Stochastic Differential
Equations
Do not worry about your problems with mathematics, I assure you
mine are far greater.
Albert Einstein.
Florian Herzog
2013
Stochastic Differential Equations (SDE)
A ordinary differential equation (ODE)
dx(t)
dt
= f (t, x) , dx(t) = f(t, x)dt , (1)
with initial conditions x(0) = x
0
can be written in integral form
x(t) = x
0
+
t
0
f(s, x(s))ds , (2)
where x(t) = x(t, x
0
, t
0
) is the solution with initial conditions x(t
0
) = x
0
. An
example is given as
dx(t)
dt
= a(t)x(t) , x(0) = x
0
. (3)
Stochastic Systems, 2013 2
Stochastic Differential Equations (SDE)
When we take the ODE (3) and assume that a(t) is not a deterministic parameter
but rather a stochastic parameter, we get a stochastic differential equation (SDE). The
stochastic parameter a(t) is given as
a(t) = f(t) + h(t)ξ(t) , (4)
where ξ(t) denotes a white noise process.
Thus, we obtain
dX(t)
dt
= f (t)X(t) + h(t)X(t)ξ(t) . (5)
When we write (5) in the differential form and use dW (t) = ξ(t)dt, where dW (t)
denotes differential form of the Brownian motion,we obtain:
dX(t) = f(t)X(t)dt + h(t)X(t)dW (t) . (6)
Stochastic Systems, 2013 3
Stochastic Differential Equations (SDE)
In general an SDE is given as
dX(t, ω) = f (t, X(t, ω))dt + g(t, X(t, ω))dW (t, ω) , (7)
where ω denotes that X = X(t, ω) is a random variable and possesses the initial
condition X(0, ω) = X
0
with probability one. As an example we have already
encountered
dY (t, ω) = µ(t)dt + σ(t)dW (t, ω) .
Furthermore, f(t, X(t, ω)) ∈ R, g(t, X(t, ω)) ∈ R, and W (t, ω) ∈ R. Similar as
in (2) we may write (7) as integral equation
X(t, ω) = X
0
+
t
0
f(s, X(s, ω))ds +
t
0
g(s, X(s, ω))dW (s, ω) . (8)
Stochastic Systems, 2013 4
Stochastic Integrals
For the calculation of the stochastic integral
T
0
g(t, ω)dW (t, ω), we assume that
g(t, ω) is only changed at discrete time points t
i
(i = 1, 2, 3, ..., N − 1), where
0 = t
0
< t
1
< t
2
< . . . < t
N−1
< t
N
< T . We define the integral
S =
T
0
g(t, ω)dW (t, ω) , (9)
as the Riemannßum
S
N
(ω) =
N
i=1
g(t
i−1
, ω)
W (t
i
, ω) − W (t
i−1
, ω)
. (10)
with N → ∞.
Stochastic Systems, 2013 5
Stochastic Integrals
A random variable S is called the Itˆo integral of a stochastic process g(t, ω) with
respect to the Brownian motion W (t, ω) on the interval [0, T ] if
lim
N→∞
E
S −
N
i=1
g(t
i−1
, ω)
W (t
i
, ω) − (W (t
i−1
, ω)
= 0 , (11)
for each sequence of partitions (t
0
, t
1
, . . . , t
N
) of the interval [0, T ] such that
max
i
(t
i
− t
i−1
) → 0. The limit in the above definition converges to the stochastic
integral in the mean-square sense. Thus, the stochastic integral is a random variable,
the samples of which depend on the individual realizations of the paths W (., ω).
Stochastic Systems, 2013 6
Stochastic Integrals
The simplest p ossible example is g(t) = c for all t. This is still a stochastic
process, but a simple one. Taking the definition, we actually get
T
0
c dW (t, ω) = c lim
N→∞
N
i=1
W (t
i
, ω) − W (t
i−1
, ω)
= c lim
N→∞
[(W (t
1
, ω)−W (t
0
, ω)) + (W (t
2
, ω)−W (t
1
, ω)) + . . .
+(W (t
N
, ω)−W (t
N−1
, ω))
= c (W (T, ω) − W (0, ω)) ,
where W (T, ω) and W (0, ω) are standard Gaussian random variables. With
W (0, ω) = 0, the last result becomes
T
0
c dW (t, ω) = c W (T, ω) .
Stochastic Systems, 2013 7
Stochastic Integrals
Example: g(t, ω) = W (t, ω)
T
0
W (t, ω) dW (t, ω) =
= lim
N→∞
N
i=1
W (t
i−1
, ω)
W (t
i
, ω) − W (t
i−1
, ω)
= lim
N→∞
1
2
N
i=1
(W
2
(t
i
, ω) − W
2
(t
i−1
, ω)) −
1
2
N
i=1
(W (t
i
, ω) − W (t
i−1
, ω))
2
= −
1
2
lim
N→∞
N
i=1
(W (t
i
, ω) − W (t
i−1
, ω))
2
+
1
2
W
2
(T, ω) , (12)
where we have used the following algebraic relationship y(x − y) = yx − y
2
+
1
2
x
2
−
1
2
x
2
=
1
2
x
2
−
1
2
y
2
−
1
2
(x − y)
2
.
Stochastic Systems, 2013 8
Stochastic Integrals
We take now a detailed look at :lim
N→∞
N
i=1
(W (t
i
, ω) − W (t
i−1
, ω))
2
.
E[ lim
N→∞
N
i=1
(W (t
i
, ω) − W (t
i−1
, ω))
2
] = lim
N→∞
N
i=1
E[(W (t
i
, ω) − W (t
i−1
, ω))
2
]
= lim
N→∞
N
i=1
(t
i
− t
i−1
)
= T
Var[ lim
N→∞
N
i=1
(W (t
i
, ω) − W (t
i−1
, ω))
2
] = lim
N→∞
N
i=1
Var[(W (t
i
, ω) − W (t
i−1
, ω))
2
]
= 2 lim
N→∞
N
i=1
(t
i
− t
i−1
)
2
.
Stochastic Systems, 2013 9
Stochastic Integrals
By reducing the partition, the variance becomes zero,
lim
N→∞
N
i=1
(t
i
− t
i−1
)
2
≤ max
i
(t
i
− t
i−1
) lim
N→∞
N
i=1
(t
i
− t
i−1
)
= max
i
(t
i
− t
i−1
) T
= 0 , (13)
since t
i−1
− t
i
→ 0. Since the expected value of
N
i=1
(t
i
− t
i−1
)
2
is T and the
variance becomes zero, we get
N
i=1
(W (t
i
, ω) − W (t
i−1
, ω))
2
= T (14)
Stochastic Systems, 2013 10
Stochastic Integrals
The stochastic integral has the solution
T
0
W (t, ω) dW (t, ω) =
1
2
W
2
(T, ω) −
1
2
T (15)
This is in contrast to our intuition from standard calculus. In the case of a deterministic
integral
T
0
x(t)dx(t) =
1
2
x
2
(t), whereas the Itˆo integral differs by the term −
1
2
T .
— This example shows that the rules of differentiation (in particular the chain rule)
and integration need to be re-formulated in the stochastic calculus.
Stochastic Systems, 2013 11
Stochastic Integrals
Properties of Itˆo Integrals.
E[
T
0
g(t, ω) dW (t, ω)] = 0 .
Proof:
E[
T
0
g(t, ω)dW (t, ω)] = E[ lim
N→∞
N
i=1
g(t
i−1
, ω)
W (t
i
, ω) − W (t
i−1
, ω)
]
= lim
N→∞
N
i=1
E[g(t
i−1
, ω)] E[
W (t
i
, ω) − W (t
i−1
, ω)
]
= 0 .
The expectation of stochastic integrals is zero. This is what we would expect anyway.
Stochastic Systems, 2013 12
Stochastic Integrals
Properties of Itˆo Integrals.
Var
T
0
g(t, ω)dW (t, ω)
=
T
0
E[g
2
(t, ω)]dt . (16)
Proof:
Var
T
0
g(t, ω)dW (t, ω)
= E
(
T
0
g(t, ω)dW (t, ω))
2
= E
lim
N→∞
N
i=1
g(t
i−1
, ω)
W (t
i
, ω) − W (t
i−1
, ω)
2
Stochastic Systems, 2013 13
Stochastic Integrals
= lim
N→∞
N
i=1
N
j=1
E[g(t
i−1
, ω)g(t
j−1
, ω)
(W (t
i
, ω) − W (t
i−1
, ω))(W (t
j
, ω) − W (t
j−1
, ω))]
= lim
N→∞
N
i=1
E[g
2
(t
i−1
, ω)] E[
W (t
i
, ω) − W (t
i−1
, ω)
2
]
= lim
N→∞
N
i=1
E[g
2
(t
i−1
, ω)] (t
i
− t
i−1
)
=
T
0
E[g
2
(t, ω)]dt . (17)
Stochastic Systems, 2013 14
Stochastic Integrals
The calculation of the variance of the Itˆo Integrals shows two important properties:
• E
T
0
g(t, ω)dW (t, ω)
2
=
T
0
E
g
2
(t, ω)
dt
•
T
0
E[g
2
(t, ω)]dt < ∞
The second property is the condition of existence for Itˆo integrals. The next property is
the linearity of Itˆo integrals:
T
0
[a
1
g
1
(t, ω) + a
2
g
2
(t, ω)]dW (t, ω)
= a
1
T
0
g
1
(t, ω)dW (t, ω) + a
2
T
0
g
2
(t, ω)dW (t, ω) , (18)
for numbers a
1
, a
2
and stochastic functions g
1
(t, ω), g
2
(t, ω).
Stochastic Systems, 2013 15
Itˆo’s lemma
As mentioned shown in the second example, the rules of classical calculus are not valid
for stochastic integrals and differential equations. It is the equivalent to the chain rule
in classical calculus. The problem can be stated as follows:
Given a stochastic differential equation
dX(t) = f(t, X(t))dt + g(t, X(t))dW (t) , (19)
and another process Y (t ) which is a function of X(t),
Y (t) = ϕ(t, X(t)) ,
where the function ϕ(t, X(t)) is continuously differentiable in t and twice continuously
differentiable in X, find the stochastic differential equation for the process Y (t):
dY (t) =
˜
f(t, X(t))dt + ˜g(t, X(t))dW (t) .
Stochastic Systems, 2013 16
Itˆo’s lemma
In the case when we assume that g(t, X(t)) = 0, we know the result: the chain rule
for standard calculus. The result is given by
dy(t) = (ϕ
t
(t, x) + ϕ
x
(t, x)f(t, x))dt . (20)
In the case of stochastic problems, we reason as follows: The Taylor expansion of
ϕ(t, X(t)) yields
dY (t) = ϕ
t
(t, X)dt +
1
2
ϕ
tt
(t, X)dt
2
+ ϕ
x
(t, X)dX(t)
+
1
2
ϕ
xx
(t, X)(dX(t))
2
+ h.o.t . (21)
Stochastic Systems, 2013 17
Itˆo’s lemma
We use (19) for dX(t) and get
dY (t) = ϕ
t
(t, X)dt + ϕ
x
(t, X)[f (t, X(t))dt + g(t, X(t))dW (t)]
+ϕ
tt
(t, X)dt
2
+
1
2
ϕ
xx
(t, X)
f
2
(t, X(t))dt
2
+ g
2
(t, X(t))dW
2
(t)
+2f(t, X(t))g(t, X(t))dt dW (t)
+ h.o.t . (22)
The differentials of higher order (dt, dW ) become fast zero, dt
2
→ 0 and
dtdW (t) → 0. The stochastic term dW
2
(t) according to the rules of Brownian
motion is given as
dW
2
(t, ω) = dt . (23)
Stochastic Systems, 2013 18
Itˆo’s lemma
Omitting higher order terms and using the properties of Brownian motion, we arrive at
dY (t) = [ϕ
t
(t, X) + ϕ
x
(t, X)f (t, X(t)) +
1
2
ϕ
xx
(t, X)g
2
(t, X(t))]dt
+ϕ
x
(t, X)g(t, X(t))dW (t) . (24)
Reordering the terms yields the scalar version of Itˆo’s Lemma:
dY (t) =
˜
f(t, X(t))dt + ˜g(t, X(t))dW (t) , (25)
˜
f(t, X(t)) = ϕ
t
(t, X) + ϕ
x
(t, X)f (t, X(t))
+
1
2
ϕ
xx
(t, X)g
2
(t, X(t)) , (26)
˜g(t, X(t)) = ϕ
x
(t, X)g(t, X(t)) . (27)
Stochastic Systems, 2013 19
Itˆo’s lemma
The term
1
2
ϕ
xx
(t, X)g
2
(t, X(t)) is often called the Itˆo corretion term, since this
does not occur in the det. case.
We apply Itˆos formula for the following problem: ϕ(t, X) = X
2
with the SDE
dX(t) = dW (t). From the SDE, we get X(t) = W (t) and calculate the partial
derivatives of
∂ϕ(t,X)
∂X
= 2X,
∂
2
ϕ(t,X)
∂X
2
= 2, and
∂ϕ(t,X)
∂t
= 0. The Itˆo lemma yields
d(W
2
(t)) = 1dt + 2W (t)dW (t) . (28)
We rewrite the equation and use W (0) = 0
W
2
(t) = 1t + 2
t
0
W (t)dW (t) ,
t
0
W (t)dW (t) =
1
2
W
2
(t) −
1
2
t . (29)
Stochastic Systems, 2013 20
Itˆo’s lemma
We now allow that the process X(t) is in R
n
. We let W (t) be an m-dimensional
standard Brownian motion and f (t, X(t)) ∈ R
n
and g(t, X(t)) ∈ R
n×m
. Consider
a scalar process Y (t) defined by Y (t) = ϕ(t, X(t)), where ϕ(t, X) is a scalar
function which is continuously differentiable with respect to t and twice continuously
differentiable with respect to X. The Itˆo formula can be written in vector notation as
follows:
dY (t) =
˜
f(t, X(t))dt + ˜g(t, X(t))dW (t) , (30)
˜
f(t, X(t)) = ϕ
t
(t, X(t)) + ϕ
x
(t, X(t)) · f(t, X(t))
+
1
2
tr
ϕ
xx
(t, X(t))g(t, X(t))g
T
(t, X(t)))
, (31)
˜g(t, X(t)) = ϕ
x
(t, X(t)) · g(t, X(t)) , (32)
where “tr” denotes the trace operator.
Stochastic Systems, 2013 21
Itˆo’s lemma
Consider the following stochastic differential equation:
dS(t) = µ S(t)dt + σ S(t)dW (t) , (33)
We want to find the SDE for the process Y related to S as follows: Y (t) = ϕ(t, S) =
ln(S(t)) . The partial derivatives are:
∂ϕ(t,S)
∂S
=
1
S
,
∂
2
ϕ(t,S)
∂S
2
= −
1
S
2
, and
∂ϕ(t,S)
∂t
= 0.
Therefore, according to Itˆo we get,
dY (t) =
∂ϕ(t, S)
∂t
+
∂ϕ(t, S)
∂S
µS(t) +
1
2
∂
2
ϕ(t, S)
∂S
2
σ
2
S
2
(t)
dt
+
∂ϕ(t, S)
∂S
σS(t)
dW (t) , (34)
dY (t) = (µ −
1
2
σ
2
)dt + σdW (t) . (35)
Stochastic Systems, 2013 22
Itˆo’s lemma
Since the right hand side of (35) is independent of Y (t), we are able to compute the
stochastic integral:
Y (t) = Y
0
+
t
0
(µ −
1
2
σ
2
)dt +
t
0
σdW , (36)
Y (t) = Y
0
+ (µ −
1
2
σ
2
)t + σW (t) . (37)
Since Y (t) = ln S(t) we have found a solution for S(t) :
ln(S(t)) = ln(S(0)) + (µ −
1
2
σ
2
)t + σW (t) , (38)
S(t) = S(0)e
(µ−
1
2
σ
2
)t+σW (t)
, (39)
where W (t) is a standard BM.
Stochastic Systems, 2013 23
Itˆo’s lemma
Show for U(t) = X
1
(t)X
2
(t) with
dX
1
(t) = f
1
(t, X
1
)dt + g
1
(t, X
1
)dW (t) ,
dX
2
(t) = f
2
(t, X
2
)dt + g
2
(t, X
2
)dW (t) ,
that following formula is valid:
dU(t) = dX
1
(t)X
2
(t) + X
1
(t)dX
2
(t) + g
1
(t, X
1
)g
2
(t, X
2
)dt (40)
We show that we obtain the same result as in the previous formula by apply Itˆo’s
lemma. By (40) liefert
dU(t) = [ X
2
(t)f
1
(t, X
1
) + X
1
(t)f
2
(t, X
2
) + g
1
(t, X
1
)g
2
(t, X
2
)]dt
+[X
2
(t)g
1
(t, X
1
) + X
1
(t)g
2
(t, X
2
)]dW (t)
Stochastic Systems, 2013 24
Itˆo’s lemma
The partial derivatives of U are :
∂U
∂X
= (X
2
(t), X
1
(t))
T
,
∂
2
U
∂X
2
=
0 1
1 0
and
∂U
∂t
= 0.
dU(t) = [
∂U
∂t
+
∂U
∂X
[f
1
(t, X
1
), f
2
(t, X
2
)]
T
+
1
2
tr
∂
2
U
∂X
2
g
1
(t, X
1
)
2
g
1
(t, X
1
)g
2
(t, X
2
)
g
1
(t, X
1
)g
2
(t, X
2
) g
2
(t, X
2
)
2
]dt
+
∂U
∂X
[g
1
(t, X
1
), g
2
(t, X
2
)]
T
dW (t)
= [X
2
(t)f
1
(t, X
1
) + X
1
(t)f
2
(t, X
2
) + g
1
(t, X
1
)g
2
(t, X
2
)]dt
+[X
2
(t)g
1
(t, X
1
) + X
1
(t)g
2
(t, X
2
)]dW (t)
Stochastic Systems, 2013 25
Stochastic Differential Equations (SDE)
We classify SDEs into two large groups, linear SDEs and non-linear SDEs. Furthermore,
we distinguish between scalar linear and vector-valued linear SDEs.
We start with the easy case, the scalar linear linear SDEs. An SDE
dX(t) = f(t, X(t))dt + g(t, X(t))dW (t) , (41)
for a one-dimensional stochastic process X(t) is called a linear (scalar) SDE if and
only if the functions f(t, X(t)) and g(t, X(t)) are affine functions of X(t) ∈ R and
thus
f(t, X(t)) = A(t)X(t) + a(t) ,
g(t, X(t)) = [B
1
(t)X(t) + b
1
(t), ··· , B
m
(t)X(t) + b
m
(t)] ,
where A(t), a(t) ∈ R, W (t) ∈ R
m
is an m-dimensional Brownian motion, and
B
i
(t), b
i
(t) ∈ R, i = 1, ··· , m. Hence, f (t, X(t)) ∈ R and g(t, X(t)) ∈ R
1×m
.
Stochastic Systems, 2013 26
Stochastic Differential Equations (SDE)
The linear SDE possesses the following solution
X(t) = Φ(t)
x
0
+
t
0
Φ
−1
(s)
a(s) −
m
i=1
B
i
(s)b
i
(s)
ds
+
m
i=1
t
0
Φ
−1
(s)b
i
(s)dW
i
(s)
, (42)
where we denote Φ(t) as the fundamental matrix, which we obtain from
Φ(t) = exp
t
0
A(s) −
m
i=1
B
2
i
(s)
2
ds +
m
i=1
t
0
B
i
(s)dW
i
(s)
, (43)
The solution is similar to the solution of ODEs.
Stochastic Systems, 2013 27
Stochastic Differential Equations (SDE)
Let us assume that W (t) ∈ R, a(t) = 0, b(t) = 0, A(t) = A, B(t) = B. We
want to compute the solution of the SDE
dX(t) = AX(t)dt + BX(t)dW (t) , X(t) = x
0
, (44)
We can solve it using (42) and (43):
Φ(t) = e
(A−
1
2
B
2
)t+BW (t)
, (45)
and (42) is easy to calculate since
x(t) = Φ(t)x
0
= x
0
e
(A−
1
2
B
2
)t+BW (t)
. (46)
Stochastic Systems, 2013 28
Stochastic Differential Equations (SDE)
The expectation m(t) = E[X(t)]and the second moment P (t) = E[X
2
(t)] for
dX(t) = (A(t)X(t) + a(t))dt +
m
i=1
(B
i
(t)X(t) + b(t))dW
i
(t) . (47)
can be calculated by solving the following system of ODEs:
˙m(t) = A(t)m(t) + a(t) , m(0) = x
0
, (48)
˙
P (t) =
2A(t) +
m
i=1
B
2
i
(t)
P (t) + 2m(t)
a(t) +
m
i=1
B
i
(t)b
i
(t)
+
m
i=1
b
2
i
(t)
, P (0) = x
2
0
. (49)
Stochastic Systems, 2013 29
Stochastic Differential Equations (SDE)
The ODE for the expectation is derived by applying the expectation operator on both
sides of (42).
E[dX(t)] = E[(A(t)X(t) + a(t))dt +
m
i=1
(B
i
(t)X(t) + b
i
(t))dW
i
(t) ]
E[dX(t)]
dm(t)
= (A(t) E[X(t)]
=m(t)
+a(t))dt
+
m
i=1
E[(B
i
(t)X(t) + b
i
(t))] E[dW
i
(t) ]
=0
dm(t) = (A(t)m(t) + a(t))dt . (50)
Stochastic Systems, 2013 30
Stochastic Differential Equations (SDE)
In order to compute the second moment, we need to derive the SDE for Y (t) = X
2
(t):
dY (t) =
2X(t)(A(t)X(t) + a(t)) +
m
i=1
B
i
(t)X(t) + b
i
(t)
2
dt
+2X(t)
m
i=1
B
i
(t)X(t) + b
i
(t)
dW
i
(t) (51)
dY (t) =
2A(t)X
2
(t) + 2X(t)a(t) +
m
i=1
B
2
i
(t)X
2
(t) + 2B
i
(t)b
i
(t)X(t)
+b
2
i
(t)
dt + 2X(t)
m
i=1
B
i
(t)X(t) + b
i
(t)
dW
i
(t) (52)
Stochastic Systems, 2013 31
Stochastic Differential Equations (SDE)
Furthermore, we apply the expectation operator to (52) and use P (t) = E[X
2
(t)] =
E[Y (t)] and m(t) = E[X(t)].
E[dY (t)] =
2A(t)E[X
2
(t)] + 2a(t)E[X(t)] +
m
i=1
B
2
i
(t)E[X
2
(t)]
+2B
i
(t)b
i
(t)E[X(t)] + b
2
i
(t)
dt
+E
2X(t)
m
i=1
B
i
(t)X(t) + b
i
(t)
dW
i
(t)
dP (t) =
2A(t)P (t) + 2a(t)m(t)
+
m
i=1
B
2
i
(t)P (t) + 2B
i
(t)b
i
(t)m(t) + b
2
i
(t)
dt
Stochastic Systems, 2013 32
Stochastic Differential Equations (SDE)
In the case that B
i
(t) = 0, i = 1, . . . , m, we are able to directly compute the
distribution. The scalar linear SDE
dX(t) = (A(t)X(t) + a(t))dt +
m
i=1
b
i
(t)dW
i
(t), (53)
with X(0) = x
0
is normaly distributed
P (X(t)|x
0
) ∼ N(m(t), V (t)) with expected value m(t) and variance V (t), which
are solutions of the following ODEs,
˙m(t) = A(t)m(t) + a(t) , m(0) = x
0
, (54)
˙
V (t) = 2A(t)V (t) +
m
i=1
b
2
i
(t) , V (0) = 0 . (55)
Stochastic Systems, 2013 33
Stochastic Differential Equations (SDE)
There are some specific scalar linear SDEs which are found to be quite useful in practice.
The simplest case of SDE is where the drift and the diffusion coefficients are independent
of the information received over time
dS(t) = µdt + σdW (t) , S(0) = S
0
. (56)
This model has been used to simulate commodity prices, such as metals or agricultural
products.
The mean is E[S(t)] = µt + S
0
and the variance Var[S(t)] = σ
2
t. S(t) possesses
a behavior of fluctuations around the straight line S
0
+ µt.The process is normally
distributed with the given mean and variance.
Stochastic Systems, 2013 34
Stochastic Differential Equations (SDE)
The standard model of stock prices is the geometric Brownian motion as given by
dS(t) = µS(t)dt + σS(t)dW (t, ω) , S(0) = S
0
.
The mean is given by E[S(t)] = S
0
e
µt
and its variance by Var[S(t)] = S
2
0
e
2µt
(e
σ
2
t
−
1). This model forms the starting point for the famous Black-Scholes formula for option
pricing. The geometric Brownian motion has two main features which make it popular
for stock
The first property is that S(t) > 0 for all t ∈ [0, T ] and the second is that all returns
are in scale with the current price. This process has a log-normal probability density
function.
Stochastic Systems, 2013 35
Stochastic Differential Equations (SDE)
Another very popular class of SDEs are mean reverting linear SDEs. The model is
obtained by
dS(t) = κ[µ − S(t)]dt + σ dW (t, ω) , S(0) = S
0
. (57)
A special case of this SDE where µ = 0 is called Ohrnstein-Uhlenbeck process.
Equation (57) models a process which naturally falls back to its equilibrium level of µ.
The expected price is E[S(t)] = µ − (µ − S
0
)e
−κ t
and the variance is
Var[S(t)] =
σ
2
2κ
1 − e
−2κ t
.
Stochastic Systems, 2013 36
Stochastic Differential Equations (SDE)
In the long run, the following (unconditional) approximations are valid
lim
t→∞
E[S(t)] = µ
and
lim
t→∞
Var[S(t)] =
σ
2
2κ
.
This analysis shows that the process fluctuates around µ and has a variance of
σ
2
2κ
which depends on the parameter κ: the higher κ, the lower the variance.
This is obvious since the higher κ, the faster the process reverts back to its mean
value.
This process is a stationary process which is normally distributed.
Stochastic Systems, 2013 37
Stochastic Differential Equations (SDE)
A popular extension is where the diffusion term is in scale with the current value, i.e.,
the geometric mean reverting process:
dS(t) = κ[µ − S(t)]dt + σS(t)dW (t, ω) , S(0) = S
0
.
In this model S(t) ≥ 0, if S
0
≥ 0, µ > 0, and κ > 0.
The first mean reversion model(57) may produce negative values even for µ > 0.
Since the second mean-reversion model has always positive realizations, it is also
called log-normal mean reversion. This type of model is used to model interest rate or
volatilities.
Stochastic Systems, 2013 38
Stochastic Differential Equations (SDE)
In control engineering science, the most important (scalar) case is
dX(t) = (A(t)X(t) + C(t)u(t)) dt +
m
i=1
b
i
(t) dW
i
. (58)
In this equation, X(t) is normally distributed because the Brownian motion is just
multiplied by time-dependent factors.
When we compute an optimal control law for this SDE, the deterministic optimal control
law (ignoring the Brownian motion) and the stochastic optimal control law are the same.
This feature is called certainty equivalence. For this reason, the stochastics are often
ignored in control engineering.
Stochastic Systems, 2013 39
Stochastic Differential Equations (SDE)
The logical extension of scalar SDEs is to allow X(t) ∈ R
n
to be a vector. The rest of
this section proceeds in a similar fashion as for scalar linear SDEs. A stochastic vector
differential equation
dX(t) = f(t, X(t))dt + g(t, X(t))dW (t)
with the initial condition X(0) = x
0
∈ R
n
for an n-dimensional stochastic process
X(t) is called a linear SDE if the functions f (t, X(t)) ∈ R
n
and g(t, X(t)) ∈ R
n×m
are affine functions of X(t) and thus
f(t, X(t)) = A(t)X(t) + a(t) ,
g(t, X(t)) = [B
1
(t)X(t) + b
1
(t), ··· , B
m
(t)X(t) + b
m
(t)] ,
where A(t) ∈ R
n×n
, a(t) ∈ R
n
, W (t) ∈ R
m
is an m-dimensional Brownian motion,
and B
i
(t) ∈ R
n×n
, b
i
(t) ∈ R
n
.
Stochastic Systems, 2013 40
Stochastic Differential Equations (SDE)
Alternatively, the vector-valued linear SDE can be written as
dX(t) = (A(t)X(t) + a(t))dt +
m
i=1
(B
i
(t)X(t) + b
i
(t))dW
i
(t) . (59)
A common extension of the above equation is the following form of a controlled
stochastic differential equation as given by
dX(t) = (A(t)X(t) + C(t)u(t) + a(t)) dt
+
m
i=1
(B
i
(t)X(t) + D
i
(t)u(t) + b
i
(t)) dW
i
, (60)
where u (t) ∈ R
k
, C(t) ∈ R
n×k
, D
i
(t) ∈ R
n×k
.
Stochastic Systems, 2013 41
Stochastic Differential Equations (SDE)
The linear SDE (59) has the following solution:
X(t) = Φ(t)
x
0
+
t
0
Φ
−1
(s)
a(s) −
m
i=1
B
i
(s)b
i
(s)
ds
+
m
i=1
t
0
Φ
−1
(s)b
i
(s)dW
i
(s)
, (61)
where the fundamental matrix Φ(t) ∈ R
n×n
is the solution of the homogenous
stochastic differential equation.
Stochastic Systems, 2013 42
Stochastic Differential Equations (SDE)
The fundamental matrix Φ(t) ∈ R
n×n
is the solution of the homogenous stochastic
differential equation:
dΦ(t) = A(t)Φ(t)dt +
m
i=1
B
i
(t)Φ(t)dW
i
(t) , (62)
with initial condition Φ(0) = I, I ∈ R
n
×
n
e now prove that (61) and (62) are
solutions of (59). We rewrite (61) as
X(t) = Φ(t)
x
0
+
t
0
Φ
−1
(t)dY (t)
dY (t) =
a(t) −
m
i=1
B
i
(t)b
i
(t)
dt +
m
i=1
b
i
(t)dW
i
(t) .
Stochastic Systems, 2013 43
Stochastic Differential Equations (SDE)
X(t) = Φ(t)Z(t) , Z(t) =
x
0
+
t
0
Φ
−1
(t)dY (t)
dZ(t) = Φ
−1
(t)dY (t)
We use the Itˆo formula to calculate X(t) = Φ(t )Z(t):
dX(t) = Φ(t)dZ(t) + dΦ(t)Z(t) +
m
i=1
B
i
(t)Φ(t)Φ(t)
−1
b
i
(t)dt
= dY (t) + A(t)Φ(t)Z(t)dt +
m
i=1
B
i
(t)Φ(t)Z(t)dW
i
(t) +
m
i=1
B
i
(t)b
i
(t)dt
Stochastic Systems, 2013 44
Stochastic Differential Equations (SDE)
Noting that Z(t) = Φ
−1
(t)X(t) and using the SDE for Y (t), we get
dX(t) = dY (t) + A(t)Φ(t)Z(t)dt +
m
i=1
B
i
(t)Φ(t)Z(t)dW
i
(t) +
m
i=1
B
i
(t)b
i
(t)dt
=
a(t) −
m
i=1
B
i
(t)b
i
(t)
dt +
m
i=1
b
i
(t)dW
i
(t) + A(t)X(t)dt
+
m
i=1
B
i
(t)X(t)dW
i
(t) +
m
i=1
B
i
(t)b
i
(t)dt
= [a(t ) + A(t)X(t)]dt +
m
i=1
(B
i
(t)X(t) + b
i
(t))dW
i
(t) .
This completes the proof.
Stochastic Systems, 2013 45
Stochastic Differential Equations (SDE)
The expectation m(t) = E[X(t)] ∈ R
n
and the second moment matrix P (t) =
E[X(t)X
T
(t)] ∈ R
n×n
can be computed as follows:
˙m(t) = A(t)m(t) + a(t) , m(0) = x
0
, (63)
˙
P (t) = A(t)P (t) + P (t)A
T
(t) + a(t)m
T
(t) + m(t)a
T
(t)
+
m
i=1
[B
i
(t)P (t)B
T
i
(t) + B
i
(t)m(t)b
T
i
(t)
+b
i
(t)m
T
(t)B
T
i
(t) + b
i
(t)b
i
(t)
T
] , P (0) = x
0
x
T
0
. (64)
The covariance matrix for the system of linear SDEs is given by als
V (t) = Var{x(t)} = P (t) − m(t)m
T
(t) . (65)
Stochastic Systems, 2013 46
Stochastic Differential Equations (SDE)
The special case
dX(t) = (A(t)X(t) + a(t))dt +
m
i=1
b
i
(t)dW
i
(t)
with the initial condition X(0) = x
0
∈ R
n
is normally distributed, i.e.,
P (X(t)|x
0
) ∼ N(m(t), V (t))
where
˙m(t) = A(t)m(t) + a(t) m(0) = x
0
˙
V (t) = A(t)V (t) + V (t)A
T
(t) +
m
i=1
b
i
b
T
i
(t) V (0) = 0 .
Stochastic Systems, 2013 47
Stochastic Differential Equations (SDE)
As first example of a linear vector valued SDE, we consider a two dimensional geometric
Brownian motion:
dS
1
(t) = µ
1
S
1
(t)dt + S
1
(t)
σ
11
dW
1
(t) + σ
12
dW
2
(t)
, (66)
dS
2
(t) = µ
2
S
2
(t)dt + S
2
(t)
σ
21
dW
1
(t) + σ
22
dW
2
(t)
. (67)
Written in matrix form S = (S
1
, S
2
)
T
, the same SDE is given as:
A(t) =
µ
1
0
0 µ
1
a(t) =
0
0
B
1
(t) =
σ
11
0
0 σ
21
B
2
(t) =
σ
12
0
0 σ
22
Both processes S
1
(t) and S
2
(t) are correlated if σ
12
= σ
21
̸= 0. This model can be
easily extended to n processes.
Stochastic Systems, 2013 48
Stochastic Differential Equations (SDE)
The observed volatility for real existing price processes, such as stocks or bonds is itself
a stochastic process. The following model describes this observation:
dP (t) = µdt + σ(t)dW
1
(t) , P (0) = P
0
,
dσ(t) = κ(θ − σ(t))dt + σ(t)σ
1
dW
2
(t) , σ(0) = σ
0
.
where θ is the average volatility, σ
1
a volatility, and κ the mean reversion rate of
the volatility process σ(t). If this model is used for stock prices, the transformation
P (t) = ln(S(t)) is useful. The two Brownian motions dW
1
(t) and dW
2
(t) are
correlated, hence corr[dW
1
(t), dW
2
(t)] = ρ. This model captures the behavior of
real existing prices better and its distribution of returns shows “fatter tails”.
Stochastic Systems, 2013 49
Stochastic Differential Equations (SDE)
Die system (68) can be rewritten as linear SDE:
A(t) =
0 0
0 −κ
a(t) =
µ
κθ
B
1
(t) =
0 1
0 σ
1
ρ
B
2
(t) =
0 0
0 σ
1
1 − ρ
2
,
wobei x(t) = (P (t), σ(t))
T
. The system (68) has the property, that the variance
of P (t) depends on the initial condition σ
0
For the parameters µ = 0.1, κ = 2,
θ = 0.2, σ
1
= 0.5 and ρ = 0.5, we calculate the standard deviation of P (t) with
σ
0
= 0.1 and alternatively with σ
0
= 0.8. The expected value of σ(t) has the
following evaluation over time m(t) = θ + (σ
0
− θ)e
−κt
and thus the variance of
P (t) depends on σ
0
.
Stochastic Systems, 2013 50
Stochastic Differential Equations (SDE)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time
Standardabweichung
σ
0
=0.1
σ
0
=0.8
Abbildung 1: Stand. dev. of P (t) for different initial conditions of σ(t)
Stochastic Systems, 2013 51
Stochastic Differential Equations (SDE)
In comparison with linear SDEs, nonlinear SDEs are less well understood. No general
solution theory exists. And there are no explicit formulae for calculating the moments.
In this section, we show some examples of nonlinear SDEs and their properties.
In general, a scalar square root process can be written as
dX(t) = f(t, X(t))dt + g(t, X(t))dW (t)
with
f(t, X(t)) = A(t)X(t) + a(t)
g(t, X(t)) = B(t)
X(t) ,
where A(t), a(t), and B(t) are real scalars. The nonlinear mean reverting SDEs differ
from the linear scalar equations by their nonlinear diffusion term. For this process, the
distribution and moments can be calculated.
Stochastic Systems, 2013 52
Stochastic Differential Equations (SDE)
For a specific square root process with A(t) = 0, a(t) = 1 and B(t) = 2 we are
able to derive the analytical solution: The SDE
dX(t) = 1dt + 2
X(t)dW (t) , X(0) = x
o
,
has the solution X(t) = (W (t) + x
0
)
2
We verify the solution using Itˆo formula. We
use Φ(t) = X(t) = (Y (t) + x
0
)
2
and dY (t) = dW (t). The partial derivatives are
Φ
t
= 0, Φ
Y
= 2(Y (t) + x
0
), and Φ
Y Y
= 2. Thus
dΦ(t) = [Φ
t
+ Φ
Y
· 0 +
1
2
Φ
Y Y
· 1]dt + Φ
Y
· 1dW (t) ,
dΦ(t) = 1dt + 2(Y (t) + x
0
)dW (t) , ⇒ dX(t) = 1dt + 2
X(t)dW (t) ,
since
X(t) = Y (t) + x
0
.
Stochastic Systems, 2013 53
Stochastic Differential Equations (SDE)
Another widely used mean reversion model is obtained by
dS(t) = κ[µ − S(t)]dt + σ
S(t)dW (t) , S(0) = S
0
. (68)
This model is also known as the Cox-Ross-Ingersol processes.The process shows a
less volatile behavior than its linear geometric counterpart and it has a non-central
chi-square distribution. The process is often used to model short-term interest rates or
stochastic volatility processes for stock prices. Another often used square root process
is similar to the geometric Brownian motion, but with a square root diffusion term
instead of the linear diffusion term. Its model is given by
dS(t) = µS(t)dt + σ
S(t)dW (t) , S(0) = S
0
. (69)
Stochastic Systems, 2013 54
Stochastic Differential Equations (SDE)
The expected value for (69) is E[S(t)] = S
0
e
µt
and the variance is obtained by
Var[S(t)] =
σ
2
S
0
µ
e
2µt
− e
µt
.
Another widely used mean reversion model is obtained by
dS(t) = κS(t)[µ − ln(S(t))]dt + S(t)σdW (t) . (70)
Using the transformation P (t) = ln(S(t)) yields the linear mean reverting and
normally distributed process P (t):
dP (t) = κ[(µ −
σ
2
2κ
) − P (t)]dt + σdW (t) , (71)
Because of the transformation, S(t) is log-normally distributed. This model is used
to model stock prices, stochastic volatilities, and electricity prices. Because S(t) is
log-normally distributed, S(t) is always positive.
Stochastic Systems, 2013 55
Stochastic Differential Equations (SDE)
In this part, we introduce three major methods to compute solution of SDEs.
• The first method is based on the Itˆo integral and has already been used for linear
solutions.
• We introduce numerical methods to compute path-wise solutions of SDEs.
• The third method is based on partial differential equations, where the problem of
finding the probability density function of the solution is transformed into solving a
partial differential equation.
Stochastic Systems, 2013 56
Stochastic Differential Equations (SDE)
The stochastic process X(t) governed by the stochastic differential equation
dX(t) = f(t, X(t))dt + g(t, X(t))dW (t)
X(0) = X
0
is explicitly described by the integral form
X(t, ω) = X
0
+
t
0
f(s, X(s)) ds +
t
0
g(s, X(s)) dW (s) ,
where the first integral is a path-wise Riemann integral and the second integral is an
Itˆo integral.
In this definition, it is assumed that the functions f(t, X(t)) and g(t, X(t)) are
sufficiently smooth in order to guarantee the existence of the solution X(t).
Stochastic Systems, 2013 57
Stochastic Differential Equations (SDE)
There are several ways of finding analytical solutions. One way is to guess a soluti-
on and use the Itˆo calculus to verify that it is a solution for the SDE under consideration.
We assume that the following nonlinear SDE
dX(t) = dt + 2
X(t) dW (t) ,
has the solution
X(t) = (W (t) +
X
0
)
2
.
In order to verify this claim, we use the Itˆo calculus. We have X(t) = ϕ(W ) where
ϕ(W ) = (W (t) +
√
X
0
)
2
, so that ϕ
′
(W ) = 2(W (t) +
√
X
0
) and ϕ
′′
(W ) = 2.
Stochastic Systems, 2013 58
Stochastic Differential Equations (SDE)
Using Itˆo’s rule, we get
dX(t) =
f(t, X)dt + g(t, X)dW (t)
f(t, X) = ϕ(W )
′
1 +
1
2
ϕ
′′
(W )(2
X(t))
2
= 1
g(T, X) = ϕ
′
(W )(2
X(t)) = 2(W (t) +
X
0
) .
Since X(t) = (W (t) +
√
X
0
)
2
we know that (W (t) +
√
X
0
) =
X(t) and thus
the Itˆo calculation generated the original SDE where we started at.
Stochastic Systems, 2013 59
Stochastic Differential Equations (SDE)
For some classes of SDEs, analytical formulas exist to find the solution, e.g. consider
the following SDE:
dX(t) = f(t, X(t))dt + σ(t)dW (t) , X(0) = x
0
(72)
where X(t) ∈ R
n
, f(t, X(t)) ∈ R
n
is an arbitrary function, σ(t) ∈ R
n×m
and
dW (t) ∈ R
m
. This class of SDEs has the following general solution:
X(t) = Y (t) + F (t) (73)
dY (t) = f (t, Y (t) + F (t))dt , Y (0) = x
0
(74)
dF (t) = σ(t)dW (t) , F (0) = 0 . (75)
The SDE for F (t) can be integrated, i.e. F (t) =
t
0
σ(s)dW (s). When σ(t) = σ
than F (t) = σW (t).
Stochastic Systems, 2013 60
Stochastic Differential Equations (SDE)
SinceF (t) is know,, we are able to solve for Y (t) in in function of F (t).
Using Itˆo lemman, we show that X(t) = Y (t) + F (t) and this solves the SDE
dX(t) = dY (t) + dF (t) = f (t, Y (t) + F (t))dt + σ(t)dW (t)
= f (t, X(t))dt + σ(t)dW (t) (76)
This solution is not very suprising, since X(t) is the sum of the process of Y (t) and
the BM of F (t).
Stochastic Systems, 2013 61
Stochastic Differential Equations (SDE)
For another class of SDEs, exist an analytical formula for their solution:
dX(t) = f(t, X(t))dt + c(t)X(t)dW (t) , X(0) = x
0
, (77)
where f (t, X(t)) ∈ R, c(t) ∈ R and dW ∈ R. DThe solution can be derived as
follows:
X(t) = F
−1
(t)Y (t) (78)
dF (t) = F (t)c
2
(t)dt − F (t)c(t)dW (t) , F (0) = 1 (79)
dY (t) = F (t)f(t, F
−1
Y (t))dt (80)
The proof is similar to the first case, sice the diffusion is linear.
Stochastic Systems, 2013 62
Stochastic Differential Equations (SDE)
Calculate the analytical solution for
dX(t) =
dt
X(t)
+ αX(t)dW (t) , X(0) = x
0
.
F (t) = e
1
2
α
2
t−αW (t)
, dY (t) =
F (t)
F
−1
(t)Y
dt =
F
2
(t)
Y
dt
dY (t)Y (t) = F
2
(t)dt ,
1
2
Y
2
(t) =
t
0
F
2
(s)ds + C
0
Y (t) =
x
2
0
+ 2
t
0
e
α
2
s−2αW (s)
ds
1
2
X(t) = e
−
1
2
α
2
t+αW (t)
x
2
0
+ 2
t
0
e
α
2
s−2αW (s)
ds
1
2
Stochastic Systems, 2013 63
Stochastic Differential Equations (SDE)
However, most SDEs, especially nonlinear SDEs, do not have analytical solutions so
that one has to resort to numerical approximation schemes in order to simulate sample
paths of solutions to the given equation.
The simplest scheme is obtained by using a first-order approximation. This is called the
Euler scheme
X(t
k
) = X(t
k−1
) + f(t
k−1
, X(t
k−1
))∆t + g(t
k−1
, X(t
k−1
))∆W (t
k
) .
The Brownian motion term can be approximated as follows:
∆W (t
k
) = ϵ(t
k
)
√
∆t ,
where the ϵ(.) is a discrete-time Gaussian white process with mean 0 and standard
deviation 1.
Stochastic Systems, 2013 64
