one-dimensional system (3.3) with a = b = 1.The corresponding deterministic trajectory is shown by thedot-dashed curve. In Graph (b) σ = 1.0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
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 FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
If the variance is independent of the state x(t), namelyv = σ = const ., then the SDE (3.2) becomes
dx(t) = x(t)(
[b + ax(t)]dt + σdB(t)). (3.4)
How is this SDE different from others?
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
For this SDE, we require a < 0 in order to have noexplosion.If b < 0.5σ2 then limt→∞ x(t) = 0 with probability 1,namely the noise makes the population extinct.If b > 0.5σ2, the population is persistent and
limT→∞
1T
∫ T
0x(t)dt =
b − 0.5σ2
|a|a.s.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
For this SDE, we require a < 0 in order to have noexplosion.If b < 0.5σ2 then limt→∞ x(t) = 0 with probability 1,namely the noise makes the population extinct.If b > 0.5σ2, the population is persistent and
limT→∞
1T
∫ T
0x(t)dt =
b − 0.5σ2
|a|a.s.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
For this SDE, we require a < 0 in order to have noexplosion.If b < 0.5σ2 then limt→∞ x(t) = 0 with probability 1,namely the noise makes the population extinct.If b > 0.5σ2, the population is persistent and
limT→∞
1T
∫ T
0x(t)dt =
b − 0.5σ2
|a|a.s.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
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.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
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.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
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.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Exponential growth modelLogistic ModelSummary
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.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Most of SDEs used in practice do not have explicit solutions.How can we use these SDEs to forecast?One of the important techniques is the method of Monte Carlosimulations. There are two main motivations for suchsimulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Most of SDEs used in practice do not have explicit solutions.How can we use these SDEs to forecast?One of the important techniques is the method of Monte Carlosimulations. There are two main motivations for suchsimulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Most of SDEs used in practice do not have explicit solutions.How can we use these SDEs to forecast?One of the important techniques is the method of Monte Carlosimulations. There are two main motivations for suchsimulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Question: Can we trust the Monte Carlo simulations?
Test problemThe linear SDE
dX (t) = 2X (t)dt + X (t)dB(t), X (0) = 1
has the explicit solution
x(t) = exp(1.5t + B(t)).
The Monte Carlo simulation can be carried out based on theEuler-Maruyama (EM) method
x(0) = 1, x(i + 1) = x(i)[1 + 2∆ + ∆Bi ], i ≥ 0,
where ∆Bi = B((i + 1)∆)− B(i∆) ∼ N(0,∆).
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
0.0 0.5 1.0 1.5 2.0
24
68
1012
t
X(t)
or x
(t)
true solnEM soln
0.0 0.5 1.0 1.5 2.0
05
1015
2025
t
X(t)
or x
(t)
true solnEM soln
0.0 0.5 1.0 1.5 2.0
020
6010
014
0
t
X(t)
or x
(t)
true solnEM soln
0.0 0.5 1.0 1.5 2.0
24
68
10
t
X(t)
or x
(t)
true solnEM soln
x
x
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Typically, let us consider the square root process
dS(t) = rS(t)dt + σ√
S(t)dB(t), 0 ≤ t ≤ T .
A numerical method, e.g. the Euler–Maruyama (EM) methodapplied to it may break down due to negative values beingsupplied to the square root function. A natural fix is to replacethe SDE by the equivalent, but computationally safer, problem
dS(t) = rS(t)dt + σ√|S(t)|dB(t), 0 ≤ t ≤ T .
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Discrete EM approximation
Given a stepsize ∆ > 0, the EM method applied to the SDEsets s0 = S(0) and computes approximations sn ≈ S(tn), wheretn = n∆, according to
sn+1 = sn(1 + r∆) + σ√|sn|∆Bn,
where ∆Bn = B(tn+1)− B(tn).
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Continuous-time EM approximation
s(t) := s0 + r∫ t
0s̄(u))du + σ
∫ t
0
√|s̄(u)|dB(u),
where the “step function” s̄(t) is defined by
s̄(t) := sn, for t ∈ [tn, tn+1).
Note that s(t) and s̄(t) coincide with the discrete solution at thegridpoints; s̄(tn) = s(tn) = sn.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:
Theorem
lim∆→0
E(
sup0≤t≤T
|s(t)−S(t)|2)
= lim∆→0
E(
sup0≤t≤T
|s̄(t)−S(t)|2)
= 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:
Theorem
lim∆→0
E(
sup0≤t≤T
|s(t)−S(t)|2)
= lim∆→0
E(
sup0≤t≤T
|s̄(t)−S(t)|2)
= 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline
1 Stochastic Modelling2 Well-known Models
Linear SDE modelsNon-linear SDE models
3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary
4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Bond
If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff
β := E exp
(−∫ T
0S(t)dt
)from a bond. A natural approximation based on the EM methodis
β∆ := E exp
(−∆
N−1∑n=0
|sn|
), where N = T/∆.
Theorem
lim∆→0|β − β∆| = 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Bond
If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff
β := E exp
(−∫ T
0S(t)dt
)from a bond. A natural approximation based on the EM methodis
β∆ := E exp
(−∆
N−1∑n=0
|sn|
), where N = T/∆.
Theorem
lim∆→0|β − β∆| = 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
European call option
A European call option with the exercise price K at expiry timeT pays S(T )− K if S(T ) > K otherwise 0.
TheoremLet r be the risk-free interest rate and define
C = e−rT E[(S(T )− K )+
],
C∆ = e−rT E[(s̄(T )− K )+
].
Thenlim
∆→0|C − C∆| = 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
European call option
A European call option with the exercise price K at expiry timeT pays S(T )− K if S(T ) > K otherwise 0.
TheoremLet r be the risk-free interest rate and define
C = e−rT E[(S(T )− K )+
],
C∆ = e−rT E[(s̄(T )− K )+
].
Thenlim
∆→0|C − C∆| = 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Example - the Black-Scholes model
Consider a BS model
dS(t) = 0.05S(t)dt + 0.03S(t)dB(t), S(0) = 10
and a European call option with the exercise price K = 10.05 atexpiry time T = 1, where 0.05 is the risk-free interest rate and0.03 is the volatility. By the well-known Black-Scholes formulaon the option, we can compute the value of a European calloption at time zero is
C = 0.4487318.
On the other hand, we can let ∆ = 0.001, simulate 1000 pathsof the SDE, compute the mean payoff at T = 1, discounting itby e−0.05, we get the estimated option value
C∆ = 0.4454196
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
To be more reliable, we can carry out such simulation, say 10times, to get 10 estimated values:
0.4454196, 0.4611569, 0.4512847, 0.4490462,⋃.4294038,
0.4618921, 0.4556195, 0.4559547, 0.4399189, 0.4489594.
Their mean valueC̄∆ = 0.4498656
gives a better estimation for C.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Up-and-out call option
An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.
TheoremDefine
V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}
],
V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤c, 0≤t≤T}
].
Thenlim
∆→0|V − V∆| = 0.
Xuerong Mao FRSE SDEs
Stochastic ModellingWell-known Models
Stochastic verse DeterministicForecasting and Monte Carlo Simulations
Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Up-and-out call option
An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.
TheoremDefine
V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}
],
V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤c, 0≤t≤T}
].
Thenlim
∆→0|V − V∆| = 0.
Xuerong Mao FRSE SDEs