Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Basics of simulation and statistic of dynamic
SystemsDiffusion processes and linear stochastic equations
by
Dany DJEUDEU
TU Dortmund
Faculty of Statistics
November 2014
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
1
Introduction
2
Diffusion processes: Definition
General definition
Some remarks
Some examples of diffusion processes
3
The Stochastic differential equation
Solution of the stochastic differential equation: The integral
form
Existence and uniqueness of solutions to the diffusion
equation
4
5
properties of diffusion processes and applications
Markov property of a diffusion process
Ito Formula
Applications
The Lamperti transform
Summary
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Introduction
differential equations are known to describe the time evolution
of some phenomenon: diseases for instance
It is frequently the case that economic and financial
considerations will suggest that a stock price, exchange rate,
interest rate, or other economic variables evolve in time
according to a stochastic differential equation of the form
dXt = b(t, Xt )dt + σ(t, Xt )dBt
(1)
Bt is the standard Brownian motion and b and σ are given
functions of time t and the current state x.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
General definition
Definition 2.1 ([4])
A stochastic real-valued process (Xt )t≥0 is said to be a diffusion
process if it satisfies the following conditions:
A) (Xt )t≥0 is a markov-process
B) There exist limits:
b(x, t) = lim
∆→0
1
E ((X (t+∆)−X (t))|X (t) = x) (2)
∆
1
E {(X (t + ∆) − X (t))2 |X (t) = x}
∆→0 ∆
(3)
σ 2 (x, t) = lim
C) X (t)t≥0 is a continuous process
(P(|Xt − Xs | ≥ |Xs = x) = o(t − s))
b(x, t) is called The drift (coefficient, parameter) and σ 2 (x, t) is
the diffusion (coefficient, parameter)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Some remarks
Non differentiability of diffusion processes
The definition of a diffusion process suggests a relationship of
the following form ([4]):
dXt = b(s, Xs )dt + σ(s, Xs )dWt
(4)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Some examples of diffusion processes
The standard Wiener process is a diffusion process with drift
0 and diffusion parameter 1.
The class of Gaussian processes is an important class of
diffusion processes ([5])
Consider the stochastic differential equation
1
2
dXt = 3Xt3 dt + 3Xt3 dWt
(5)
with the initial condition X0 = 0.
Clearly, the process Xt ≡ 0 is a solution. But so is Xt = Wt3 .
The problem in this example is that the coefficients
1
2
b(t) = 3Xt3 and σ(t) = 3Xt3 although continuous in x, are
not smooth enough at x = 0.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Solution to the stochastic differential equation in integral
form
We concentrate on the dynamic of diffusion processes
We will be interested in solving stochastic differential
equations (or SDE s) of the following form:
dXt = b(s, Xs )dt + σ(s, Xs )dWt
with initial conditions X0 = Z .
W : [0, ∞)xΩ → R with (t, ω) → W (t)(ω) the Brownian
motion and Z a random variable with distribution µ,
independent of the σ-Algebra generated by W and with a
finite second moment.
b with (t, ω) → b(t, Xt ω) and σ with (t, ω) → σ(t, Xt (ω))
are deterministic measurable applications.
(6)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Solution to the stochastic differential equation in integral
form
A solution X of the SDE (6) is a continuous stochastic
process which satisfies the integral equation
Z
Xt = X0 +
t
Z
0
t
σ(s, Xs )dWs f .s. ∀t ≥ 0 (7)
b(s, Xs )ds +
0
Rt
the integral 0 b(s, Xs )ds is the usual Riemann-Integral and
Rt
0 σ(s, Xs )dWs the Ito-Integral.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
The stochastic differential equation: Existence and
uniqueness of solutions
We consider the SDE
dXt = b(s, Xs )dt + σ(s, Xs )dWt
(8)
with initial conditions X0 = Z .
with b and σ defined like in the previous section.
We suppose that
Z
P{
0
T
sup (|b(t, x)| + σ 2 (t, x))dt < ∞} = 1
x≤R
for all T , R ∈ [0, ∞) because (7) is an Ito process.
(9)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Assumptions
Assumption 3.1 (Global Lipschitz condition)
There exists a constant K < +∞ such that for all x, y ∈ R and
t ∈ [0, T ],
|b(t, x) − b(t, y )| + |σ(t, x) − σ(t, y )| < K |x − y |.
(10)
Assumption 3.2 (Linear growth)
There exists a constant C < +∞ such that for all x, y ∈ R and
t ∈ [0, T ],
|b(t, x)| + |σ(t, x)| < C (1 + |x|).
(11)
The linear growth condition controls the behavior of the solution
so that Xt does not explode in a finite time.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Existence and uniqueness
Theorem 3.1 ([1])
Under the previous assumptions, the stochastic differential
equation (8) has a unique, continuous, and adapted strong solution
such that
Z
T
|Xt |2 dt
E
0
<∞
(12)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Weak solution
Assumption 3.3 (Local assumptions)
For any x0 ∈ R , there exists a constant L(x0 ) > 0 and Bx0 > 0
such that: for all x ∈ R
Local Lipschitz condition
|b(x) − b(x0 )| + |σ(x) − σ(x0 )| ≤ Lx0 |x − x0 |
(13)
Local linear growth
2xb(x) + σ 2 (x) ≤ Bx0 (1 + x 2 )
Theorem 3.2
Under the previous assumptions, the stochastic differential
equation (8) has a unique weak solution [1].
(14)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Transition probability
From the Markov property of the diffusion process, it is also
possible to define the transition density from value x at time s to
value y at time t by p(t, y |s, x) or, when convenient, as
p(t − s, y |x).
The transition density satisfies the Kolmogorov forward equation:
∂
1 ∂2 2
∂
p(t, y |s, x) = − (b(t, y )p(t, y |s, x))+
(σ (t, x)p(t, y |s, x))
∂t
∂y
2 ∂y 2
(15)
and Kolmogorov backward equation
−
∂
∂
1
∂2
p(t, y |s, x) = b(t, x) p(t, y |s, x) + σ 2 (t, x) 2 p(t, y |s, x)
∂s
∂x
2
∂x
(16)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Ito Formula
Useful in simulations
It is the foundation of mathematical finance and stochastic
calculus
This formula can be seen as the stochastic version of a Taylor
expansion of f (X ) stopped at the second order, where X is a
diffusion process.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Ito formula
Lemma 4.1 (Ito formula)
If f is a function that is two times differentiable on both t and x
(f = f (t, x)), then
Z t
Z t
f (t, Xt ) = f (0, X0 ) +
ft (s, Xs )ds +
fx (s, Xs )dXs (17)
0
0
Z
1 t
fxx (s, Xs )(dXs )2
+
2 0
or, in differential form
1
df (t, Xt ) = ft (t, Xt )dt + fx (t, Xt )dXt + fxx (t, Xt )(dXt )2
2
fx (t, x) =
∂f (t, x)
∂ 2 f (t, x)
∂f (t, x)
, ft (t, x) =
, fxx (t, x) =
∂x
∂t
∂x 2
(18)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Proof of the Ito formula
Proof.
We give here the steps of the proof. The Taylor expansion in one
dimension is used, say
f (t, x) − f (s, x0 ) = (f (t, x) − f (s, x)) + (f (s, x) − f (s, x0 ))
Taylor
= ft (s + α(t − s), x)(t − s) + fx (s, x0 )(x − x0 )
1
fxx (s, x0 + β(x − x0 ))(x − x0 )2
+
2
We first use the partition of the intervals,
f (t, Xt ) − f (0, X0 ) =
n
X
f (ti , Xti ) − f (ti−1 , Xti−1 )
i=1
Then we use the Taylor expansion and the definition of the
integrals
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Application of the Ito formula on the Brownian motion
Lemma 4.2
If Xt is the Brownian motion, this simplifies to the following
Z t
Z t
1
f (t, Wt ) = f (0, 0)+ (ft (s, Ws )+ fxx (s, Ws ))ds+
fx (s, Ws )dWs
2
0
0
(19)
or, in differential form
1
df (t, Wt ) = (ft (t, Wt ) + fxx (t, Wt ))dt + fx (t, Wt )dWt
2
(20)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Steps of the proof
(dt · dWt) and (dt)2 are of order O(dt),
(dWt)2 behave like dt for the properties of the Brownian
motion
(Wti − Wti−1 ) ' (ti − ti−1 )
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Ito integral
Application 4.1
consider the function f (t, x) = f (x) = x 2 . The Ito formula applied
to f (Wt) then leads to
Wt2 = 02 +
Z
2Ws dWs +
0
and therefore
Z
0
t
t
1
2
Z
1
1
Ws dWs = Wt2 − t
2
2
t
2ds
0
(21)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Linear stochastic differential equations
Linear in the sense that both the drift and the diffusion
coefficient are affine functions of the solution.
dXt = b1 (t)Xt dt + σ1 (t)Xt dWt
(22)
This equation is called a stochastic differential equation with
multiplicative noise. We aim to solve this equation. We use
the Ito formula to do so.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Linear stochastic differential equations
Theorem 4.1
Choosing f (x) = log (x) , the solution to (22) is given by
Z t
Z t
1 2
σ1 (s)dWs
Xt = X0 · exp
(b1 (s) − σ1 (s))ds +
2
0
0
(23)
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Application
Application 4.2 (geometric Brownian motion)
It is now easy to derive the stochastic differential equation for the
geometric Brownian motion. We have the Black-Scholes
differential equation
dSt = rSt dt + σSt dWt
b1 and σ1 are considered here constants.
Consider the geometric Brownian motion
σ2
St = S0 exp (r − )t + σWt , t > 0.
2
2
(24)
(25)
choosing f (t, x) == S0 exp{(r − σ2 )t + σx, we easily prove with
the Ito formula that the geometric Brownian motion is solution
to the Black-Scholes equation (24).
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
The Lamperti transform
Theorem 4.2
Suppose we have the stochastic differential equation
dXt = b(t, Xt )dt + σ(Xt )dWt
(26)
where the diffusion coefficient depends only on the state variable.
Such a stochastic differential equation can always be transformed
into one with a unitary diffusion coefficient by applying the
Lamperti transform
Z
Yt = F (Xt ) =
z
Xt
1
ds
σ(s)
(27)
Here z is any arbitrary value in the state space of X . Indeed, the
process Yt solves the stochastic differential equation
dYt = (
b(t, Xt ) 1
− σx (Xt ))dt + dWt
σ(X )
2
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Application: A trick for simulations
Application 4.3
Trick for simulation
If possible, it is better to apply the Lamperti transform F to X
and simulate Y = F (X ) by Euler scheme. This makes the
simulation more stable and usually efficient ([6]). Then apply
F −1 (Y ) to obtain X on the original scale.
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Summary
The diffusion process in one dimension
General definition of a diffusion process: Solution to a
stochastic differential equations
Conditions for the resolution of those stochastic processes
have been presented
Problematic : Strictly makes the differential equation
considered no sense ( the Brownian motion for example is
not differentiable)
Trick: Solution in the integral form, with the advantage that
there is no differential term in the expression.
Properties of diffusion processes: Markov property with
Kolmogorov forward and backward equations
Stress on the Ito form: basis of mathematical finance,
Brownian motion solution to the Black-Scholes equation,
Ito integral
The Lamperti transform: Trick for simulations
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
Stefano M. Iacus, Simulation and Inference for Stochastic
Differential Equations. Springer- Verlag. New York.
K. Webel, D. Wied (2011): Stochastische Prozesse. Eine
Einführung für Statistiker und Datenwissenschaftler,2011, XVI,
270S. 39 Abb..,ISBN 978-3-8349-2809-2
Øksendal, B. (1998): Stochastic Differential Equations. An
Introduction with Applications,5th ed., Springer-Verlag, Berlin.
http://kurser.math.su.se/moodle19/file.php/827/SP-Lecture12.pdf
R. Coleman Stochastic Processes 1974 Lecture in
Mathematics, Imperial College, University of London.
Stefano M. Iacus 18 − 1 − 2008) Wien: Simulation and
inference for
SDEs,http://www.wu.ac.at/firm/talks/ws0708/sde.pdf
Introduction Diffusion processes: Definition The Stochastic differential equation
properties of diffusion processes and applications
J. R. Movellan (2011) Tutorial on Stochastic Differential
Equations, MPLab Tutorials Version 06.1.