Download Tutorial on Stochastic Differential Equations

Please cite as J. R. Movellan (2011) Tutorial on Stochastic Differential Equations,
MPLab Tutorials Version 06.1.
Tutorial on Stochastic Differential
Equations
Javier R. Movellan
c 2003, 2004, 2005, 2006, Javier R. Movellan
Copyright This document is being reorganized. Expect redundancy, inconsistencies, disorganized presentation ...
1
Motivation
There is a wide range of interesting processes in robotics, control, economics, that
can be described as a differential equations with non-deterministic dynamics. Suppose the original processes is described by the following differential equation
dXt
= a(Xt )
(1)
dt
with initial condition X0 , which could be random. We wish to construct a mathematical model of how the may behave in the presence of noise. We wish for this
noise source to be stationary and independent of the current state of the system.
We also want for the resulting paths to be continuous.
As it turns out building such a model is tricky. An elegant mathematical solution
to this problem may be found by considering a discrete time versions of the process
and then taking limits in some meaningful way. Let π = {0 = t0 ≤ t1 · · · ≤ tn = t}
be a partition of the interval [0, t]. Let ∆π tk = tk+1 − tk . For each partition π we
can construct a continuous time process X π defined as follows
Xtπ0 = X0
Xtπk+1
=
(2)
Xtπk
+
a(Xtπk )∆π tk
+
c(Xtπk )(Ntk+1
− Ntk )
(3)
where N is a noise process whose properties remain to be determined and b is a
function that allows us to have the amount of noise be a function of time and of the
state. To make the process be continuous in time, we make it piecewise constant
between the intervals defined by the partition, i.e.
Xtπ = Xtπk for t ∈ [tk , tk+1 )
(4)
We want for the noise Nt to be continuous and for the increments Ntk+1 − Ntk to
have zero mean, and to be independently and identically distributed. It turns out
that the only noise source that satisfies these requirements is Brownian motion.
Thus we get
Xtπ = X0 +
n−1
X
a(Xtπk )∆tk +
k=0
n−1
X
c(Xtπk )∆Bk
(5)
k=0
where ∆tk = tk+1 − tk , and ∆Bk = Btk+1 − Btk where B is Brownian Motion. Let
kπk = max{∆tk } be the norm of the partition π. It can be shown that as π → 0
the X π processes converge in probability to a stochastic process X. It follows that
Z t
n−1
X
lim
a(Xtπk )∆k =
a(Xs )ds
(6)
kπk→0
0
k=0
and that
n−1
X
c(Xtπk )∆Bk
(7)
k=0
converges to a process It
It = lim
kπk→0
n−1
X
k=0
c(Xtπk )∆Bk
(8)
Note It looks like an integral where the integrand is a random variable c(Xs ) and
the integrator ∆Bk is also a random variable. As we will see later, It turns out to
be an Ito Stochastic Integral. We can now express the limit process X as a process
satisfying the following equation
Z
t
Xt = X0 +
a(Xs )ds + It
(9)
0
Sketch of Proof of Convergence: Construct a sequence of partitions π1 , π2 , · · · each
one being a refinement of the previous one. Show that the corresponding Xtπ−i
form a Cauchy sequence in L2 and therefore converge to a limit. Call that process
X.
In order to get a better understanding of the limit process X there are two things
we need to do: (1) To study the properties of Brownian motion and (2) to study
the properties of the Ito stochastic integral.
2
Standard Brownian motion
By Brownian motion we refer to a mathematical model of the random movement
of particles suspended in a fluid. This type of motion was named after Robert
Brown that observed it in pollens of grains in water. The processes was described
mathematically by Norbert Wiener, and is is thus also called a Wiener Processes.
Mathematically a standard Brownian motion (or Wiener Process) is defined by the
following properties:
1. The process starts at zero with probability 1, i.e., P (B0 = 0) = 1
2. The probability that a randomly generated Brownian path be continuous
is 1.
3. The path increments are independent Gaussian, zero mean, with variance
equal to the temporal extension of the increment. Specifically for 0 ≤ s1 ≤
t1 ≤ s2 , ≤ t2
Bt1 − Bs1 ∼ N (0, s1 − t1 )
Bt2 − Bs2 ∼ N (0, s2 − t2 )
(10)
(11)
and Bt2 − Bs2 is independent of Bt1 − Bs1 .
Wiener showed that such a process exists, i.e., there is a stochastic process that does
not violate the axioms of probability theory and that satisfies the 3 aforementioned
properties.
2.1
2.1.1
Properties of Brownian Motion
Statistics
From the properties of Gaussian random variables,
E(Bt − Bs ) = 0
Var(Bt − Bs ) = E[(Bt − Bs ) ] = t − s
2
E((Bt − Bs )4 ] = 3(t − s)
Var[(Bt − Bs )2 ] = E[(Bt − Bs )4 ] − E[(Bt − Bs )2 ]2 = 2(t − s)2
Cov(Bs , Bt ) = s, for t > s
r
s
, for t > s.
Corr(Bs , Bt ) =
t
(12)
(13)
(14)
(15)
(16)
(17)
Proof: For the variance of (Bt − Bs )2 we used the that for a standard random
variable Z
E(Z 4 ) = 3
(18)
Note
Var(BT ) = Var(BT − B0 ) = T
(19)
since P (B0 = 0) and for all ∆t ≥ 0
Var(Bt+∆t − Bt ) = ∆t
(20)
Moreover,
Cov(Bs , Bt ) = Cov(Bs , Bs + (Bt − Bs )) = Cov(Bs , Bs ) + Cov(Bs , (Bt − Bs ))
= Var(Bs ) = s
(21)
since Bs and Bt − Bs are uncorrelated.
2.1.2
Distributional Properties
Let B represent a standard Brownian motion (SBM) process.
• Self-similarity:
For any c 6= 0, Xt =
√1 Bct
c
is SBM.
We can use this property to simulate SBM in any given interval [0,T] if we
know how to simulate in the interval [0, 1]:
√
If B is SBM in [0,1], c = T1 then Xt = T B T1 t is SBM in [0,T].
• Time Inversion: Xt = tB 1t is SBM
• Time Reversal: Xt = BT − BT −t is SBM in the interval [0, T ]
• Symmetry: Xt = −Bt is SBM
2.1.3
Pathwise Properties
• Brownian motion sample paths are non-differentiable with probability 1
This is the basic why we need to develop a generalization of ordinary calculus to
handle stochastic differential equations. If we were to define such equations simply
as
dBt
dXt
= a(Xt ) + c(Xt )
(22)
dt
dt
we would have the obvious problem that the derivative of Brownian motion does
not exist.
Proof: Let X be a real valued stochastic process. For a fixed t let π = {0 = t0 ≤
t1 , · · · ≤ tn = t} be a partition of the interval [0, t]. Let kπk be the norm of the
partition. The quadratic variation of X at t is a random variable represented as
< X, X >2t and defined as follows
n
X
< X, X >2t = lim
kπk→0
|Xtk+1 − Xtk |2
(23)
k=1
We will show that the quadratic variation of SBM is larger than zero with probability
one, and therefore the quadratic paths are not differentiable with probability 1.
Let B be a Standard Brownian Motion. For a partition π = {0 = t0 ≤ t1 , · · · ≤
tn = t} let Bkπ be defined as follows
Bkπ = Btk
Let
π
S =
n
X
(24)
2
(∆Bkπ )
(25)
tk+1 − tk = t
(26)
k=1
Note
E(S π ) =
n−1
X
k=0
and
0 ≤ Var(S π ) =
n−1
X
Var (∆Bkπ )2
k=0
=2
n−1
X
(tk+1 − tk )2
k=0
≤ 2kπk
n−1
X
(tk+1 − tk ) = 2kπkt
(27)
k=0
Thus

lim Var(S π ) = lim
kπk→0
kπk→0
n−1
X
E
!2 
(∆Bkπ )2 − t
=0
(28)
k=0
This shows mean square convergence, which implies convergence in probability, of
S π to t. (I think) almost sure convergence can also be shown.
Comments:
Rt
• If we were to define the stochastic integral 0 (dBs )2 as
Z t
(dBs )2 = lim S π
kπk→0
0
(29)
Then
Z
t
2
Z
(dBs ) =
0
t
ds = t
0
(30)
• If a path Xt (ω) were differentiable almost everywhere in the interval [0, T ]
then
< X, X >2t (ω)) ≤ lim
n−1
X
∆t →0
= ( max Xt0 (ω)2 ) lim
(∆t Xt0k (ω))2
X
n→∞
t∈[0,T ]
∆2t
= ( max Xt0 (ω)2 ) lim (n)(T /n)2 = 0
n→∞
t∈[0,T ]
(31)
k=0
(32)
(33)
where X 0 = dX/dt. Since Brownian paths have non-zero quadratic variation with probability one, they are also non-differentiable with probability
one.
2.2
Simulating Brownian Motion
Let π = {0 = t0 ≤ t1 · · · ≤ tn = t} be a partition of the interval [0, t]. Let
{Z1 , · · · , Zn }; be i.i.d Gaussian random variables E(Zi ) = 0; Var(Zi ) = 1. Let
the stochastic process B π as follows,
Btπ0 = 0
Btπ1
=
Btπ0
=
Btπk−1
+
√
(34)
t1 − t0 Z1
..
.
Btπk
(35)
(36)
p
+ tk − tk−1 Zk
(37)
Btπ = Btπk−1 for t ∈ [tk−1 , tk )
(38)
Moreover,
For each partition π this defines a continuous time process. It can be shown that as
kπk → 0 the process B π converges in distribution to Standard Brownian Motion.
2.2.1
Exercise
Simulate Brownian motion and verify numerically the following properties
E(Bt ) = 0
Var(Bt ) = t
Z t
Z t
dBs2 =
ds = t
0
3
(39)
(40)
(41)
0
The Ito Stochastic Integral
We want to give meaning to the expression
Z t
Ys dBs
(42)
0
where B is standard Brownian Motion and Y is a process that does not anticipate
the future of Brownian motion. For example, Yt = Bt+2 would not be a valid
integrand. A random process Y is simply a set of functions f (t, ·) from an outcome
space Ω to the real numbers, i.e for each ω ∈ Ω
Yt (ω) = f (t, ω)
(43)
We will first study the case in which f is piece-wise constant. In such case there is
a partition π = {0 = t0 ≤ t1 · · · ≤ tn = t} of the interval [0, .t] such that
fn (t, ω) =
n−1
X
Ck (ω)ξk (t)
(44)
k=0
where
ξk (t) =
1 if t ∈ [tk , tk+1 )
0 else
(45)
where Ck is a non-anticipatory random variable, i.e., a function of X0 and the
Brownian noise up to time tk . For such a piece-wise constant process Yt (ω) =
fn (t, ω) we define the stochastic integral as follows. For each outcome ω ∈ Ω
Z
t
Ys (ω)dBs (ω) =
0
n−1
X
Ck (ω) Btk+1 (ω) − Btk (ω)
(46)
k=0
More succinctly
Z
t
Ys dBs =
0
n−1
X
Ck Btk+1 − Btk
(47)
k=0
This leads us to the more general definition of the Ito integral
Definition of the Ito Integral Let f (t, ·) be a non-anticipatory function from an
outcome space Ω to the real numbers. Let {f1 , f2 , · · · } be a sequence of elementary
non-anticipatory functions such that
Z t
2
lim E[
f (s, ω) − fn (s, ω) ds] = 0
(48)
n→∞
0
Let the random process Y be defined as follows: Yt (ω) = f (t, ω) Then the Ito
integral
Z t
Ys dBs
(49)
0
is a random variable defined as follows. For each outcome ω ∈ Ω
Z t
Z t
f (s, ω)dBs (ω) = lim
fn (t, ω)dBs (ω)
0
n→∞
(50)
0
where the limit is in L2 (P ). It can be shown that an approximating sequence
f1 , f2 · · · satisfying (48) exists. Moreover the limit in (50) also exists and is independent of the choice of the approximating sequence.
Comment Strictly speaking we need for f to be measurable, i.e., induce a proper
random variable. We also need for f (t, ·) to be Ft adapted. This basically means
that Yt must be a function of Y0 and the Brownian motion
R t up to time t It cannot
be a function of future values of B. Moreover we need E[ 0 f (t, ·)2 dt] ≤ ∞.
3.1
Properties of the Ito Integral
•
E(It ) = 0
(51)
•
Var(It ) = E
(It2 )
Z
=
0
t
E(Xs2 )ds
(52)
•
Z
t
Z
t
Z
(53)
0
0
0
t
Ys dBs
Xs dBs +
(Xs + Ys ) dBs =
•
Z
T
t
Z
Xs dBs =
0
T
Z
Xs dBs for t ∈ (0, T )
Xs dBs +
0
(54)
t
• The Ito integral is a Martingale process
E(It | Fs ) = Is lfor all t > s
(55)
where E(It | Fs ) is the least squares prediction of It based on all the information available up to time s.
4
Stochastic Differential Equations
In the introduction we defined a limit process X which was the limit process of
a dynamical system expressed as a differential equation plus Brownian noise perturbation in the system dynamics. The process was a solution to the following
equation
Z t
Xt = X0 +
a(Xs )ds + It
(56)
0
where
It = lim c(Xtπk )∆Bk
kπk→0
It should now be clear that It is in fact an Ito Stochastic Integral
Z t
It =
c(Xs )dBs
(57)
(58)
0
and thus X can be expressed as the solution of the following stochastic integral
equation
Z t
Z t
Xt = X0 +
a(Xs )ds +
c(Xs )dBs
(59)
0
0
It is convenient to express the integral equation above using differential notation
dXt = a(Xt )dt + c(Xt )dBt
(60)
with given initial condition X0 . We call this an Ito Stochastic Differential Equation
(SDE). The differential notation is simply a pointer, and thus acquires its meaning
from, the corresponding integral equation.
4.1
Second order differentials
The following rules are useful
Z t
Xt (dt)2 = 0
0
Z t
Xt dBt dt = 0
0
Z t
Xt dBt dWt = 0 if B, W are independent Brownian Motions
0
Z t
Z t
Xt (dBt )2 =
Xt dt
0
(61)
(62)
(63)
(64)
0
(65)
Symbolically this is commonly expressed as follows
dt2 = 0
dBt dt = 0
dBt dWt = 0
(66)
(67)
(68)
(dBt )2 = dt
(69)
Sketch of proof:
Let π = {0 = t0 ≤ t1 · · · ≤ tn = t} a partition of the [0, t] with equal intervals, i.e.
tk+1 − tk = ∆t.
• Regarding dt2 = 0 note
lim
∆t →0
n−1
X
Xtk ∆t2 = lim ∆t
∆t →0
k=0
t
Z
Xs ds = 0
(70)
0
• Regarding dBt dt = 0 note
lim
∆t →0
n−1
X
Z
Xtk ∆t∆Bk = lim ∆t
∆t →0
k=0
t
Xs dBs = 0
(71)
0
• Regarding dBt2 = dt note
n−1
h n−1
2 i
h n−1
2
X
X
X
Xtk ∆Bk2 −
E
Xtk ∆t
Xtk (∆Bk2 − ∆t) ]
=E
k=0
=
k=0
n−1
X n−1
X
E[Xt
k
k=0
Xt0k (∆Bk2 − ∆t)(∆Bk20 − ∆t)]
(72)
k=0 k0 =0
If k > k 0 then (∆Bk2 − ∆t) is independent of Xtk Xtk0 (∆Bk20 − ∆t), and
therefore
E[Xt
k
Xtk0 (∆Bk2 − ∆t)(∆Bk20 − ∆t)]
= E[Xtk Xtk0 (∆Bk20 − ∆t)]E[∆Bk2 − ∆t)] = 0
0
Equivalently, if k > k then
∆t) and therefore
E[Xt
k
(∆Bk20
(73)
− ∆t) is independent of Xtk Xtk0 (∆Bk2 −
Xtk0 (∆Bk2 − ∆t)(∆Bk20 − ∆t)]
= E[Xtk Xtk0 (∆Bk2 − ∆t)]E[∆Bk20 − ∆t)] = 0
(74)
Thus
n−1
X n−1
X
E[Xt
Xtk0 (∆Bk2 − ∆t)(∆Bk20 − ∆t)] =
k
k=0 k0 =0
n−1
X
E[Xt2 (∆Bk2 − ∆t)2 ]
k
k=0
(75)
Note since ∆Bk is independent of Xtk then
E[Xt2 (∆Bk2 − ∆t)2 = E[Xt2 ]E[(∆Bk2 − ∆t)2 ]
= E[Xt2 ]Var(∆Bk2 ) = 2E[Xt2 ]∆t2
k
(76)
k
k
(77)
k
Thus
E
h n−1
X
Xtk ∆Bk2 −
k=0
n−1
X
Xtk ∆t
2 i
n−1
X
=
E[Xt2 ]∆2t
k
k=0
(78)
k=0
which goes to zero as ∆t → 0. Thus, in the limit as ∆t → 0
lim
n−1
X
∆t→0
Xtk ∆Bk2 = lim
n−1
X
∆t→0
k=0
Xtk ∆t
(79)
k=0
where the limit is taken in the mean square sense. Thus
Z t
Z t
Xs dBs2 =
Xs ds
0
(80)
0
• Regarding dBt dWt = 0 note
E
h n−1
X
Xtk ∆Bk ∆Wk
2 i
=
n−1
X n−1
X
E[Xt
k
Xtk0 ∆Bk ∆Wk ∆Bk0 ∆Wk0 ]
k=0 k0 =0
k=0
(81)
0
If k > k then ∆Bk , ∆Wk are independent of Xtk Xtk0 ∆Bk0 ∆Wk0 and therefore
E[Xt
k
Xtk0 ∆Bk ∆Wk ∆Bk0 ∆Wk0 ] = E[Xtk Xtk0 ∆Bk0 ∆Wk0 ]E[∆Bk ]E[∆Wk ] = 0
(82)
Equivalently, if k 0 > k then ∆Bk0 , ∆Wk0
Xtk Xtk0 ∆Bk ∆Wk and therefore
E[Xt
k
are independent of
Xtk0 ∆Bk ∆Wk ∆Bk0 ∆Wk0 ] = E[Xtk Xtk0 ∆Bk ∆Wk ]E[∆Bk0 ]E[∆Wk0 ] = 0
(83)
Finally, for k = k 0 , ∆Bk , ∆Wk and Xtk are independent, thus
E[Xt2 ∆Bk2 ∆Wk2 ] = E[Xt2 ]E[∆Bk2 ]E[∆Wk2 ] = E[Xt2 ]∆t2
k
k
k
(84)
Thus
E
h n−1
X
Xtk ∆Bk ∆Wk
2 i
k=0
=
n−1
X
E[Xt2 ]∆t2
k
(85)
k=0
which converges to 0 as ∆t → 0. Thus
Z
t
Xs dBs dWs = 0
0
(86)
4.2
Vector Stochastic Differential Equations
The form
dXt = a(Xt )dt + c(Xt )dBt
(87)
is also used to represent multivariate equations. In this case Xt represents an ndimensional random vector, Bt an m-dimensional vector of m independent standard
Brownian motions, and c(Xt is an n × m matrix. a is commonly known as the drift
vector and b the dispersion matrix.
5
Ito’s Rule
The main thing with Ito’s calculus is that for the general case a differential carries
quadratic and linear components. For example suppose that Xt is an Ito process.
Let
Yt = f (t, Xt )
(88)
dYt = ∇f (t, Xt )T dXt + 12 dXtT ∇2 f (t, Xt )dXt
(89)
then
where ∇, ∇2 are the gradient and Hessian with respect to (t, x). Note basically this
is the second order Taylor series expansion. In ordinary calculus the second order
terms are zero, but in Stochastic calculus, due to the fact that these processes have
non-zero quadratic variation, the quadratic terms do not go away. This is really all
you need to remember about Stochastic calculus, everything else derives from this
basic fact.
The most important consequence of this fact is Ito’s rule. Let Xt be governed by
an SDE
dXt = a(Xt , t)dt + c(Xt , t)dBt
(90)
Let Yt = f (Xt , t). Ito’s rule tells us that Yt is governed by the following SDE
def
dYt = ∇t f (t, Xt )dt + ∇x f (t, x)T dXt + 21 dXtT ∇2x f (t, Xt )dXt
(91)
where
def
dBi,t dBj,t = δ(i, j) dt
def
dXdt = 0
2 def
dt = 0
(92)
(93)
(94)
Equivalently
dYt = ∇t f (Xt , t)dt + ∇x f (Xt , t)T a(Xt , t)dt + ∇x f (Xt , t)T c(Xt , t)dBt
+ 12 trace c(Xt , t)c(Xt , t)T ∇2x f (Xt , t) dt
(95)
where
∇x f (x, t)T a(x, t) =
X ∂f (x, t)
ai (x, t)
∂xi
XX
∂ 2 f (x, t)
trace c(x, t)c(x, t)T ∇2x f (x, t) =
(c(x, t)c(x, t)T )ij
∂xi ∂xj
i
j
(96)
i
(97)
Note b is a matrix. Sketch of Proof: To second order
∆Yt = f (Xt+∆t , t + ∆t) − f (Xt , t) = ∇t f (Xt , t)∆t + ∇x f (Xt , t)T ∆Xt
1
1
+ ∆t2 ∇2t f (Xt , t) + ∆XtT ∇2x f (Xt , t)∆Xt + ∆t(∇x ∇t f (Xt , t))T ∆Xt ∆t
2
2
(98)
where ∇t , ∇x are the gradients with respect to time and state, and ∇2t is the second
derivative with respect to time, ∇2x the Hessian with respect to time and ∇x ∇t the
gradient with respect to state of the gradient with respect to time. Integrating over
time
n−1
X
Yt = Y0 +
∆Ytk
(99)
k=00
and taking limits
Z t
∇t f (Xs , s)ds +
∇x f (Xs , s)T dXs
0
0
0
Z
Z
1 t
1 t 2
∇t f (Xs , s)(ds)2 +
dXsT ∇2x f (Xs , s)dXs
+
2 0
2 0
Z t
+
(∇x ∇t f (Xs , s))T dXs ds
Z
Yt =Y0 +
t
Z
t
dYs = Y0 +
(100)
0
In differential form
dYt =∇t f (Xt , t)dt + ∇x f (Xt , t)T dXt
1
1
+ ∇2t f (Xt , t)(dt)2 + dXtT ∇2x f (Xt , t)dXt
2
2
+ (∇x ∇t f (Xt , t))T dXt dt
(101)
Expanding dXt
(∇x ∇t f (Xt , t))T dXt dt = (∇x ∇t f (Xt , t))T a(Xt , t)(dt)2
+ (∇x ∇t f (Xt , t))T c(Xt , t)dBt dt = 0
(102)
where we used the standard rules for second order differentials
(dt)2 = 0
(dBt )dt = 0
(103)
(104)
(105)
Moreover
dXtT ∇2x f (Xt , t)dXt
= (a(Xt , t)dt + c(Xt , t)dBt )T ∇2x f (Xt , t)(a(Xt , t)dt + c(Xt , t)dBt )
= a(Xt , t)T ∇2x f (Xt , t)a(Xt , t)(dt)2
+ 2a(Xt , t)T ∇2x f (Xt , t)c(Xt , t)(dBt )dt
+ dBtT c(Xt , t)T ∇2x f (Xt , t)c(Xt , t)(dBt )
(106)
Using the rules for second order differentials
(dt)2 = 0
(dBt )dt = 0
(107)
(108)
dBtT K(Xt , t)dBt =
XX
i
Ki,j (Xt , t)dBi,t dBj,t =
j
X
Ki,i dt
(109)
i
where
K(Xt , t) = c(Xt , t)T ∇2x f (Xt , t)c(Xt , t)
(110)
Thus
dYt =∇t f (Xt , t)dt + ∇x f (Xt , t)T a(Xt , t)dt + ∇x f (Xt , t)T c(Xt , t)dBt
1
+ trace c(Xt , t)c(Xt , t)T ∇2x f (Xt , t) dt
(111)
2
where we used the fact that
X
Kii (Xt , t)dt = trace(K)dt
i
= trace c(Xt , t)T ∇2x f (Xt , t)c(Xt , t)
= trace c(Xt , t)c(Xt , t)T ∇2x f (Xt , t)
5.1
(112)
Product Rule
Let X, Y be Ito processes then
d(Xt Yt ) = Xt dYt + Yt dXt + dXt dYt
(113)
Proof: Consider X, Y as a joint Ito process and take f (x, y, t) = xy. Then
∂f
=0
∂t
∂f
=y
∂x
∂f
=x
∂y
∂2f
=1
∂x∂y
∂2f
∂2f
=
=0
∂x2
∂y 2
(114)
(115)
(116)
(117)
(118)
Applying Ito’s rule, the Product Rule follows.
RT
Exercise: Solve 0 Bt dBt symbolically.
Let a(Xt , t) = 0, c(Xt , t) = 1, f (x, t) = x2 . Thus
dXt = dBt
Xt = Bt
(119)
(120)
and
∂f (t, x)
=0
∂t
∂f (t, x)
= 2x
∂x
2
∂ f (t, x)
=2
∂x2
(121)
(122)
(123)
Applying Ito’s rule
df (Xt , t) =
∂f (Xt , t)
∂f (Xt , t)
∂f (Xt , t)
dt +
a(Xt , t)dt +
c(Xt , t)dBt
∂t
∂x
∂x
1
∂ 2 f (x, t) + trace c(Xt , t)c(Xt , t)T
2
∂x2
(124)
we get
dBt2 = 2Bt dBt + dt
(125)
Equivalently
Z t
Z t
dBs2 = 2
Bs dBs +
ds
0
0
0
Z t
Bt2 = 2
Bs dBs + t
Z
t
(126)
(127)
0
Therefore
Z
t
Bs dBs =
0
1 2 1
B − t
2 t
2
(128)
NOTE: dBt2 is different from (dBt )2 .
Exercise:
Get
E[eβ Bt ]
Let Let a(Xt , t) = 0, c(Xt , t) = 1, i.e., dXt = dBt . Let Yt = f (Xt , t) = eβBt , and
dXt = dBt . Using Ito’s rule
1
dYt = βeβBt dBt + β 2 eβBt dt
2
Z t
Z
β 2 t βBs
βBs
Yt = Y0 + β
e
dBs +
e
ds
2 0
0
(129)
(130)
Taking expected values
Z
β2 t
(131)
E[Yt ] = E[Y0 ] + 2 E[Ys ]ds
0
Rt
where we used the fact that E[ 0 eβBs dBs ] = 0 because for any non anticipatory
Rt
random variable Yt , we know that E[ 0 Ys dBs ] = 0. Thus
and since
E[Y0 ] = 1
dE[Yt ]
β2
=
E[Yt ]
dt
2
E[eβB ] = e
t
β2
2
t
(132)
(133)
Exercise:
Solve the following SDE
dXt = αXt dt + βXt dBt
(134)
In this case a(Xt , t) = αXt , c(Xt , t) = βXt . Using Ito’s formula for f (x, t) = log(x)
∂f (t, x)
=0
∂t
∂f (t, x)
1
=
∂x
x
1
∂ 2 f (t, x)
=− 2
2
∂x
x
(135)
(136)
(137)
Thus
d log(Xt ) =
1
1
1 2 2
β2
αXt dt +
βXt dBt −
β Xt dt = (α −
)dt + βdBt
2
Xt
Xt
2Xt
2
(138)
Integrating over time
log(Xt ) = log(X0 ) + (α −
β2
)t + βBt
2
(139)
2
Xt = X0 exp((α −
β
)t) exp(βBt )
2
(140)
E[exp(αBt )] = E[X0 ]eαt
(141)
Note
E[Xt ] = E[X0 ]e(α−
6
β2
2
)t
Moment Equations
Consider an SDE of the form
dXt = a(Xt )dt + c(Xt )dBt
(142)
Taking expected values we get the differential equation for first order moments
dE[Xt ]
= E[a(Xt )]
(143)
dt
seems weird that c has no effect. Double check with generator of ito diffusion result
With respect to second order moments, let
Yt = f (Xt ) = Xi,t Xj,t
(144)
using Ito’s product rule
dYt = d(Xi,t Xj,t ) =Xi,t dXj,t + Xj,t dXi,t + dXi,t dXj,t
=Xi,t (aj (Xt )dt + (c(Xt )dBt )j ) + Xj,t (ai (Xt )dt + (c(Xt )dBt )i )
+ (ai (Xt )dt + (c(Xt )dBt )i )(aj (Xt )dt + (c(Xt )dBt )j )
= Xi,t (aj (Xt )dt + (c(Xt )dBt )j ) + Xj,t (ai (Xt )dt + (c(Xt )dBt )i )
+ ci (Xt )cj (Xt )dt
(145)
Taking expected values
dE[Xi,t Xj,t ]
= E[Xi,t aj (Xt )] + E[Xj,t ai (Xt )] + E[ci (Xt )cj (Xt )]
(146)
dt
In matrix form
dE[Xt Xt0 ]
= E[Xt a(Xt )0 ] + E[a(Xt )Xt0 ] + E[c(Xt )c(Xt )0 ]
(147)
dt
The moment formulas are particularly useful when a, c are constant with respect to
Xt , in such case
dE[Xt ]
= aE[Xt ]
dt
dE[Xt Xt0 ]
= E[Xt Xt0 ]a0 + aE[Xt Xt0 ] + cc0
dt
Var[Xt ]
= E[Xt Xt0 ]a0 + aE[Xt Xt0 ] − aE[Xt ]E[Xt ]0 a0 + cc0
dt
(148)
(149)
(150)
(151)
Example
Calculate the equilibrium mean and variance of the following process
dXt = −Xt + cdBt
(152)
The first and second moment equations are
dE[Xt ]
= −E[Xt ]
dt
dE[Xt2 ]
= −2E[Xt ]2 + c2
dt
(153)
(154)
Thus
lim
t→∞
E[Xt ] = 0
(155)
2
c
E[Xt2 ] = t→∞
lim Var[Xt ] =
t→∞
2
lim
7
(156)
Generator of an Ito Diffusion
The generator Gt of the Ito diffusion
dXt = a(Xt , t)dt + c(Xt , t)dBt
(157)
is a second order partial differential operator. For any function f it provides the
directional derivative of f averaged across the paths generated by the diffusion. In
particular given the function f , the function Gt [f ] is defined as follows
dE[f (Xt ) | Xt = x]
E[f (Xt+∆t ) | Xt = x] − f (x)
= lim
∆t→0
dt
∆t
E
[df (Xt ) | Xt = x]
=
dt
Note using Ito’s rule
Gt [f ](x) =
d f (Xt ) =∇x f (Xt , t)T a(Xt , t)dt + ∇x c(Xt , t)T dBt
1
+ trace c(Xt , t)c(Xt , t)T ∇2x f (Xt , t) dt
2
(158)
(159)
Taking expected values
Gt [f ](x) =
E[df (Xt ) | Xt = x] = ∇
dt
T
x f (x) a(x, t)
1
+ trace c(x, t)c(x, t)T ∇2x f (x)
2
(160)
In other words
Gt [·] =
X
ai (x, t)
i
8
∂
1 XX
∂2
[·] +
(c(x, t)c(x, t)T )i,j
[·]
∂xi
2 i j
∂xi ∂xj
(161)
Adjoints
Every linear operator G on a Hilbert space H with inner product < ·, · > has a
corresponding adjoint operator G∗ such that
< Gx, y >=< x, G∗ y > for all x, y ∈ H
(162)
In our case the elements of the Hilbert space are functions f, g and the inner product
will be of the form
Z
< f, g >= f (x) · g(x) dx
(163)
Using partial integrations it can be shown that if
G[f ](x) =
X ∂f (x)
i
1
ai (x, t) + trace c(x, t)c(x, t)T ∇2x f (x)
∂xi
2
(164)
(165)
then
G∗ [f ](x) = −
9
X ∂
1 X ∂2
[f (x)ai (x, t)] +
[(c(x, t)c(x, t)T )ij f (x)]
∂x
2
∂x
∂x
i
i
j
i
i,j
(166)
The Feynman-Kac Formula (Terminal Condition Version)
Let X be an Ito diffusion
dXt = a(Xt , t)dt + c(Xt , t)dBt
with generator Gt
X
∂ 2 v(x, t)
∂v(x, t) 1 X X
+
(c(x, t)c(x, t)T )i,j
Gt [v](x) =
ai (x, t)
∂xi
2 i j
∂xi ∂xj
i
(167)
(168)
Let v be the solution to the following pde
∂v(x, t)
= Gt [v](x, t) − v(x, t)f (x, t)
(169)
∂t
with a known terminal condition v(x, T ), and function f . It can be shown that the
solution to the pde (169) is as follows
Z T
h
i
v(x, s) = E v(XT , T ) exp −
f (Xt )dt | Xs = x
(170)
−
s
We can think of v(XT , T ) as a terminal reward and of
factor.
RT
s
f (Xt )dt as a discount
Informal Proof:
Rt
Let s ≤ t ≤ T let Yt = v(Xt , t), Zt = exp(− s f (Xτ )dτ ), Ut = Yt Zt . It can be
shown (see Lemma below) that
dZt = −Zt f (Xt )dt
(171)
Using Ito’s product rule
dUt = d(Yt Zt ) = Zt dYt + Yt dZt + dYt dZt
(172)
Since dZt has a dt term, it follows that dYt dZt = 0. Thus
dUt = Zt dv(Xt , t) − v(Xt , t)Zt f (Xt )dt
(173)
Using Ito’s rule on dv we get
dv(Xt , t) =∇t v(Xt , t)dt + (∇x v(Xt , t))T a(Xt , t)dt + (∇x v(Xt , t))T c(Xt , t)dBt
1
+ trace c(Xt , t)c(Xt , t)T ∇2x v(Xt , t) dt
(174)
2
Thus
h
dUt =Zt ∇t v(Xt , t) + (∇x v(Xt , t))T a(Xt , t)
i
1
+ trace c(Xt , t)c(Xt , t)T ∇2x v(Xt , t) − v(Xt , t)f (Xt ) dt
2
+ Zt (∇x v(Xt , t))T c(Xt , t)dBt
and since v is the solution to (169) then
dUt = (∇x v(Xt , t))T c(Xt , t)dBt
(175)
(176)
Integrating
Z
UT − Us =
T
Yt (∇x v(Xt , t))T c(Xt , t)dBt
(177)
s
taking expected values
E[UT | Xs = x] − E[Us | Xs = x] = 0
(178)
where we used the fact that the expected values of integrals with respect to Brownian
motion is zero. Thus, since Us = Y0 Z0 = v(Xs , s)
E[UT | Xs = x] = E[Us | Xs = x] = v(x, s)
(179)
Using the definition of UT we get
v(x, s) = E[v(XT , T )e− s f (Xt )dt | Xs = x]
We end the proof by showing that
dZt = −Zt f (Xt )dt
Rt
First let Yt = s f (Xτ )dτ and note
Z t+∆t
∆Yt =
f (Xτ )dτ ≈ f (Xt )∆t
RT
(180)
(181)
(182)
t
dYt = f (Xt )dt
Let Zt = exp(−Yt ). Using Ito’s rule
1
dZt = ∇e−Yt dYt + ∇2 e−Yt (dYt )2 = −e−Yt f (Xt )dt = −Zt f (Xt )dt
2
where we used the fact that
(dYt )2 = Zt2 f (Xt )2 (dt)2 = 0
(183)
(184)
(185)
10
Kolmogorov Backward equation
The Kolmogorov backward equation tells us at time s whether at a future time t
the system will be in the target set A. We let ξ be the indicator function of A, i.e,
ξ(x) = 1 if x ∈ A, otherwise it is zero. We want to know for every state x at time
s < T what is the probability of ending up in the target set A at time T . This is
call the the hit probability.
Let X be an Ito diffusion
dXt = a(Xt , t)dt + c(Xt , t)dBt
X0 = x
(186)
(187)
The hit probability p(x, t) satisfies the Kolmogorov backward pde
−
∂p(x, t)
= Gt [p](x, t)
∂t
(188)
i.e.,
− ∂p(x,t)
=
∂t
P
i
ai (x, t) ∂p(x,t)
∂xi +
1
2
P
i,j (c(x, t)c(x, t)
T
2
)ij ∂∂xp(x,t)
i ∂xj
(189)
subject to the final condition p(x, T ) = ξ(x). The equation can be derived from
the Feynman-Kac formula, noting that the hit probability is an expected value over
paths that originate at x at time s ≤ T , and setting f (x) = 0, q(x) = ξ(x) for all x
p(x, t) = p(XT ∈ A | Xt = x) = E[ξ(XT ) | Xt = x] = E[q(XT )e
11
RT
t
f (Xs )ds
]
(190)
The Kolmogorov Forward equation
Let X be an Ito diffusion
dXt = a(Xt , t)dt + c(Xt , t)dBt
X0 = x0
(191)
(192)
with generator G. Let p(x, t) represent the probability density of Xt evaluated at
x given the initial state x0 . Then
∂p(x, t)
= G∗ [p](x, t)
(193)
∂t
where G∗ is the adjoint of G, i.e.,
∂p(x,t)
∂t
=−
∂
i ∂xi [p(x, t)ai (x, t)]
P
+
1
2
∂2
T
i,j ∂xi ∂xj [(c(x, t)c(x, t) )ij p(x, t)]
P
(194)
It is sometimes useful to express the equation in terms of the negative divergence
(inflow) of a probability current J, caused by a probability velocity V
X ∂Ji (x, t)
∂p(x, t)
= −∇ · J(x, t) = −
(195)
∂t
∂xi
i
J(x, t) = p(x, t)V (x, t)
1X
∂
Vi (x, t) = ai (x, t) −
k(x, t)i,j
log(p(x, t)ki,j (x))
2 i
∂xj
(196)
(197)
k(x) = c(x, t)c(x, t)T
(198)
From this point of view the Kolmogorov forward equation is just a law of conservation of probability (the rate of accumulation of probability in a state x equals the
inflow of probability due to the probability field V ).
11.1
Example: Discretizing an SDE in state/time
Consider the following SDE
dXt = a(Xt )Xt dt + c(Xt )dBt
(199)
The Kolmogorov Forward equation looks as follows
∂p(x, t)
∂a(x)p(x, t) 1 ∂ 2 c(x)2 p(x, t)
(200)
=−
+
∂t
∂x
2
∂x2
Discretizing in time and space, to first order
∂p(x, t)
1
=
(p(x, t + ∆t ) − p(x−, t))
(201)
∂t
∆t
and
1
∂a(x)p(x, t)
=
(a(x + ∆x )p(x + ∆x , t) − a(x − ∆x )p(x − ∆x , t))
∂x
2∆x
1 ∂a(x)
∂a(x)
=
)p(x + ∆x , t) − (a(x) − ∆x
)p(x − ∆x , t)
(a(x) + ∆x
2∆x
∂x
∂x
a(x) ∂a(x)
a(x) ∂a(x)
= p(x + ∆x , t)(
) − p(x − ∆x , t)(
)
(202)
+
−
2∆x
2∂x
2∆x
2∂x
and
∂ 2 c2 (x)p(x, t)
1 2
2
2
=
c
(x
+
∆
)p(x
+
∆
,
t)
+
c
(x
−
∆
)p(x
−
∆
,
t)
−
2c
(x)p(x,
t)
x
x
x
x
∂x2
∆2x
(203)
1
∂c(x)
∂c(x)
= 2 (c2 (x) + 2∆x c(x)
)p(x + ∆x , t) + (c2 (x) − 2∆x c(x)
)p(x − ∆x , t)
∆x
∂x
∂x
− 2c2 (x)p(x, t)
c(x) 2
c(x) ∂c(x)
+2
)
= p(x + ∆x , t)(
∆x
∆x ∂x
c(x) 2
− 2p(x, t)
∆x
c(x) 2
c(x) ∂c(x)
+ p(x − ∆x , t)(
)
(204)
−2
∆x
∆x ∂x
Putting it together, the Kolmogorov Forward Equation can be approximated as
follows
p(x, t + ∆t ) − p(x, t)
a(x) ∂a(x)
=p(x − ∆x , t)(
−
)
∆t
2∆x
2∂x
a(x) ∂a(x)
− p(x + ∆x , t)(
+
)
2∆x
2∂x
1 c(x) 2 c(x) ∂c(x)
+ p(x + ∆x , t)(
+
)
2 ∆x
∆x ∂x
c(x) 2
− p(x, t)
∆x
1 c(x) 2 c(x) ∂c(x)
+ p(x − ∆x , t)(
−
)
(205)
2 ∆x
∆x ∂x
Rearranging terms
∆t c2 (x) p(x, t + ∆t ) =p(x, t) 1 −
∆2x
∂a(x) ∆t c2 (x)
∂c(x)
+ p(x − ∆x , t)
− 2c(x)
+ a(x) − ∆x
2∆x ∆x
∂x
∂x
∂c(x)
∂a(x) ∆t c2 (x)
(206)
+ 2c(x)
− a(x) − ∆x
+ p(x + ∆x , t)
2∆x ∆x
∂x
∂x
Considering in a discrete time, discrete state system
X
p(Xt+∆t = x) =
p(Xt = x0 )p(Xt+∆t = x | Xt = x0 )
(207)
x0
we make the following discrete time/discrete state approximation

2
c (x)
∂c(x)
∂a(x)
∆t

−
a(x)
+
2c(x)
−
∆

x
 2∆x ∆x
∂x
∂x if xt+∆t = xt − ∆x


 ∆t c2 (x)
∂c(x)
∂a(x)
+ a(x) − 2c(x) ∂x − ∆x ∂x
if xt+∆t = xt + ∆x
p(xt+∆t | xt ) = 2∆x ∆2x

c (x)

1 − ∆t∆
if xt+∆t = xt

2

x

0
else
(208)
Note if the derivative of the drift function is zero, i.e., ∂a(x)/∂x = 0 the conditional
probabilities add up to one. Not sure how to deal with the case in which the
derivative is not zero.
11.2
Girsanov’s Theorem (Version I)
Let (Ω, F, P) be a probability space. Let B be a standard m-dimensional Brownian
motion adapted to the filtration Ft . Let X, Y be defined by the following SDEs
dXt = a(Xt )dt + c(Xt )dBt
dYt = (c(Yt )Ut + a(Yt ))dt + c(Yt )dBt
X0 = Y0 = x
(209)
(210)
(211)
where Xt ∈ Rn , Bt ∈ Rm and a, c satisfy the necessary conditions
R t for the SDEs to
be well defined and Ut is an Ft adapted process such that P( 0 kc(Xs )Us k2 ds <
∞) = 1. Let
Z
Z t
1 t 0
0
U Us ds
(212)
Zt =
Us dBs +
2 0 s
0
Λt = e−Zt
(213)
and
i.e., for all A ∈ Ft
dQt = Λt dP
(214)
Qt (A) = E P [Λt IA ]
(215)
Then
Z
Wt =
t
Us ds + Bt
0
(216)
Qt .
is a standard Brownian motion with respect to
Informal Proof: We’ll provide a heuristic argument for the discrete time case. In
discrete time the equation for W = (W1 , · · · , Wn ) would look as follows
√
∆Wk = Uk ∆t + ∆tGk
(217)
where G1 , G2 , · · · are independent standard Gaussian vectors under P. Thus, under
is as follows
P the log-likelihodd of W
log p(W ) = h(n, ∆t) −
n−1
1 X
(∆Wk − Uk ∆t)0 (∆Wk − Uk ∆t)
2∆t
(218)
k=1
where h(n, ∆t) is constant with respect to W, U . For W to behave as Brownian
motion under Q we need the probability density of W under Q to be as follows
n−1
1 X
log q(W ) = h(n, ∆t) −
∆Wk0 ∆Wk
2∆t
(219)
k=1
Let the random variable Z be defined as follows
p(W )
Z = log
q(W )
(220)
where q, p represent the probability densities of W under
tively. Thus
Z=
n−1
X
n−1
1X 0
Uk Uk ∆t
2
Uk0 (∆Wk − Uk ∆t) +
k=1
=
n−1
X
Q and under P respec(221)
k=1
Uk0
√
k=1
n−1
1X 0
∆tGt +
Uk Uk ∆t
2
(222)
k=1
(223)
Note as ∆t → 0
Z
Z→
0
t
1
Us dBs +
2
Z
t
Us0 Us ds
(224)
0
q(W )
→ e−Z
p(W )
(225)
Remark 11.1.
dYt = (Ht + a(Xt ))dt + c(Yt )dBt
= a(Xt )dt + c(Yt )(Ut + dBt )
= a(Xt )dt + c(Yt )dWt
(226)
(227)
(228)
Therefore the distribution of Y under Qt is the same as the distribution of X under
P, i.e., for all A ∈ Ft .
P(X ∈ A) = Qt (Y
or more generally
∈ A)
E P [f (X0:t )] = E Qt [f (Y0:t )] = E P [f (Y0:t )Λt ]
(229)
(230)
Remark 11.2. Radon Nykodim derivative Λt is the Radon Nykodim derivative
of Qt with respect to P. This is typically represented as follows
dQt
Λt = e−Zt =
(231)
dP
We can get the derivative of P with respect to Qt by inverting Λt , i.e.,
dP
(232)
= eZ t
dQt
Remark 11.3. Likelihood Ratio This tells us that Λt is the likelihood ratio
between the process X0:t and the process Y0:t , i.e., the equivalent of pX (X)/pY (Y )
where pX , pY are the probability densities of X and Y .
Remark 11.4. Importance Sampling Λt can be used in importance sampling
schemes. Suppose (y [1] , λ[1] ), · · · , (y [n] , λ[n] ) are iid samples from (Y0:t , Λt ), then we
can estimate E P [f (X0:t )] as follows
E P [f (X0:t )] ≈
11.2.1
n
1X
f (y [i] )λ[i]
n i=1
(233)
Girsanov Version II
Let
dXt = c(Xt )dBt
dYt = b(Yt )dt + c(Yt )dBt
X0 = Y0 = x
(234)
(235)
(236)
Let
Z
Zt =
t
b(Ys )0 (c(Ys )−1 )0 dBs
0
Z
1 t
b(Ys )0 k(Ys )b(Ys )ds
+
2 0
dP
= eZt
dQt
(237)
(238)
where
Then under
Qt the process Y
k(Ys ) = (c(Ys )0 c(Ys ))−1
(239)
has the same distribution as the process X under
P.
Proof. We apply Girsanov’s version I with a(Xt ) = 0, Ut = c(Yt )−1 b(Xt ). Thus
Z
Z t
1 t 0
U Us ds
Zt =
Us0 dBs +
2 0 s
0
Z t
b(Ys )0 (c(Ys )−1 )0 dBs
=
0
Z
1 t
+
b(Ys )0 k(Ys )b(Ys )ds
(240)
2 0
dP
= eZt
(241)
dQt
And under Qt the process Y looks like the process X, i.e., a process with zero
drift.
11.2.2
Girsanov Version III
Let
dXt = c(Xt )dBt
dYt = b(Yt )dt + c(Yt )dBt
X0 = Y0 = x
(242)
(243)
(244)
Let
t
Z
b(Ys )0 k(Ys )dYt
Zt =
0
−
1
2
Z
t
b(Ys )0 k(Ys )b(Ys )ds
(245)
0
dP
= eZt
dQt
(246)
k(Ys ) = (c(Ys )0 c(Ys ))−1
(247)
where
Then under
Qt the process Y
has the same distribution as the process X under
Proof. We apply Girsanov’s version II
Z t
Zt =
b(Ys )0 (c(Ys )−1 )0 dBs
0
Z
1 t
b(Ys )0 (c(Ys )−1 )0 c(Ys )−1 b(Ys )ds
+
2 0
Z t
=
b(Ys )0 (c(Ys )−1 )0 c(Ys )−1 (dYt − b(Ys )ds)
0
Z
1 t
b(Ys )0 (c(Ys )−1 )0 c(Ys )−1 b(Ys )ds
+
2 0
Z t
=
b(Ys )0 k(Ys )dYt
0
Z
1 t
b(Ys )0 k(Ys )b(Ys )ds
−
2 0
dP
= eZt
dQt
P.
(248)
(249)
(250)
(251)
Informal Discrete Time Based Proof: For a given path x the ratio of the probability
density of x under P and Q can be approximated as follows
Y p(xtk+1 − xtk | xtk )
dP(x)
≈
(252)
dQt (x)
q(xtk+1 − xtk | xtk )
k
were π = {0 = t0 < t1 · · · < tn = T } is a partition of [0, T ] and
p(xtk+1 | xtk ) = G(∆xtk | a(xtk )∆tk , ∆tk k(xt )−1 )
−1
q(xtk+1 | xtk ) = G(∆xtk | 0, ∆tk k(xt )
)
(253)
(254)
where G(·, µ, σ) is the multivariate Gaussian distribution with mean µ and covariance matrix c. Thus
X
dP(x)
1 (∆xtk − a(xtk )∆tk )0 k(xt )(∆xtk − a(xtk )∆tk )
log
≈
−
dQt (x)
2∆tk
k=0
− ∆x0tk k(xt )∆xtk
n−1
n−1
X
=
k=0
1
a(xtk )0 k(xt )∆xtk − a(xtk )0 k(xtk )a(xtk )∆tk
2
(255)
taking limits as |π| → 0
log
dP(x)
=
dQt (x)
Z
T
a(Xt )0 k(Xt ) dXt −
0
1
2
Z
T
a(Xt )0 k(Xt )a(Xt ) dt
(256)
0
Theorem 11.1. The dΛt differential
Let Xt be an Ito process of the form
dXt = a(t, Xt )dt + c(t, Xt )dBt
(257)
Let
Λt = eZt
Z
Z t
1 t
a(s, Xs )0 k(s, Xs )a(s, Xs )ds
Zt =
a(s, Xs )0 k(t, Xt )dXs −
2
0
0
(258)
(259)
where k(t, x) = (c(t, x)c(t, x)0 )−1 .Then
dΛt = Λt a(t, Xt )0 k(t, Xt )dXt
(260)
Proof. From Ito’s product rule
d(Λt f (Xt )) = f (Xt )dΛt + Λt f (Xt ) + (dΛt )(df (Xt )
(261)
Note
dXt0 k(t, X)a(t, Xt )a0 (t,k(t, Xt )dXt
1
= dΛt = (∇z Λt )0 dZt + dZt0 (∇2z Λt )dZt
2
1
= Λt dZt + dZt0 dZt
2
(262)
(263)
(264)
Moreover, from the definition of Zt
1
dZt = a(t, Xt )0 k(t, Xt )dXt − a(t, Xt )0 k(t, Xt )a(t, Xt )dt
2
(265)
Thus
dZt0 dZt = dXt0 k(t, X)a(t, Xt )a0 (t, Xt )k(t, Xt )dXt
=dBt0 c(t, Xt )0 k(t, X)a(t, Xt )a0 (t, Xt )k(t, Xt )c(t, Xt )dBt
(266)
(267)
= trace c(t, Xt )0 k(t, Xt )a(t, Xt )a0 (t, Xt )k(t, Xt )c(t, Xt ) dt
= trace c(t, Xt )c(t, Xt )0 k(t, Xt )a(t, Xt )a0 (t, Xt )k(t, Xt ) dt
= trace a(t, Xt )a0 (t, Xt )k(t, Xt ) dt
= trace a(t, Xt )a0 (t, Xt )k(t, Xt ) dt
= trace a0 (t, Xt )k(t, Xt )a(t, Xt ) dt
(268)
= a0 (t, Xt )k(t, Xt )a(t, Xt )dt
(273)
(269)
(270)
(271)
(272)
Thus
dΛt = Λt a(t, Xt )0 k(t, Xt )dXt
12
(274)
Zakai’s Equation
Let
dXt = a(Xt )dt + c(Xt )dBt
dYt = g(Xt )dt + h(Xt )dWt
(275)
(276)
Λt = eZt
Z
Z t
1 t
0
g(Yt )0 k(Yt )g(Yt )dt
Zt =
g(Yt ) k(Yt )dYt +
2 0
0
k(Yt ) = (h(Yt )h(Yt )0 )−1
(277)
(278)
(279)
Using Ito’s product rule
d(f (Xt )Λt ) = Λt df (Xt ) + f (Xt )dΛt + df (Xt )dΛt
(280)
1
df (Xt ) = ∇x f (Xt )0 dXt + trace c(Xt )c0 (Xt )∇2x f (Xt ) dt
2
= Gt [f ](x) + ∇x f (Xt )c(Xt )dBt
dΛt = Λt g(Yt )k(Yt )dYt
(281)
(282)
where
Following the rules of Ito’s calculus we note dXt dYt0 is an n × m matrix of zeros.
Thus
d(f (Xt )Λt ) = Λt Gt [f ](x) + ∇x f (Xt )c(Xt )dBt + g(Yt )k(Yt )dYt
(283)
13
Solving Stochastic Differential Equations
Let dXt = a(t, Xt )dt + c(t, Xt )dBt .
(284)
dBt
dBt
t
Conceptually, this is related to dX
dt = a(t, Xt ) + c(t, Xt ) dt where dt is white
dBt
noise. However, dt does not exist in the usual sense, since Brownian motion is
nowhere differentiable with probability one.
We interpret solving for (284), as finding a process Xt that satisfies
Zt
Xt = M +
Zt
a(s, Xs ) ds +
0
c(s, Xs ) dBs.
(285)
0
for a given standard Brownian process B. Here Xt is an Ito process with a(s, Xs ) =
Ks and c(s, Xs ) = Hs .
a(t, Xt ) is called the drift function.
c(t, Xt ) is called the dispersion function (also called diffusion or volatility function).
Setting b = 0 gives an ordinary differential equation.
Example 1: Geometric Brownian Motion
dXt = aXt dt + bXt dBt
X0 = ξ > 0
(286)
(287)
Using Ito’s rule on log Xt we get
2
1
1
1
(dXt )2
d log Xt =
− 2
dXt +
Xt
2
Xt
dXt
1
=
− b2 dt
Xt
2
1
= aXt − b2 dt + αdBt
2
(289)
1
log Xt = log X0 + a − b2 t + bBt
2
(291)
1 2
Xt = X0 e(a− 2 b )t+bBt
(292)
Yt = Y0 eαt+βBt
(293)
(288)
(290)
Thus
and
Processes of the form
where α and β are constant, are called Geometric Brownian Motions. Geometric
Brownian motion Xt is characterized by the fact that the log of the process is
Brownian motion. Thus, at each point in time, the distribution of the process is
log-normal.
Let’s study the dynamics of the average path. First let
Yt = ebBt
(294)
Using Ito’s rule
1
dYt = bebBt dBt + b2 ebBt (dBt )2
2
Z t
Z
1 2
Yt = Y0 + b
Ys dBs + b
+0t Ys ds
2
0
Z
1 2 t
E(Yt ) = E(Y0 ) + 2 b E(Ys )ds
0
1 2
dE(Yt )
= b E(Yt )
dt
2
E(Yt ) = E(Y0 )e
Thus
E(Xt ) = E(X0 )e(a−
1 2
2b t
=e
1 2
2b t
(295)
(296)
(297)
(298)
(299)
)t E(Y ) = E(X )e(a− 21 b2 )t
(300)
t
0
Thus the average path has the same dynamics as the noiseless system. Note the
result above is somewhat trivial considering
1 2
2b
E(dXt ) = dE(Xt ) = E(a(Xt ))dt + E(c(Xt )dBt ) = E(a(Xt ))dt + E(c(Xt ))E(dBt )
(301)
(302)
(303)
dE(Xt ) = E(a(Xt ))dt
and in the linear case
E(a(Xt ))dt = E(at Xt + ut )dt = at E(Xt ) + ut )dt
(304)
These symbolic operations on differentials trace back to the corresponding integral
operations they refer to.
14
14.1
Linear SDEs
The Deterministic Case (Linear ODEs)
Let xt ∈ <n be defined by the following ode
dxt
= axt + u
dt
The solution takes the following form:
Constant Coefficients
(305)
xt = eat x0 + a−1 (eat − I)u
(306)
dxt
= aeat x0 + eat u
dt
(307)
axt + u = aeat x0 + eat u − u + u = dxt dt
(308)
To see why note
and
Example:
Let xt be a scalar such that
dxt
= α (u − xt )
dt
(309)
Thus
1 −αt
(e
− 1)αu
α
= e−αt x0 + (1 − e−αt )u
xt = e−αt x0 −
(310)
Time variant coefficients
Let xt ∈ <n be defined by the following ode
dxt
= at xt + ut
dt
xo = ξ
(311)
(312)
where ut is known as the driving, or input, signal. The solution takes the following
form:
Z
t
xt = Φt x0 +
Φ−1
s us ds
(313)
0
where Φt is an n × n matrix, known as the fundamental solution, defined by the
following ODE
dΦt
= at Φt
(314)
dt
Φ0 = I n
(315)
14.2
The Stochastic Case
Linear SDEs have the following form
dXt = (at Xt + ut )dt +
m
X
(ci,t Xt + vi,t ) dBi,t
(316)
i=1
= (at Xt + ut )dt + vt dBt +
m
X
ci,t Xt dBi,t
(317)
i=1
X0 = ξ
(318)
where Xt is an n dimensional random vector, Bt = (B1 , · · · , Bm ), bi,t are n × n
matrices, and vi,t are the n-dimensional column vectors of the n × m matrix vt . If
bi,t t = 0 for all i, t we say that the SDE is linear in the narrow sense. If vt = 0 for
all t we say that the SDE is homogeneous. The solution has the following form
!
!
Z t
Z t
m
m
X
X
−1
−1
Xt = Φt X0 +
Φs
us −
bi,s vi,s ds +
Φs
vi,s dBi,s
(319)
0
0
i=1
i=1
where Φt is an n × n matrix satisfying the following matrix differential equation
m
X
dΦt = at Φt dt +
bi,s Φs dBi,s
(320)
i=1
Φ0 = In
(321)
One property of the linear Ito SDEs is that the trajectory of the expected value
equals the trajectory of the associated deterministic system with zero noise. This is
due to the fact that in the Ito integral the integrand is independent of the integrator
dBt :
E(dXt ) = dE(Xt ) = E(a(Xt ))dt + E(c(Xt )dBt ) = E(a(Xt ))dt + E(c(Xt ))E(dBt )
(322)
dE(Xt ) = E(a(Xt ))dt
(323)
(324)
and in the linear case
E(a(Xt ))dt = E(at Xt + ut )dt = at (E(Xt ) + ut )dt
(325)
14.3
Solution to the Linear-in-Narrow-Sense SDEs
In this case
dXt = (at Xt + ut ) dt + vt dBt
(326)
X0 = ξ
(327)
where v1 , · · · , vm are the columns of the n×m matrix v, and dBt is an m-dimensional
Brownian motion. In this case the solution has the following form
Z t
Z t
−1
Xt = Φt X0 +
Φ−1
u
ds
+
Φ
v
dB
(328)
s
s
s
s
s
0
0
where Φ is defined as in the ODE case
dΦt
= at
dt
Φ0 = In
(329)
(330)
White Noise Interpretation This solution can be interpreted using a “symbolic” view of white noise as
dBt
Wt =
(331)
dt
and thinking of the SDE as an ordinary ODE with a driving term given by ut +vt Wt .
We will see later that this interpretation breaks down for the more general linear
case with bt 6= 0.
Moment Equations
Let
def
ρr,s = E ((Xr − mr )(Xs − ms ))
2 def
ρt = ρt,t = Var(Xt )
Then
dE(Xt )
= at dE(Xt ) + ut
dt
Z t
E(Xt ) = Φt E(X0 ) + Φ−1
u
ds
s
s
0
Z t
T
2
2
−1
−1
ρt = Φt ρ0 +
Φs vs Φs vs ds ΦTt
(332)
(333)
(334)
(335)
(336)
0
dρ2t
= at ρ2t + ρ2t aTt + vt v T
dt
Z r∧s
T
−1
ρr,s = Φr ρ20 +
Φ−1
v
Φ
v
dt
ΦTs
t
t
t
t
(337)
(338)
0
where r ∧ s = min{r, s}. Note the mean evolves according to the equivalent ODE
with no driving noise.
Constant Coefficients:
dXt = aXt + u + vdBt
(339)
In this case
Φt = eat
(340)
t
E(Xt ) = eat E(X0 ) + e−as uds
0
Z t
T
2
at
2
−as
T
−as T
e vv e
ds eat
Var(Xt ) = ρt = e
ρ0 +
Z
0
(341)
(342)
Example: Linear Boltzmann Machines (Multidimensional OU Process)
√
dXt = θ(γ − Xt )dt + 2τ dBt
(343)
where θ is symmetric and τ > 0. Thus in this case
a = −θ,
u = θγ,
(344)
(345)
Φt = e−tθ
(346)
and
Xt = e
−tθ
t
Z
sθ
X0 +
e dsθγ +
√
Z
t
sθ
2τ
e dBs
√ Z sθ
t
=e−tθ X0 + θ−1 esθ 0 θγ + 2τ
e dBs
0
Z t
√
=e−tθ X0 + (etθ − I)γ + 2τ
e−as dBs
0
(347)
0
t
(348)
(349)
0
Thus
E(Xt ) = e−tθ E(X0 ) +
lim E(Xt ) = γ
t→∞
ρ2t
−tθ
ρ20
I − e−tθ γ
(351)
Z
t
Var(Xt ) =
=e
+ 2τ
e
0
θ−1 2sθ t
e
= e−2tθ ρ20 + 2τ
0
2
−2ts
2
−1
2tθ
=e
ρ0 + τ θ
e −I
= e−2tθ ρ20 + τ θ−1 I − e−2tθ
= τ θ−1 + e−2tθ ρ20 − τ θ−1
lim Var(Xt ) = τ θ
(350)
2sθ
ds e−tθ
−1
t→∞
(352)
(353)
(354)
(355)
(356)
(357)
R
where we used the fact that a, eat are symmetric matrices, and eat dt = a−1 eat . If
the distribution of X0 is Gaussian, the distribution continues being Gaussian at all
times.
Example: Harmonic Oscillator Let Xt = (X1,t , X2,t )T represent the location
and velocity of an oscillator
0
0
0
1
dXt = aXt +
dBt =
dXt +
dBt
(358)
b
−α −β
b
thus
14.4
Z t
0
Xt = eat X0 +
e−as
dBs
b
0
(359)
Solution to the General Scalar Linear Case
Here we will sketch the proof for the general solution to the scalar linear case. We
have Xt ∈ R defined by the following SDE
dXt = (at Xt + ut ) dt + (bt Xt + vt ) dBt
(360)
In this case the solution takes the following form
Z t
Z t
−1
−1
Xt = Φt X0 +
Φs (us − vs bs )ds +
Φs vs dBs
0
(361)
0
dΦt = at Φt dt + bt Φt dBt
Φ0 = 1
(362)
(363)
Sketch of the proof
• Use Ito’s rule to show
−1
−1
2
dΦ−1
t = (b − at )Φt dt − bt Φt dBt
(364)
• Use Ito’s rule to show that if Y1,t , Y2,t are scalar processes defined as follows
dY1,t = a1 (t, Y1,t )dt + b1 (t, Y1,t )dBt ,
for i = 1, 2
(365)
(366)
d(Y1,t Y2,t ) = Y1,t dY2,t + Y2,t dY1,t + b1 (t, Y1,t )b2 (t, Y2,t )dt
(367)
then
• Use the property above to show that
−1
d(Xt Φ−1
t ) = Φt ((ut − vt bt )dt + vt dBt )
(368)
• Integrate above to get (361)
• Use Ito’s rule to get
1 2
d log Φ−1
=
b
−
a
t dt − bt dBt
t
2 t
Z t
Z t
1
b
)ds
+
log Φt = − log Φ−1
=
bs dBs .
(a
−
s
s
t
2
0
0
(369)
(370)
White Noise Interpretation Does not Work The white noise interpretation
of the general linear case would be
dXt = (at Xt + ut )dt + (bt Xt + vt )Wt dt
= (at + bt Wt )Xt dt + (ut + vt Wt )dt
(371)
(372)
If we interpret this as an ODE with noisy driving terms and coefficients, we would
have a solution of the form
Z t
−1
Xt = Φt X0 +
Φs (us + vt Wt ) dt
(373)
0
Z t
Z t
−1
= Φt X0 +
Φ−1
(u
−
v
b
)ds
+
Φ
v
dBs
(374)
s
s
s
s
s
s
0
0
(375)
with
dΦt = (at + bt Wt )Φt dt = at Φt dt + bt Φt dBt
Φ0 = 1
(376)
(377)
The ODE solution to Φ would be of the form
Z t
Z t
Z t
log Φt =
(as + bs Ws )ds =
as ds +
bs dBs
0
0
(378)
0
which differs from the Ito SDE solution in (370) by the term −
Rt
0
bs ds /2.
14.5
Solution to the General Vectorial Linear Case
Linear SDEs have the following form
dXt = (at Xt + ut )dt +
m
X
(bi,t Xt + vi,t ) dBi,t
(379)
i=1
= (at Xt + ut )dt + vdBt +
m
X
bi,t Xt dBi,t
(380)
i=1
X0 = ξ
(381)
where Xt is an n dimensional random vector, Bt = (B1 , · · · , Bm ), bi,t are n × n
matrices, and vi,t are the n-dimensional column vectors of the n × m matrix vt . The
solution has the following form
!
!
Z t
Z t
m
m
X
X
−1
−1
Xt = Φt X0 +
Φs
us −
bi,s vi,s ds +
Φs
vi,s dBi,s
(382)
0
i=1
0
i=1
where Φt is an n × n matrix satisfying the following matrix differential equation
dΦt = at Φt dt +
m
X
bi,s Φs dBi,s
Φ0 = In
(383)
i=1
An explicit solution for Φ cannot be found in general even when a, bi are constant.
However if in addition of being constant they pairwise commute, agi = gi a, gi gj =
gj gi for all i, j then
(
!
)
m
m
X
1X 2
Φt = exp
a−
bi Bi,t
b t+
(384)
2 i=1 i
i=1
Moment Equations
Let mt = E(Xt ), st = E(Xt XtT ), then
dmt
= at mt + ut ,
dt
m
X
dst
= at st + st aTt +
bi,t mt bTi,t + ut mTt + mt uTt
dt
i=1
+
m
X
T
T
bi,t mt vi,t
+ vi,t mTt bTi,t + vi,t vi,t
,
(385)
(386)
(387)
i=1
The first moment equation can be obtained by taking expected values on (379). Note
it is equivalent to the differential equation one would obtain for the deterministic
part of the original system.
For the second moment equation apply Ito’s product rule
dXt X T = Xt dXtT + (dXt )XtT + (dXt )dXtT
m
X
T
T
= Xt dXt + (dXt ) Xt +
(bi,t Xt + vi,t ) XtT bTi,t + vi,t dt
i=1
substitute the dXt for its value in (379) and take expected values.
(388)
Example: Multidimensional Geometric Brownian Motion
dXi,t = ai Xi,t dt + Xi,t
n
X
bi,j dBj,t
(389)
j=1
for i = 1, · · · , n. Using Ito’s rule
1 −1 2
dXi,t
2
+
(dXi,t ) =
d log Xi,t =
Xi,t
2 Xi , t


X1
X
ai −
b2i,j  +
bi,j dBi,j
2
j
j
Thus



n
n


X
X
1
Xi,t = Xi,0 exp ai −
b2i,j  t +
bi,j Bj,t


2 j=1
j=1




n
n
X
X
1
b2  t + log 
bi,j Bj,t 
log Xi,t = log(Xi,0 ) + ai −
2 j=1 i,j
j=1
(390)
(391)
(392)
(393)
and thus Xi,t has a log-normal distribution.
15
Important SDE Models
• Stock Prices: Exponential Brownian Motion with Drift
dXt = aXt dt + bXt dBt
(394)
• Vasiceck( 1977) Interest rate model: OU Process
dXt = α(θ − Xt )dt + bdBt
• Cox-ingersol-Ross (1985) Interest rate model
p
dXt = α(θ − Xt )dt + b Xt dBt
(395)
(396)
• Constant Elasticity of Variance process
dX= µXt + σXtγ dBt , γ ≥ 0
(397)
• Generalized Cox Ingersoll Ross Model for short term interest rate, proposed
by Chan et al (1992).
dXt = (θ0 − θ1 Xt )dt + γXtΨ dBt , for Ψ, γ > 0
(398)
Let
X̃s =
Xs1−Ψ
γ(1 − Ψ)
(399)
Thus
dX̃s = a(X̃s )ds + dBs
(400)
where
a(x) =
Ψγ Ψ−1
(θ0 + θ1 x̂)x̂−Ψ
−
x̂
γ
2
(401)
where
−1
x̂ = (γ(1 − Ψ)x)(1−Ψ)
(402)
A special case is when Ψ = 0.5. In this case the process increments are
known to have a non-central chi-squared distribution (Cox, Ingersoll, Ross,
1985)
• Logistic Growth
– Model I
dXt = aXt (1 − Xt /k)dt + bXt dBt
(403)
X0 exp (a − b2 /2)t + bBt
Xt =
Rt
1 + Xk0 a 0 exp {(a − b2 /2)s + bBs } ds
(404)
dXt = aXt (1 − Xt /k)dt + bXt (1 − Xt /k)dBt
(405)
dXt = rXt (1 − Xt /k)dt − srXt2 dBt
(406)
The solution is
– Model II
– Model III
In all the models r is the Maltusian growth constant and k the carrying capacity of the environment. In model II, k is unattainable. In
the other models Xt can have arbitrarily large values with nonzero
probability.
• Gompertz Growth
dXt = αXt dt rXt log
k
Xt
dt + bXt dBt
(407)
where r is the Maltusian growth constant and k the carrying capacity. For
α = 0 we get the Skiadas version of the model, for r = 0 we get the lognormal model. Using Ito’s rule on Yt = eβt log Xt we can get expected value
follows the following equation
2
n
o
b
E(Xt ) = exp log(x0 )e−rt exp γr (1 − e−rt ) exp 4r
(1 − e−2rt (408)
where
b2
2
Something fishy in expected value formula. Try α = b = 0!
γ =α −
16
(409)
Stratonovitch and Ito SDEs
Stochastic differential equations are convenient pointers to their corresponding
stochastic integral equations. The two most popular stochastic integrals are the
Ito and the Stratonovitch versions. The advantage of the Ito integral is that the
integrand is independent of the integrator and thus the integral is a Martingale.
The advantage of the Stratonovitch definition is that it does not require changing
the rules of standard calculus. The Ito interpretation of
dXt = f (t, Xt )dt +
m
X
j=1
gj (t, Xt )dBj,t
(410)
is equivalent to the Stratonovitch equation


m X
m m
X
X
1
∂g
i,j
dXt = f (t, Xt )dt −
gi j (t, Xt ) dt +
gj (t, Xt )dBj,t
2 i=1 j=1 ∂xi
j=1
(411)
and the Stratonovitch interpreation of
dXt = f (t, Xt )dt +
m
X
gj (t, Xt )dBj,t
(412)
j=1
is equivalent to the Ito equation

m m X
m
X
X
∂g
1
i,j
gi j (t, Xt ) dt +
gj (t, Xt )dBj,t
dXt = f (t, Xt )dt +
2 i=1 j=1 ∂xi
j=1

17
(413)
SDEs and Diffusions
• Diffusions are rocesses governed by the Fokker-Planck-Kolmogorov equation.
• All Ito SDEs are diffusions, i.e., they follow the FPK equation.
• There are diffusions that are not Ito diffusions, i.e., they cannot be described
by an Ito SDE. Example: diffusions with reflection boundaries.
18
Appendix I: Numerical methods
18.1
Simulating Brownian Motion
18.1.1
Infinitesimal “piecewise linear” path segments
Get n independent standard Gaussian variables {Z1 , · · · , Zn };
E(Zi ) = 0; Var(Zi ) = 1. Define the stochastic process B̂ as follows,
B̂t0 = 0
B̂t1 = B̂t0 +
..
.
B̂tk = B̂tk−1
√
(414)
t1 − t0 Z1
(415)
(416)
p
+ tk − tk−1 Zk
(417)
Moreover,
B̂t = B̂tk−1 for t ∈ [tk−1 , tk )
(418)
This defines a continuous time process that converges in distribution to Brownian
motion as n → ∞.
18.1.2
Linear Interpolation
Same as above but linearly interpolating the starting points of path segments.
B̂t = B̂tk−1 + (t − tk )(B̂tk−1 − B̂tk )/(tk − tk−1 ) for t ∈ [tk−1 , tk )
(419)
Note this approach is non-causal, in that it looks into the future. I believe it is
inconsistent with Ito’s interpretation and converges to Stratonovitch solutions
18.1.3
Fourier sine synthesis
Pn−1
B̂t (ω) = k=0 Zk (ω)φk (t)
where Zk (ω)
random variables as in previous approach, and φk (t) =
are same
√
(2k+1)Rt
2 2T
(2k+1)R sin
2T
As n → ∞ B converges to BM in distribution. Note this approach is non-causal,
in that it looks into the future. I believe it is inconsistent with Ito’s interpretation
and converges to Stratonovitch solutions
18.2
Simulating SDEs
Our goal is to simulate
dXt = a(Xt )dt + c(Xt )dBt ,
0 ≤ t ≤ T X0 = ξ
(420)
Order of Convergencce Let 0 = t1 < t2 · · · < tk = T
A method is said to have strong oder of convergence α if there is aconstant K such
that
sup E Xtk − X̂k < K(∆tk )α
(421)
tk
A method is said to have week oder of convergence α if there is a constant K such
that
sup E[Xtk ] − E[X̂k ] < K(∆tk )α
(422)
tk
Euler-Maruyama Method
X̂k = X̂k−1 + a(X̂k−1 )(tk − tk−1 ) + c(X̂k−1 )(Bk − Bk−1 )
p
Bk = Bk−1 + tk − tk−1 Zk
(423)
(424)
where Z1 , · · · , Zn are independent standard Gaussian random variables.
The Euler-Maruyama method has strong convergence of order α = 1/2, which is
poorer of the convergence for the Euler method in the deterministic case, which is
order α = 1. The Euler-Maruyama method has week convergence of order α = 1.
Milstein’s Higher Order Method: It is based on a higher order truncation of
the Ito-Taylor expansion
X̂k = X̂k−1 + a(X̂k−1 )(tk − tk−1 ) + c(X̂k−1 )(Bk − Bk−1 )
1
+ c(Xk−1 )∇x c(Xk−1 ) (Bk − Bk−1 )2 − (tk − tk−1 )
2
p
Bk = Bk−1 + tk − tk−1 Zk
where Z1 , · · · , Zn are independent standard Gaussian random variables.
method has strong convergence of order α = 1.
19
(425)
(426)
(427)
This
History
• The first version of this document, which was 17 pages long, was written
by Javier R. Movellan in 1999.
• The document was made open source under the GNU Free Documentation
License 1.1 on August 12 2002, as part of the Kolmogorov Project.
• October 9, 2003. Javier R. Movellan changed the license to GFDL 1.2 and
included an endorsement section.
• March 8, 2004: Javier added Optimal Stochastic Control section based on
Oksendals book.
• September 18, 2005: Javier. Some cleaning up of the Ito Rule Section. Got
rid of the solving symbolically section.
• January 15, 2006: Javier added new stuff to the Ito rule. Added the Linear
SDE sections. Added Boltzmann Machine Sections.
• January/Feb 2011. Major reorganization of the tutorial.