Brownian motion
(cont.)
18.S995 - L05
http://web.mit.edu/mbuehler/www/research/f103.jpg
order terms, it follows from Eq. (1.15) that
Basic
idea
⌧ @t p(t, x) ' (1
⇢) p(t, x) +
2
`
Split dynamics ⇢into
[p(t, x) `@x p(t, x) + @xx
2
• deterministic part (drift)
2
`
(diffusion)
x) + `@x p(t, x) + @xx
• random part[p(t,
2
tains the advection-di↵usion equationSDE
@t p =
u @x p + D @xx p
5
PDE
the higher-order terms, it follows from Eq. (1.15) that
p(t, x) + ⌧ @t p(t, x) ' (1
⇢) p(t, x) +
`2
⇢ [p(t, x) `@x p(t, x) + @xx p(t, x)] +
1.2 Brownian motion
2
`2
1.2.1 SDEs and
discretization
rules
[p(t,
x) + `@x p(t, x)
+ @xx p(t, x)].
2
(1.18)
The continuous
stochastic
process X(t)
Diffusion
equation
with constant
drift described by Eq. (1.19a) or, equivalently, Eq. (1.20)
y ⌧ , one obtains
the beadvection-di↵usion
equation
can also
represented by the stochastic
di↵erential equation
@t p =
velocity u
u @x p +dX(t)
D @=xxupdt +
p
2D dB(t).
(1.25)
(1.19a)
Here, dX(t) = X(t + dt) X(t) is increment
of the stochastic particle trajectory X(t),
2
andwhilst
di↵usion
given
by an increment of the standard Brownian motion
dB(t) =constant
B(t + dt) DB(t)
denotes
3
(or Wiener) process B(t), uniquely defined by the
following
properties
:
2
`
`
u(i):=B(0)
(⇢ = 0 with
) representation
,probability
D :=
.of
1. (⇢ + )
Path-wise
⌧
2⌧
(ii) B(t) is stationary, i.e., for t > s
di↵usion
equation
from
distribution
as B(t (1.12)
s).
(1.19b) ?
typical trajectories
0 the increment B(t)
B(s) has the same
the classical
Eq. (1.19a) for ⇢ = = 0.5. The
dent fundamental solution of Eq. (1.19a) reads
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
r variables B(t
✓n ) B(tn 1 ), .2.◆
the random
. , B(t2 ) B(t1 ), B(t1 ) are independently
1 their joint distribution
(x ut) factorizes).
distributed (i.e.,
p(t, x) =
exp
(1.20)
4⇡Dt
4Dt
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
Note that Eqs. (1.12) and Eq. (1.19a) can both be written in the current-form
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
@ t p + @ x jx = 0
(1.21)
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
the higher-order terms, it follows from Eq. (1.15) that
p(t, x) + ⌧ @t p(t, x) ' (1
⇢) p(t, x) +
`2
⇢ [p(t, x) `@x p(t, x) + @xx p(t, x)] +
1.2 Brownian motion
2
`2
1.2.1 SDEs and
discretization
rules
[p(t,
x) + `@x p(t, x)
+ @xx p(t, x)].
2
(1.18)
The continuous
stochastic
process X(t)
Diffusion
equation
with constant
drift described by Eq. (1.19a) or, equivalently, Eq. (1.20)
y ⌧ , one obtains
the beadvection-di↵usion
equation
can also
represented by the stochastic
di↵erential equation
@t p =
velocity u
u @x p +dX(t)
D @=xxupdt +
p
2D dB(t).
(1.25)
(1.19a)
Here, dX(t) = X(t + dt) X(t) is increment
of the stochastic particle trajectory X(t),
2
andwhilst
di↵usion
given
by an increment of the standard Brownian motion
dB(t) =constant
B(t + dt) DB(t)
denotes
3
(or Wiener) process B(t), uniquely defined by the
following
properties
:
2
`
`
u(i):=B(0)
(⇢ = 0 with
) representation
,probability
D :=
.of
1. (⇢ + )
Path-wise
⌧
2⌧
(ii) B(t) is stationary, i.e., for t > s
classical
equation
(1.12)
from
1.2di↵usion
Brownian
distribution
as B(t motion
s).
(1.19b) ?
typical trajectories
0 the increment B(t)
B(s) has the same
the
Eq. (1.19a) for ⇢ = = 0.5. The
dent fundamental solution of Eq. (1.19a) reads
(iii) B(t)
has independent
increments. That
is, for all tn > tn 1 > . . . > t2 > t1 ,
1.2.1
SDEs
and
discretization
rules
r variables B(t
✓n ) B(tn 1 ), .2.◆
the random
. , B(t2 ) B(t1 ), B(t1 ) are independently
1 theirprocess
(x described
ut) factorizes).
distributed
(i.e.,
joint distribution
Thep(t,
continuous
stochastic
X(t)
by Eq. (1.19a) or, equivalently,
Eq. (1.20)
x) =
exp
(1.20)
can also be represented
4⇡Dtby the stochastic
4Dtdi↵erential equation
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
= u dt
(v) B(t) is continuous with dX(t)
probability
1. +
p
2D dB(t).
Note that Eqs. (1.12) and Eq. (1.19a) can both be written in the current-form
(1.25)
ThedX(t)
probability
distribution
P governing
the driving
process
B(t) is particle
commonly
known as X(t),
Here,
= X(t
+ dt) X(t)
is increment
of the
stochastic
trajectory
the Wiener
measure.
@t p++dt)@x jxB(t)
= 0denotes an increment of the standard Brownian
(1.21)motion
whilst
dB(t) =
B(t
Although
the derivative
⇠(t) = dB/dt
is not
mathematically,
(or Wiener)
process
B(t), uniquely
defined
by well-defined
the following
properties3 : Eq. (1.25) is
in the physics literature often written in the form
p
(i) B(0) = 0 with probability 1.
Ẋ(t) = u + 2D ⇠(t).
(1.26)
1.2.1
SDEs and discretization rules
Wiener process
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
(i) B(0) = 0 with probability 1.
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
0 the increment B(t)
B(s) has the same
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
The random driving function ⇠(t) is then referred to as Gaussian white noise, characterized
by
1.2.1
SDEs and discretization rules
Wiener process
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
(i) B(0) = 0 with probability 1.
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
0 the increment B(t)
B(s) has the same
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
The random driving function ⇠(t) is then referred to as Gaussian white noise, characterized
by
1.2.1
SDEs and discretization rules
Wiener process
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
(i) B(0) = 0 with probability 1.
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
0 the increment B(t)
B(s) has the same
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
The random driving function ⇠(t) is then referred to as Gaussian white noise, characterized
by
1.2.1
SDEs and discretization rules
Wiener process
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
(i) B(0) = 0 with probability 1.
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
0 the increment B(t)
B(s) has the same
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
The random driving function ⇠(t) is then referred to as Gaussian white noise, characterized
by
1.2.1
SDEs and discretization rules
Wiener process
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
(i) B(0) = 0 with probability 1.
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
0 the increment B(t)
B(s) has the same
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
The random driving function ⇠(t) is then referred to as Gaussian white noise, characterized
by
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
1.2
0 the increment B(t)
B(s) has the same
Brownian motion
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
randomand
variables
B(tn ) B(tn 1rules
), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
1.2.1theSDEs
discretization
distributed (i.e., their joint distribution factorizes).
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can
be has
represented
the stochastic
(iv)also
B(t)
Gaussianbydistribution
withdi↵erential
variance t equation
for all t 2 (0, 1).
p
(v) B(t) is continuous withdX(t)
probability
= u dt 1.
+ 2D dB(t).
(1.25)
SDEs in physicist’s notation
The probability
distribution
P governing
the driving
B(t)
is commonly
known
as
Here,
dX(t) = X(t
+ dt) X(t)
is increment
of the process
stochastic
particle
trajectory
X(t),
the Wiener
whilst
dB(t) measure.
= B(t + dt) B(t) denotes an increment of the standard Brownian motion
3
Althoughprocess
the derivative
⇠(t) = dB/dt
notthe
well-defined
mathematically,
Eq. (1.25) is
(or Wiener)
B(t), uniquely
definedisby
following properties
:
in the physics literature often written in the form
(i) B(0) = 0 with probability 1.
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
(ii) B(t) is stationary, i.e., for t > s
0 the increment B(t) B(s) has the same
The random
driving
function
distribution
as B(t
s).⇠(t) is then referred to as Gaussian white noise, characterized
by
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables
B(t=n )0 , B(tnh⇠(t)⇠(s)i
h⇠(t)i
= 2 )(t B(t
s), 1 ), B(t1 ) are independently
(1.27)
1 ), . . . , B(t
distributed (i.e., their joint distribution factorizes).
with h · i denoting an average with respect to the Wiener measure.
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
3
Note that, since X has dimensions of length and D has dimensions length2 /time, the Wiener process
1/2
B(v)
in Eq.
(1.25)
has units time
. probability 1.
B(t)
is continuous
with
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
2.1
SDEs and discretization rules
Stochastic differential calculus
e continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
n also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
re, dX(t) = X(t
+ dt)
X(t)Note
is increment
of the
trajectory
Ito’s
formula
that property
(iv) stochastic
implies thatparticle
E[dB 2 ] =
dt. This X(t),
justifies the
ilst dB(t) = B(t
+ dt) heuristic
B(t) denotes
increment
standard change
Brownian
motion
following
derivationan
of Ito’s
formula of
forthe
the di↵erential
of some
real-valued
3
Wiener) process
B(t),Funiquely
defined by the following properties :
function
(x)
dF (X(t))
i) B(0) = 0 with probability
1. := F (X(t + dt))
i) B(t) is stationary, i.e., for t >= s
distribution as B(t s).
=
F (X(t))
1
F 0 (X(t)) dX + F 00 (X(t)) dX 2 + . . .
0 the increment
B(t) B(s) has the same
2
h
i2
p
1
F 0 (X(t)) dX + F 00 (X(t)) u dt + 2D dB + . . .
2
0
F That
(X(t))is,
dX for
+ DF
all00 (X(t))
tn >dB
tn2 +1 O(dt
> .3/2
. .); > t2 > t1 , (1.28)
=
i) B(t) has independent increments.
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
hence, in a probabilistic sense, one has to leading order in dt
distributed (i.e., their joint distribution factorizes).
dF (X(t)) = F 0 (X(t)) dX + D F 00 (X(t)) dt
v) B(t) has Gaussian distribution with0 variance t for
all t 2 (0, 1).
00
0
= [u F (X(t)) + D F (X(t))] dt + F (X(t))
p
2D dB(t).
(1.29)
v) B(t) is continuous with probability 1.
0
It is crucial toPnote
that, duethe
to the
choiceprocess
of the expansion
the coefficient
e probability distribution
governing
driving
B(t) is point,
commonly
known Fas(X) in
front of dB(t) is to be evaluated at X(t). This convention is the so-called Ito integration
e Wiener measure.
rule. In particular, it is important to keep in mind that nonlinear transformations of Ito
Although the SDEs
derivative
⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
must feature second-order derivatives.
the physics literature often written in the form
Discretization dilemma Topclarify the importance of discretization rules when dealing
= ua +
2Dgeneralization
⇠(t).
with SDEs, let usẊ(t)
consider
simple
of Eq. (1.25), where drift u (1.26)
and di↵usion
2.1
SDEs and discretization rules
Stochastic differential calculus
e continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
n also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
re, dX(t) = X(t
+ dt)
X(t)Note
is increment
of the
trajectory
Ito’s
formula
that property
(iv) stochastic
implies thatparticle
E[dB 2 ] =
dt. This X(t),
justifies the
ilst dB(t) = B(t
+ dt) heuristic
B(t) denotes
increment
standard change
Brownian
motion
following
derivationan
of Ito’s
formula of
forthe
the di↵erential
of some
real-valued
3
Wiener) process
B(t),Funiquely
defined by the following properties :
function
(x)
dF (X(t))
i) B(0) = 0 with probability
1. := F (X(t + dt))
i) B(t) is stationary, i.e., for t >= s
distribution as B(t s).
=
F (X(t))
1
F 0 (X(t)) dX + F 00 (X(t)) dX 2 + . . .
0 the increment
B(t) B(s) has the same
2
h
i2
p
1
F 0 (X(t)) dX + F 00 (X(t)) u dt + 2D dB + . . .
2
0
F That
(X(t))is,
dX for
+ DF
all00 (X(t))
tn >dB
tn2 +1 O(dt
> .3/2
. .); > t2 > t1 , (1.28)
=
i) B(t) has independent increments.
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
hence, in a probabilistic sense, one has to leading order in dt
distributed (i.e., their joint distribution factorizes).
dF (X(t)) = F 0 (X(t)) dX + D F 00 (X(t)) dt
v) B(t) has Gaussian distribution with0 variance t for
all t 2 (0, 1).
00
0
= [u F (X(t)) + D F (X(t))] dt + F (X(t))
p
2D dB(t).
(1.29)
v) B(t) is continuous with probability 1.
0
It is crucial toPnote
that, duethe
to the
choiceprocess
of the expansion
the coefficient
e probability distribution
governing
driving
B(t) is point,
commonly
known Fas(X) in
front of dB(t) is to be evaluated at X(t). This convention is the so-called Ito integration
e Wiener measure.
rule. In particular, it is important to keep in mind that nonlinear transformations of Ito
Although the SDEs
derivative
⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
must feature second-order derivatives.
the physics literature often written in the form
Discretization dilemma Topclarify the importance of discretization rules when dealing
= ua +
2Dgeneralization
⇠(t).
with SDEs, let usẊ(t)
consider
simple
of Eq. (1.25), where drift u (1.26)
and di↵usion
= [u F (X(t)) + D F (X(t))] dt + F (X(t))
2D dB(t).
(1.29)
It is crucial to note that, due to the choice of the expansion point, the coefficient F 0 (X) in
front of dB(t) is to be evaluated at X(t). This convention is the so-called Ito integration
rule. In particular, it is important to keep in mind that nonlinear transformations of Ito
SDEs must feature second-order derivatives.
Numerical integration
Discretization dilemma To clarify the importance of discretization rules when dealing
with SDEs, let us consider a simple generalization of Eq. (1.25), where drift u and di↵usion
coefficient D are position dependent. Adopting the Ito convention, the corresponding SDE
reads
p
dX(t) = u(X) dt + 2D(X) ⇤ dB(t),
(1.30a)
where from now on the ⇤-symbol signals that D(X) is to be evaluated at X(t). The simplest
numerical integration procedure for Eq. (1.30a) is the stochastic Euler scheme
p
p
X(t + dt) = X(t) + u(X(t)) dt + 2D(X(t)) dt Z(t),
(1.30b)
where, for each time step dt, a new random number Z(t) is drawn from a standard normal
distribution4 . If the driving process
p B(t) is Eq. (1.30a) were a regular deterministic function, such as for example B(t) = ⌧ sin(⌦t), then Eq. (1.30a) would reduce to a standard
inhomogeneous ordinary di↵erential equation (ODE). For ODEs, it typically does not matter whether one computes the coefficients5 u(x) and D(x) at the start point X(t) or the end
point X(t + dt). Mathematically, this is due to the fact that, for well-behaved deterministic driving functions, upper and lower Riemann sums yield the same value when letting
When you see an equation like (1.30a), then always ask which discretization rule has been adopted!
4
That is, a Gaussian distribution with mean µ = 0 and variance
5
Assuming the functions u and D are sufficiently smooth.
2
= 1.
where from now on the ⇤-symbol signals that D(X) is to be evaluated at X(t). The simplest
numerical integration procedure for Eq. (1.30a) is the stochastic Euler scheme
Discretization dilemma To clarify the importance
rules when dealing
p of discretization
p
+ dt) = aX(t)
+ generalization
u(X(t)) dt + of 2D(X(t))
dt Z(t),
(1.30b)
with SDEs, letX(t
us consider
simple
Eq. (1.25), where
drift u and di↵usion
Ito vs. backward-Ito
coefficient D are position dependent. Adopting the Ito convention, the corresponding SDE
where,
for each timedilemma
step dt, a To
new
random
number Z(t)
is drawn from
a when
standard
normal
Discretization
clarify
the importance
of discretization
rules
dealing
reads
4
with SDEs,
us driving
consider process
a simple B(t)
generalization
of Eq. (1.25),
driftdeterministic
u and di↵usionfuncdistribution
. Iflet
the
isp
Eq. (1.30a)
were awhere
regular
p
dX(t)
dt + then
2D(X)
D example
are position
dependent.
Adopting
the Eq.
Ito⇤ dB(t),
convention,
the corresponding
tion, coefficient
such as for
B(t)
== u(X)
⌧ sin(⌦t),
(1.30a) would
reduce to (1.30a)
a SDE
standard
reads
inhomogeneous
ordinary di↵erential equationD(X)
(ODE).
For
ODEs, itattypically
does
not matwhere from now on the ⇤-symbol signals that
is
to
be
evaluated
X(t).
The
simplest
p and D(x) at the start point X(t) or the end
5
ter whether
one
computes
the
coefficients
u(x)
numerical integration procedure
Eq.dt
(1.30a)
is the ⇤stochastic
dX(t) =for
u(X)
+ 2D(X)
dB(t), Euler scheme
(1.30a)
point X(t + dt). Mathematically, this is due top
the fact that,
p for well-behaved determinwhere from
now
on+
the
that
is
to beyield
evaluated
at X(t).
The when
simplest
istic driving
functions,
upper
and signals
Riemann
sums
same
value
letting
X(t
dt)⇤-symbol
= X(t)
+lower
u(X(t))
dtD(X)
+ 2D(X(t))
dtthe
Z(t),
(1.30b)
integration
procedure
for Eq.varying
(1.30a) isstochastic
the stochastic
Euler such
scheme
dt !numerical
0. If, however,
B(t)
is a rapidly
process,
as the Brownian
where,
for each
time step dt, a new
random
number
Z(t) is drawn
a standard
normal
p
p from
motion,
then
the
corresponding
lower
and
upper
Riemann
sums
in
general
do
not
converge
4
X(t
+
dt)
=
X(t)
+
u(X(t))
dt
+
2D(X(t))
dt
Z(t),
(1.30b)
distribution . If the driving process
is Eq. (1.30a) were a regular deterministic funcp B(t)when
to the
same value anymore. Therefore,
dealing with SDEs of the type (1.30a), it is
tion, such as for example B(t) = ⌧ sin(⌦t), then Eq. (1.30a) would reduce to a standard
important
convention.
where,to
forexplicitly
eachordinary
time specify
step
dt, athe
newintegration
random number
is drawnit from
a standard
normal
inhomogeneous
di↵erential
equation
(ODE).Z(t)
For ODEs,
typically
does not
mat4
For
instance,
the
so-called
backward
Ito
SDE
with
coefficients
u
and
D
,
denoted
5 is Eq. (1.30a) were a regularBdeterministic
. If
the driving
terdistribution
whether one
computes
the process
coefficients
and D(x) at the start point X(t)Bor thefuncend by
p B(t) u(x)
tion,X(t
such+as
for Mathematically,
example B(t) = this
⌧ sin(⌦t),
then
(1.30a)
reduce to adeterminstandard
point
dt).
is duepto
theEq.
fact
that, would
for well-behaved
dX(t)
=and
uB (X)
dtRiemann
+ (ODE).
2Dsums
•ODEs,
dB(t),
(1.31a)
inhomogeneous
ordinary
di↵erential
equation
For
itsame
typically
does
notletting
matB (X)
istic
driving functions,
upper
lower
yield
the
value
when
5
one computes
u(x) and
D(x) atprocess,
the startsuch
pointasX(t)
or
the end
dtter
!whether
0. If, however,
B(t) the
is acoefficients
rapidly
varying
stochastic
the
Brownian
6
point
X(t
+
dt).
Mathematically,
this
is upper
due toRiemann
the fact that,
well-behaved
is defined
as
the
upper
Riemann
sum
motion, then the corresponding lower and
sums for
in general
do notdeterminconverge
istic
driving
functions,
upper
and
lower
Riemann
sums
yield
the
same
value
when
letting
p
to the same value anymore. Therefore, when dealing
with SDEs of p
the type (1.30a),
it is
dt X(t
! 0.+ If,
however,
B(t)
is
a
rapidly
varying
stochastic
process,
such
as
the
Brownian
dt)
=
X(t)
+
u
(X(t
+
dt))
dt
+
2D
(X(t
+
dt))
dt
Z(t).
(1.31b)
B the integration convention.
B
important to explicitly specify
motion, then the corresponding lower and upper Riemann sums in general do not converge
For instance, the so-called backward Ito SDE with coefficients uB and DB , denoted by
to
the same
value
anymore.
Therefore,
when(1.31b)
dealing with
SDEs ofTo
thereemphasize,
type (1.30a), for
it issame
Unlike Eq.
(1.30b),
the
backward
Ito scheme
is implicit.
p
important
specify
integration
convention.
functions
u ⌘ to
uBexplicitly
and DdX(t)
⌘ D=B ,uthe
(1.30)
and
(1.31)
produce trajectories (1.31a)
that follow
(X)
dt
+ 2D
• dB(t),
BEqs.
B (X)
instance,
7 the so-called backward Ito SDE with coefficients uB and DB , denoted by
di↵erentFor
statistics
. The analog of the Ito formula (1.29) for a nonlinear transformation of
p
defined as theSDE
upperreads
Riemann
sum6
the isbackward-Ito
simply
dX(t) = uB (X) dt + 2DB (X) • dB(t),
(1.31a)
p
p
X(t + dt) = X(t)0 + uB (X(t + dt))
dt 00+ 2DB (X(t + dt)) dt Z(t).
(1.31b)
6
dF
(X)
=
F
(X)
•
dX
D
F
(X)
dt
is defined as the upper Riemann sum B
p
0
00
0
p
p reemphasize,
Unlike Eq. (1.30b),
the[ubackward
(1.31b)
isFimplicit.
To
same
=
dt
+
(X)
2D
• dB(t). for
B F (X) ItoDscheme
B F (X)]
X(t + dt) = X(t) + uB (X(t + dt)) dt + 2DB (X(t + dt)) B
dt Z(t).
(1.31b)
functions u ⌘ uB and D ⌘ DB , Eqs. (1.30) and (1.31) produce trajectories that follow
(1.32)
7
di↵erent
statistics
.
The
analog
of
the
Ito
formula
(1.29)
for
a
nonlinear
transformation
of
Unlike Eq. (1.30b), the backward Ito scheme (1.31b) is implicit. To reemphasize, for same
4 the backward-Ito SDE reads simply
2
That
is, a Gaussian
with
mean(1.30)
µ = 0 and
and (1.31)
variance
= 1. trajectories that follow
functions
u ⌘ u distribution
and D ⌘ D
, Eqs.
produce
Compare
with
do NOT give same results when dt → 0
5
B
B
7
Assuming the functions
u and D are sufficiently smooth.
where from now on the ⇤-symbol signals that D(X) is to be evaluated at X(t). The simplest
numerical integration procedure for Eq. (1.30a) is the stochastic Euler scheme
Discretization dilemma To clarify the importance
rules when dealing
p of discretization
p
+ dt) = aX(t)
+ generalization
u(X(t)) dt + of 2D(X(t))
dt Z(t),
(1.30b)
with SDEs, letX(t
us consider
simple
Eq. (1.25), where
drift u and di↵usion
Ito vs. backward-Ito
coefficient D are position dependent. Adopting the Ito convention, the corresponding SDE
where,
for each timedilemma
step dt, a To
new
random
number Z(t)
is drawn from
a when
standard
normal
Discretization
clarify
the importance
of discretization
rules
dealing
reads
4
with SDEs,
us driving
consider process
a simple B(t)
generalization
of Eq. (1.25),
driftdeterministic
u and di↵usionfuncdistribution
. Iflet
the
isp
Eq. (1.30a)
were awhere
regular
p
dX(t)
dt + then
2D(X)
D example
are position
dependent.
Adopting
the Eq.
Ito⇤ dB(t),
convention,
the corresponding
tion, coefficient
such as for
B(t)
== u(X)
⌧ sin(⌦t),
(1.30a) would
reduce to (1.30a)
a SDE
standard
reads
inhomogeneous
ordinary di↵erential equationD(X)
(ODE).
For
ODEs, itattypically
does
not matwhere from now on the ⇤-symbol signals that
is
to
be
evaluated
X(t).
The
simplest
p and D(x) at the start point X(t) or the end
5
ter whether
one
computes
the
coefficients
u(x)
numerical integration procedure
Eq.dt
(1.30a)
is the ⇤stochastic
dX(t) =for
u(X)
+ 2D(X)
dB(t), Euler scheme
(1.30a)
point X(t + dt). Mathematically, this is due top
the fact that,
p for well-behaved determinwhere from
now
on+
the
that
is
to beyield
evaluated
at X(t).
The when
simplest
istic driving
functions,
upper
and signals
Riemann
sums
same
value
letting
X(t
dt)⇤-symbol
= X(t)
+lower
u(X(t))
dtD(X)
+ 2D(X(t))
dtthe
Z(t),
(1.30b)
integration
procedure
for Eq.varying
(1.30a) isstochastic
the stochastic
Euler such
scheme
dt !numerical
0. If, however,
B(t)
is a rapidly
process,
as the Brownian
where,
for each
time step dt, a new
random
number
Z(t) is drawn
a standard
normal
p
p from
motion,
then
the
corresponding
lower
and
upper
Riemann
sums
in
general
do
not
converge
4
X(t
+
dt)
=
X(t)
+
u(X(t))
dt
+
2D(X(t))
dt
Z(t),
(1.30b)
distribution . If the driving process
is Eq. (1.30a) were a regular deterministic funcp B(t)when
to the
same value anymore. Therefore,
dealing with SDEs of the type (1.30a), it is
tion, such as for example B(t) = ⌧ sin(⌦t), then Eq. (1.30a) would reduce to a standard
important
convention.
where,to
forexplicitly
eachordinary
time specify
step
dt, athe
newintegration
random number
is drawnit from
a standard
normal
inhomogeneous
di↵erential
equation
(ODE).Z(t)
For ODEs,
typically
does not
mat4
For
instance,
the
so-called
backward
Ito
SDE
with
coefficients
u
and
D
,
denoted
5 is Eq. (1.30a) were a regularBdeterministic
. If
the driving
terdistribution
whether one
computes
the process
coefficients
and D(x) at the start point X(t)Bor thefuncend by
p B(t) u(x)
tion,X(t
such+as
for Mathematically,
example B(t) = this
⌧ sin(⌦t),
then
(1.30a)
reduce to adeterminstandard
point
dt).
is duepto
theEq.
fact
that, would
for well-behaved
dX(t)
=and
uB (X)
dtRiemann
+ (ODE).
2Dsums
•ODEs,
dB(t),
(1.31a)
inhomogeneous
ordinary
di↵erential
equation
For
itsame
typically
does
notletting
matB (X)
istic
driving functions,
upper
lower
yield
the
value
when
5
one computes
u(x) and
D(x) atprocess,
the startsuch
pointasX(t)
or
the end
dtter
!whether
0. If, however,
B(t) the
is acoefficients
rapidly
varying
stochastic
the
Brownian
6
point
X(t
+
dt).
Mathematically,
this
is upper
due toRiemann
the fact that,
well-behaved
is defined
as
the
upper
Riemann
sum
motion, then the corresponding lower and
sums for
in general
do notdeterminconverge
istic
driving
functions,
upper
and
lower
Riemann
sums
yield
the
same
value
when
letting
p
to the same value anymore. Therefore, when dealing
with SDEs of p
the type (1.30a),
it is
dt X(t
! 0.+ If,
however,
B(t)
is
a
rapidly
varying
stochastic
process,
such
as
the
Brownian
dt)
=
X(t)
+
u
(X(t
+
dt))
dt
+
2D
(X(t
+
dt))
dt
Z(t).
(1.31b)
B the integration convention.
B
important to explicitly specify
motion, then the corresponding lower and upper Riemann sums in general do not converge
For instance, the so-called backward Ito SDE with coefficients uB and DB , denoted by
to
the same
value
anymore.
Therefore,
when(1.31b)
dealing with
SDEs ofTo
thereemphasize,
type (1.30a), for
it issame
Unlike Eq.
(1.30b),
the
backward
Ito scheme
is implicit.
p
important
specify
integration
convention.
functions
u ⌘ to
uBexplicitly
and DdX(t)
⌘ D=B ,uthe
(1.30)
and
(1.31)
produce trajectories (1.31a)
that follow
(X)
dt
+ 2D
• dB(t),
BEqs.
B (X)
instance,
7 the so-called backward Ito SDE with coefficients uB and DB , denoted by
di↵erentFor
statistics
. The analog of the Ito formula (1.29) for a nonlinear transformation of
p
defined as theSDE
upperreads
Riemann
sum6
the isbackward-Ito
simply
dX(t) = uB (X) dt + 2DB (X) • dB(t),
(1.31a)
p
p
X(t + dt) = X(t)0 + uB (X(t + dt))
dt 00+ 2DB (X(t + dt)) dt Z(t).
(1.31b)
6
dF
(X)
=
F
(X)
•
dX
D
F
(X)
dt
is defined as the upper Riemann sum B
p
0
00
0
p
p reemphasize,
Unlike Eq. (1.30b),
the[ubackward
(1.31b)
isFimplicit.
To
same
=
dt
+
(X)
2D
• dB(t). for
B F (X) ItoDscheme
B F (X)]
X(t + dt) = X(t) + uB (X(t + dt)) dt + 2DB (X(t + dt)) B
dt Z(t).
(1.31b)
functions u ⌘ uB and D ⌘ DB , Eqs. (1.30) and (1.31) produce trajectories that follow
(1.32)
7
di↵erent
statistics
.
The
analog
of
the
Ito
formula
(1.29)
for
a
nonlinear
transformation
of
Unlike Eq. (1.30b), the backward Ito scheme (1.31b) is implicit. To reemphasize, for same
4 the backward-Ito SDE reads simply
2
That
is, a Gaussian
with
mean(1.30)
µ = 0 and
and (1.31)
variance
= 1. trajectories that follow
functions
u ⌘ u distribution
and D ⌘ D
, Eqs.
produce
Compare
with
In particular
5
B
B
7
Assuming the functions
u and D are sufficiently smooth.
SDE with coefficients (uB , DB ) can be transformed into a stochastically equivalent Ito SDE
oneadapting
obtains an
SDE that (u,
is stochastically
equivalent
to Eqs. (1.31).
by
theIto
coe↵ficients
D) accordingly.
More precisely,
by fixing
Another discretization convention, that is popular in the physics literature is the
Stratonovich-Fisk discretization,
u = udenoted
D = DB
(1.33)
B + @x Dby
B,
p
uS (X) dt + equivalent
2DS (X) dB(t),
(1.34a)
one obtains an Ito SDEdX(t)
that is= stochastically
to Eqs. (1.31).
Another discretization convention, that is popular in the physics literature is the
8
and defined as thediscretization,
mean value of denoted
lower and
Stratonovich-Fisk
byupper Riemann sum
p+ u (X(t + dt))
u
(X(t))
S
= uS+(X) dt + 2DSS (X) dB(t),dt +
(1.34a)
X(t + dt) dX(t)
= X(t)
p
p 2
p8
+ upper
2DS Riemann
(X(t + dt))
and defined as the mean value 2D
of lower
and
sum
S (X(t))
dt Z(t).
(1.34b)
2
uS (X(t)) + uS (X(t + dt))
X(t
+
dt)
=
X(t)
+
dt +
Similarly to Eq. (1.34), by fixing
p
p 2
p
2DS (X(t))
1 + 2DS (X(t + dt)) dt Z(t).
(1.34b)
u = uS + @x DS ,2
D = DS
(1.35)
2
Similarly
to an
Eq.Ito
(1.34),
fixing
one obtains
SDE by
that
is stochastically equivalent to Eqs. (1.31).
Stratonovich SDE
From a numerical perspective, the 1non-anticipatory Ito scheme (1.30b) is advantageous
= uSposition
+ @x Ddirectly
Dfrom
= Dthe
(1.35)
for it allows to compute theu new
previous one. For analytical
S,
S
2
calculations, the Stratonovich-Fisk scheme is somewhat preferable as it preserves the rules
9
of ordinary
one
obtains di↵erential
an Ito SDEcalculus,
that is stochastically
equivalent to Eqs. (1.31).
From a numerical perspective, the non-anticipatory Ito scheme (1.30b) is advantageous
0
dF
(X)
=
F
dX(t)
(1.36)
for it allows to compute the new position (X)
directly
from the previous one. For analytical
calculations, the Stratonovich-Fisk scheme is somewhat preferable as it preserves the rules
whilst
the backward
Itocalculus,
rule bears
9 certain conceptual advantageous from a physical point of
of
ordinary
di↵erential
view [DH09]. However, as mentioned before, in principle one can transform back and forth
0
between the di↵erent types of SDEs,
of the di↵erent discretization schemes
is
dF (X)i.e.,
= Fneither
(X) dX(t)
(1.36)
intrinsically superior.
Various
transformation
and their
generalizations
to higher
dimensions
whilst
the backward
Ito ruleformulas
bears certain
conceptual
advantageous
from aspace
physical
point of
each SDE formulation has advantages & disadvantages
X(t + dt) = X(t) + u(X(t)) dt + 2D(X(t)) dt Z(t),
(1.30b)
where, for each time step dt, a new random number Z(t) is drawn from a standard normal
4
Discretization
dilemma
To clarifyprocess
the importance
of
rules
when
dealing deterministic funcdistribution
.dt,
If athe
driving
B(t)
is discretization
Eq.
(1.30a)
were
a regular
where, for with
eachSDEs,
timeletstep
new
random
number
Z(t)
is
drawn
from
a
standard
normal
p
us
consider
a
simple
generalization
of
Eq.
(1.25),
where
drift
u
and
di↵usion
⌧(1.30a)
sin(⌦t),were
thena Eq.
(1.30a)
would reduce
to a standard
4tion, such as for example B(t) =
distribution
.
If
the
driving
process
B(t)
is
Eq.
regular
deterministic
funccoefficient D are position dependent.
Adopting the Ito convention, the corresponding SDE
p di↵erential
inhomogeneous
ordinary
(ODE).
For
ODEs,
it atypically
the importance
discretization
rules
when to
dealing
tion, such reads
asDiscretization
for exampledilemma
B(t) = To⌧ clarify
sin(⌦t),
thenequation
Eq.5 of
(1.30a)
would
reduce
standarddoes not matter SDEs,
whether
computes
the
coefficients
u(x)
and
D(x)
atu and
thedi↵usion
start
point
X(t) or the end
with
let usone
consider
a simple
generalization
of Eq.
(1.25),
where
p
inhomogeneous
ordinary
di↵erential
equation
(ODE).
For
ODEs,
it drift
typically
does not
matdX(t)
= u(X)5Adopting
dt + this
2D(X)
dB(t),
(1.30a)
D are
dependent.
the is
Ito⇤due
convention,
the
corresponding
SDE
point
X(t
+ position
dt).
Mathematically,
tothe
thestart
fact
that,X(t)
for
well-behaved
determinter whether
one
computes
the
coefficients
u(x)
and
D(x)
at
point
or
the
end
Ito coefficient
reads
driving
upper
andD(X)
lower
Riemann
sums
yield
the
same
value when letting
now on functions,
the ⇤-symbol
signals
that
is
to
be
evaluated
at
X(t).
The
simplest
point X(t where
+ istic
dt).from
Mathematically,
this
is due
to
the
fact
that,
for
well-behaved
determinp
dt
!
0.
If,
however,
B(t)
is
a
rapidly
process,
such
as the Brownian
numerical
integration
procedure
Eq.
(1.30a)
is sums
thevarying
Euler
scheme
dX(t)lower
=for
u(X)
dt
+ 2D(X)
⇤stochastic
dB(t),
(1.30a)
istic driving
functions,
upper
and
Riemann
yield stochastic
the
same
value
when
letting
motion,
then
the
corresponding
lower
sums
in general
do not converge
pand upper
p Riemann
dt ! 0. If,where
however,
B(t)
is
a
rapidly
varying
stochastic
process,
such
as
the
Brownian
from now
on+
the
thatdtD(X)
is to be evaluated
at X(t). The simplest
X(t
dt)⇤-symbol
= X(t) signals
+ u(X(t))
+ 2D(X(t))
dt Z(t),
(1.30b)
to
the
same
value
anymore.
Therefore,
when
dealing
with
SDEs
of the type (1.30a), it is
motion, then
the corresponding
lower for
and
sumsEuler
in general
numerical
integration procedure
Eq.upper
(1.30a)Riemann
is the stochastic
scheme do not converge
important
to explicitly
specify
the
integration
convention.
where,
for
each
time
step
dt, a new
random
number
Z(t)
isitdrawn
from
athe
standard
normal
pfunctions,
pitstraightforward
For
sufficiently
smooth
coefficient
functions,
is
to
backand
and
to the same
value
anymore.
Therefore,
when
dealing
with
SDEs
of
type
(1.30a),
it isback
For
sufficiently
smooth
coefficient
is
straightforward
to transform
transform
4
X(t
+
dt)
=
X(t)
+
u(X(t))
dt
+
2D(X(t))
dt
Z(t),
(1.30b)
distribution
. If the driving
process
B(t)backward
is Eq. (1.30a)
were
a regular
deterministic funcFor
instance,
the
so-called
Ito
SDE
with
coefficients
uaBgiven
and backward
D
by
B , denoted
p types
forth
between
di↵erent
types
of
SDEs
(see
Appendix
A).
That
is,
a
backward
Ito
important
to
explicitly
specify
the
integration
convention.
forth
between
di↵erent
of
SDEs
(see
Appendix
A).
That
is,
given
Ito
tion, such as for example B(t) = ⌧ sin(⌦t), then Eq. (1.30a) would reduce to a standard
p
SDE
with
(u
, DBSDE
)be
can
be transformed
into
aBastochastically
equivalent
where,
for
each
timecoefficients
step
dt,B a, D
new
random
number
Z(t)
is drawn
from
standard
normal
SDE
with
coefficients
(u
can
transformed
into
austochastically
equivalent
ItoSDE
SDE
For
instance,
the
so-called
backward
Ito
with
coefficients
and
DB
, denoted
by Ito
B )B
backward-Ito
inhomogeneous
ordinary
di↵erential
equation
(ODE).
For
ODEs,
it
typically
does
not
matdX(t)
=
u
(X)
dt
+
2D
(X)
•
dB(t),
(1.31a)
4
B
B
5 (u,
distribution
.
If
the
driving
process
B(t)
is
Eq.
(1.30a)
were
a
regular
deterministic
funcby
adapting
the
coe↵ficients
D)
accordingly.
More
precisely,
by
fixing
ter
whether
one
computes
the
coefficients
u(x)
and
D(x)
at
the
start
point
X(t)
or
the
end
by adapting the coe↵ficients p(u, D)paccordingly. More precisely, by fixing
tion,X(t
such+as
for
example
= this
⌧ sin(⌦t),
then
Eq.
(1.30a)
would
reduce to adeterminstandard (1.31a)
point
dt).
Mathematically,
is
to
the
fact
for well-behaved
dX(t)
= B(t)
uB (X)
dt
+ due2D
•that,
dB(t),
B
6(X)
inhomogeneous
di↵erential
equation
typically
does
notletting
matis driving
defined
asordinary
the upper
upper
Riemann
u =sum
u(ODE).
@xFor
Dyield
, theDitsame
=
Dvalue
(1.33)
B +
B ODEs,
B
istic
functions,
and
lower
Riemann
sums
when
u
=
u
+
@
D
,
D
=
D
(1.33)
5
B u(x) and
x B
B
ter
whether
one computes
the
coefficients
D(x) atprocess,
the
start
point
X(t)
or
the
end
6
dt
!
0.
If,
however,
B(t)
is
a
rapidly
varying
stochastic
such
as
the
Brownian
p
p
is defined aspoint
theX(t
upper
Riemann
sum
+
dt).
Mathematically,
this
is
due
to
the
fact
that,
for
well-behaved
determinone
obtains
an
Ito
SDE
that
is
stochastically
equivalent
to
Eqs.
(1.31).
X(t
dt) = X(t)lower
+ uand
+Riemann
dt)) dt +
dt))
dt Z(t).
(1.31b)
motion, then
the +
corresponding
upper
sums 2D
in general
do+not
converge
B (X(t
B (X(t
driving
functions,
upper
and is
lower
Riemann
sums
yield
same
when
letting
one
obtains
an
Ito
SDE
that
stochastically
equivalent
tovalue
Eqs.
(1.31).
p
p
Another
discretization
convention,
that
is the
popular
in
the
physics
toistic
the
same value
anymore.
Therefore,
when
dealing
with
SDEs
of
the
type
(1.30a),
it is literature is the
dtAnother
! 0.=If,X(t)
however,
is a+convention,
rapidly
stochastic
process,
the physics
Brownian (1.31b)
X(timportant
+
dt)
+ uBB(t)
(X(t
dt))
dtvarying
+ denoted
2D
+ dt)) such
dt as
Z(t).
B (X(t
discretization
that
is
popular
in
the
literature is
Stratonovich-Fisk
discretization,
by
to
explicitly
specify
the
integration
convention.
Unlikethen
Eq.the(1.30b),
the backward
Ito Riemann
scheme sums
(1.31b)
is implicit.
To reemphasize,
forthe
same
motion,
corresponding
lower and upper
in general
do not converge
For
instance,
the
so-called
backward
Ito
SDE
with
coefficients
u
and
D
,
denoted
by
B
B
Stratonovich-Fisk
discretization,
denoted
by
pSDEs
functions
u
⌘
u
and
D
⌘
D
,
Eqs.
(1.30)
and
(1.31)
produce
trajectories
to
the
same
value
anymore.
Therefore,
when
dealing
with
of
the
type
(1.30a), it
isfor same that follow
B
B
Unlike Eq. (1.30b), the backward Ito scheme
(1.31b)
is
implicit.
To
reemphasize,
dX(t)
=
u
(X)
dt
+
2D
(X)
dB(t),
(1.34a)
p
S
S
7
p
important
to
explicitly
specify
the
integration
convention.
di↵erent
statistics
. ,The
theB (X)
Ito •formula
(1.29)
for a nonlinear
transformation of
dX(t)
=
uBanalog
(X)
dt +ofand
2D
dB(t),
(1.31a)
functions
u
⌘
u
and
D
⌘
D
Eqs.
(1.30)
(1.31)
produce
trajectories
that
follow
B
B
dX(t)
=
u
(X)
dt
+
2D
(X)
dB(t),
(1.34a)
For instance, the so-called backward SIto SDE with coefficients
uB and DB , denoted
by
Stratonovich
S
8
7and defined asSDE
the
backward-Ito
reads
simply
the
mean
value
of
lower
and
upper
Riemann
sum
6 formula (1.29) for a nonlinear transformation of
di↵erent statistics . The analog of the Ito
Summary
the
p
is defined as the upper Riemann sum
dX(t) = uB (X) dt + 2DB (X) • dB(t),
8 (1.31a)
backward-Ito
SDE
simply
p
and defined
asreads
the mean
value
upper
Riemann
sum
0 of lowerpand
00
u
(X(t))
+
u
(X(t
+
dt))
S
S
(X)X(t
F (X)
•
dX
D
F
(X)
dt
B
X(t + dt)dF
= X(t)
+ u=B+(X(t
+=
dt))
dt
+
2D
(X(t
+
dt))
dt Z(t).
dt)
X(t)
+
dt +(1.31b)
B
6
p
is defined as the upper
Riemann sum 00
2
0
0 p
00 p
0
dF
(X)
=
F
(X)
•
dX
D
F
(X)
dt
u
(X(t))
+
u
(X(t
+
dt))
=
[u
F
(X)
D
F
(X)]
dt
+
F
(X)
2D
• dB(t).
B
S
S
B
B
Bsame
p
p
p
2D
(X(t))
+
2D
(X(t
+
dt))
Unlike Eq. (1.30b),
the
backward
Ito
scheme
(1.31b)
is
implicit.
To
reemphasize,
for
S
S
p
dt)
X(t)00+dt + 2DB (X(t
X(t + X(t
dt) =+
X(t)
u=
+ dt)) dt Z(t). dt +dt(1.31b)
Z(t).
(1.34b)
B (X(t + dt))
0 +
0 produce
2
functions u =
⌘ uB[uand
D
⌘
D
,
Eqs.
(1.30)
and
(1.31)
trajectories
that
follow
B
(1.32)
F
(X)
D
F
(X)]
dt
+
F
(X)
2D
•
dB(t).
B
B
B
p
p 2
7
di↵erent
statistics
. The
analog of the
Ito formula
(1.29)
for
transformation
of
p for same
+is implicit.
2DaS nonlinear
(X(t
+ dt))
Unlike
Eq.
(1.30b),
the backward
Ito2D
scheme
(1.31b)
To reemphasize,
S (X(t))
(1.32) (1.34b)
4
2
Similarly
to
Eq.
(1.34),
by
fixing
the
backward-Ito
SDE
reads
simply
That
is,
a
Gaussian
distribution
with
mean
µ
=
0
and
variance
=
1.
dt
Z(t).
functions u ⌘ uB and D ⌘ DB , Eqs. (1.30) and (1.31) produce trajectories that follow
5
7
2
Fokker-Planck equations
1.2.2 Fokker-Planck
Fokker-Planckequations
equations
1.2.2
ther types of SDEs can be transformed into an equivalent Ito SDE, we shall foc
Since other types of SDEs can be transformed into an equivalent Ito SDE, we shall focus
other types how
of SDEs
can can
be transformed
into
an equivalent Ito SDE,
we shall (FPE)
focus
part onSince
discussing
one
derive
a
Fokker-Planck
equation
for
th
in this
thispart
partonondiscussing
discussinghow
howone
onecan
canderive
derive
a Fokker-Planck
equation
(FPE)
for
the
in
a
Fokker-Planck
equation
(FPE)
for
the
r-Planck
equations
lity density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability
density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability density function (PDF) p(t, x) for a process X(t) described by the Ito SDE
p
pp
dX(t)= =
u(X)
dt
+ 2D(X)
2D(X)
⇤ dB(t). Ito SDE, we(1.37)
(1.37) focus
of SDEs can bedX(t)
transformed
into
an
equivalent
shall
dX(t)
u(X)
dt
+
⇤
dB(t).
= u(X) dt + 2D(X) ⇤ dB(t).
(1.3
scussing
one
can
derive
a Fokker-Planck equation (FPE) for the
Thehow
PDFcan
can
formally
definedbyby
The
PDF
bebeformally
defined
DF
can be formally
y function
(PDF)defined
p(t, x) by
for p(t,
a x)
process
X(t)
described by the Ito(1.38)
SDE
=
E[
(X(t)
x)].
p(t, x) = E[ (X(t) x)].
(1.38)
p
p(t,
x)
E[p, p,we
(X(t)
x)].
(1.3
To
obtain
an
evolution
equation
we
consider
To obtain an evolution equation=forfor
consider
dX(t) = u(X) dt + 2D(X) ⇤ dB(t).
(1.37)
ain an evolution equation for
ormally defined by
d d
p=
E[ (X(t)
(X(t) x)].x)].
E[consider
t=
p,@t@pwe
dt dt
(1.39)
(1.39)
To
to to
thethe
di↵erential
d[ (X(t)
andand
findfind
Toevaluate
evaluatethe
therhs.,
rhs.,weweapply
applyIto’s
Ito’s
formula
di↵erential
d[ (X(t)x)]]x)]]
dformula
@t p = E[
(X(t) x)].
⇥⇥
⇤ ⇤
2 2
E[d[
x)]]
E E(@dt
x))x))
dXdX
+ D(X)
@X @(X(t)
x) dt
E[d[(X
(X =
x)]]
(@ (X(Xx)].
+ D(X)
x) dt
p(t,
x)
E[==(X(t)
X (X(t)
⇥ ⇥X X
⇤ ⇤
2 2
== E E(@(@
u(X)
+ D(X)
@X @(X(t)
x) x)
dt. dt.
X X(X(X x))
x))
u(X)
+ D(X)
X (X(t)
(1.3
(1.38)
uate the rhs., we apply Ito’s formula to the di↵erential d[ (X(t) x)]] and find
ution equation
forused
p,that
we
consider
Here,
⇤ dB]
0, 0,
which
follows
from
thethe
non-anticipatory
Here,we
wehave
have
used
thatE[g(X(t))
E[g(X(t))
⇤ dB]= =
which
follows
from
non-anticipatory
⇥
⇤
definition
byby
recalling
that
definitionofofthe
theIto
Itointegral.
integral.Furthermore,
Furthermore,
recalling
that 2
E[d[ (X x)]] = E (@X (X x)) dX + D(X) @X (X(t) x) dt
⇥d @X (X x) = @x (X x),
⇤
(1.40)
2
@X
x)x)].
=u(X)
@x (X
x),
(1.40)(1.39)
@t p =
= E[E (@X(X(t)
(X(X x))
+ D(X)
@X (X(t) x) dt.
we
wemay
maywrite
write
dt
we have used that E[g(X(t)) ⇥⇤ dB] = 0, which follows from ⇤ the non-anticipato
Fokker-Planck equations
1.2.2 Fokker-Planck
Fokker-Planckequations
equations
1.2.2
ther types of SDEs can be transformed into an equivalent Ito SDE, we shall foc
Since other types of SDEs can be transformed into an equivalent Ito SDE, we shall focus
other types how
of SDEs
can can
be transformed
into
an equivalent Ito SDE,
we shall (FPE)
focus
part onSince
discussing
one
derive
a
Fokker-Planck
equation
for
th
in this
thispart
partonondiscussing
discussinghow
howone
onecan
canderive
derive
a Fokker-Planck
equation
(FPE)
for
the
in
a
Fokker-Planck
equation
(FPE)
for
the
r-Planck
equations
lity density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability
density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability density function (PDF) p(t, x) for a process X(t) described by the Ito SDE
p
pp
dX(t)= =
u(X)
dt
+ 2D(X)
2D(X)
⇤ dB(t). Ito SDE, we(1.37)
(1.37) focus
of SDEs can bedX(t)
transformed
into
an
equivalent
shall
dX(t)
u(X)
dt
+
⇤
dB(t).
= u(X) dt + 2D(X) ⇤ dB(t).
(1.3
scussing
one
can
derive
a Fokker-Planck equation (FPE) for the
Thehow
PDFcan
can
formally
definedbyby
The
PDF
bebeformally
defined
DF
can be formally
y function
(PDF)defined
p(t, x) by
for p(t,
a x)
process
X(t)
described by the Ito(1.38)
SDE
=
E[
(X(t)
x)].
p(t, x) = E[ (X(t) x)].
(1.38)
p
p(t,
x)
E[p, p,we
(X(t)
x)].
(1.3
To
obtain
an
evolution
equation
we
consider
To obtain an evolution equation=forfor
consider
dX(t) = u(X) dt + 2D(X) ⇤ dB(t).
(1.37)
ain an evolution equation for
ormally defined by
d d
p=
E[ (X(t)
(X(t) x)].x)].
E[consider
t=
p,@t@pwe
dt dt
(1.39)
(1.39)
To
to to
thethe
di↵erential
d[ (X(t)
andand
findfind
Toevaluate
evaluatethe
therhs.,
rhs.,weweapply
applyIto’s
Ito’s
formula
di↵erential
d[ (X(t)x)]]x)]]
dformula
@t p = E[
(X(t) x)].
⇥⇥
⇤ ⇤
2 2
E[d[
x)]]
E E(@dt
x))x))
dXdX
+ D(X)
@X @(X(t)
x) dt
E[d[(X
(X =
x)]]
(@ (X(Xx)].
+ D(X)
x) dt
p(t,
x)
E[==(X(t)
X (X(t)
⇥ ⇥X X
⇤ ⇤
2 2
== E E(@(@
u(X)
+ D(X)
@X @(X(t)
x) x)
dt. dt.
X X(X(X x))
x))
u(X)
+ D(X)
X (X(t)
(1.3
(1.38)
uate the rhs., we apply Ito’s formula to the di↵erential d[ (X(t) x)]] and find
ution equation
forused
p,that
we
consider
Here,
⇤ dB]
0, 0,
which
follows
from
thethe
non-anticipatory
Here,we
wehave
have
used
thatE[g(X(t))
E[g(X(t))
⇤ dB]= =
which
follows
from
non-anticipatory
⇥
⇤
definition
byby
recalling
that
definitionofofthe
theIto
Itointegral.
integral.Furthermore,
Furthermore,
recalling
that 2
E[d[ (X x)]] = E (@X (X x)) dX + D(X) @X (X(t) x) dt
⇥d @X (X x) = @x (X x),
⇤
(1.40)
2
@X
x)x)].
=u(X)
@x (X
x),
(1.40)(1.39)
@t p =
= E[E (@X(X(t)
(X(X x))
+ D(X)
@X (X(t) x) dt.
we
wemay
maywrite
write
dt
we have used that E[g(X(t)) ⇥⇤ dB] = 0, which follows from ⇤ the non-anticipato
Fokker-Planck equations
1.2.2 Fokker-Planck
Fokker-Planckequations
equations
1.2.2
ther types of SDEs can be transformed into an equivalent Ito SDE, we shall foc
Since other types of SDEs can be transformed into an equivalent Ito SDE, we shall focus
other types how
of SDEs
can can
be transformed
into
an equivalent Ito SDE,
we shall (FPE)
focus
part onSince
discussing
one
derive
a
Fokker-Planck
equation
for
th
in this
thispart
partonondiscussing
discussinghow
howone
onecan
canderive
derive
a Fokker-Planck
equation
(FPE)
for
the
in
a
Fokker-Planck
equation
(FPE)
for
the
r-Planck
equations
lity density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability
density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability density function (PDF) p(t, x) for a process X(t) described by the Ito SDE
p
pp
dX(t)= =
u(X)
dt
+ 2D(X)
2D(X)
⇤ dB(t). Ito SDE, we(1.37)
(1.37) focus
of SDEs can bedX(t)
transformed
into
an
equivalent
shall
dX(t)
u(X)
dt
+
⇤
dB(t).
= u(X) dt + 2D(X) ⇤ dB(t).
(1.3
scussing
one
can
derive
a Fokker-Planck equation (FPE) for the
Thehow
PDFcan
can
formally
definedbyby
The
PDF
bebeformally
defined
DF
can be formally
y function
(PDF)defined
p(t, x) by
for p(t,
a x)
process
X(t)
described by the Ito(1.38)
SDE
=
E[
(X(t)
x)].
p(t, x) = E[ (X(t) x)].
(1.38)
p
p(t,
x)
E[p, p,we
(X(t)
x)].
(1.3
To
obtain
an
evolution
equation
we
consider
To obtain an evolution equation=forfor
consider
dX(t) = u(X) dt + 2D(X) ⇤ dB(t).
(1.37)
ain an evolution equation for
ormally defined by
d d
p=
E[ (X(t)
(X(t) x)].x)].
E[consider
t=
p,@t@pwe
dt dt
(1.39)
(1.39)
To
to to
thethe
di↵erential
d[ (X(t)
andand
findfind
Toevaluate
evaluatethe
therhs.,
rhs.,weweapply
applyIto’s
Ito’s
formula
di↵erential
d[ (X(t)x)]]x)]]
dformula
@t p = E[
(X(t) x)].
⇥⇥
⇤ ⇤
2 2
E[d[
x)]]
E E(@dt
x))x))
dXdX
+ D(X)
@X @(X(t)
x) dt
E[d[(X
(X =
x)]]
(@ (X(Xx)].
+ D(X)
x) dt
p(t,
x)
E[==(X(t)
X (X(t)
⇥ ⇥X X
⇤ ⇤
2 2
== E E(@(@
u(X)
+ D(X)
@X @(X(t)
x) x)
dt. dt.
X X(X(X x))
x))
u(X)
+ D(X)
X (X(t)
(1.3
(1.38)
uate the rhs., we apply Ito’s formula to the di↵erential d[ (X(t) x)]] and find
ution equation
forused
p,that
we
consider
Here,
⇤ dB]
0, 0,
which
follows
from
thethe
non-anticipatory
Here,we
wehave
have
used
thatE[g(X(t))
E[g(X(t))
⇤ dB]= =
which
follows
from
non-anticipatory
⇥
⇤
definition
byby
recalling
that
definitionofofthe
theIto
Itointegral.
integral.Furthermore,
Furthermore,
recalling
that 2
E[d[ (X x)]] = E (@X (X x)) dX + D(X) @X (X(t) x) dt
⇥d @X (X x) = @x (X x),
⇤
(1.40)
2
@X
x)x)].
=u(X)
@x (X
x),
(1.40)(1.39)
@t p =
= E[E (@X(X(t)
(X(X x))
+ D(X)
@X (X(t) x) dt.
we
wemay
maywrite
write
dt
we have used that E[g(X(t)) ⇥⇤ dB] = 0, which follows from ⇤ the non-anticipato
Fokker-Planck equations
1.2.2 Fokker-Planck
Fokker-Planckequations
equations
1.2.2
ther types of SDEs can be transformed into an equivalent Ito SDE, we shall foc
Since other types of SDEs can be transformed into an equivalent Ito SDE, we shall focus
other types how
of SDEs
can can
be transformed
into
an equivalent Ito SDE,
we shall (FPE)
focus
part onSince
discussing
one
derive
a
Fokker-Planck
equation
for
th
in this
thispart
partonondiscussing
discussinghow
howone
onecan
canderive
derive
a Fokker-Planck
equation
(FPE)
for
the
in
a
Fokker-Planck
equation
(FPE)
for
the
r-Planck
equations
lity density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability
density
function
(PDF)
p(t,
x)
for
a
process
X(t)
described
by
the
Ito
SDE
probability density function (PDF) p(t, x) for a process X(t) described by the Ito SDE
p
pp
dX(t)= =
u(X)
dt
+ 2D(X)
2D(X)
⇤ dB(t). Ito SDE, we(1.37)
(1.37) focus
of SDEs can bedX(t)
transformed
into
an
equivalent
shall
dX(t)
u(X)
dt
+
⇤
dB(t).
= u(X) dt + 2D(X) ⇤ dB(t).
(1.3
scussing
one
can
derive
a Fokker-Planck equation (FPE) for the
Thehow
PDFcan
can
formally
definedbyby
The
PDF
bebeformally
defined
DF
can be formally
y function
(PDF)defined
p(t, x) by
for p(t,
a x)
process
X(t)
described by the Ito(1.38)
SDE
=
E[
(X(t)
x)].
p(t, x) = E[ (X(t) x)].
(1.38)
p
p(t,
x)
E[p, p,we
(X(t)
x)].
(1.3
To
obtain
an
evolution
equation
we
consider
To obtain an evolution equation=forfor
consider
dX(t) = u(X) dt + 2D(X) ⇤ dB(t).
(1.37)
ain an evolution equation for
ormally defined by
d d
p=
E[ (X(t)
(X(t) x)].x)].
E[consider
t=
p,@t@pwe
dt dt
(1.39)
(1.39)
To
to to
thethe
di↵erential
d[ (X(t)
andand
findfind
Toevaluate
evaluatethe
therhs.,
rhs.,weweapply
applyIto’s
Ito’s
formula
di↵erential
d[ (X(t)x)]]x)]]
dformula
@t p = E[
(X(t) x)].
⇥⇥
⇤ ⇤
2 2
E[d[
x)]]
E E(@dt
x))x))
dXdX
+ D(X)
@X @(X(t)
x) dt
E[d[(X
(X =
x)]]
(@ (X(Xx)].
+ D(X)
x) dt
p(t,
x)
E[==(X(t)
X (X(t)
⇥ ⇥X X
⇤ ⇤
2 2
== E E(@(@
u(X)
+ D(X)
@X @(X(t)
x) x)
dt. dt.
X X(X(X x))
x))
u(X)
+ D(X)
X (X(t)
(1.3
(1.38)
uate the rhs., we apply Ito’s formula to the di↵erential d[ (X(t) x)]] and find
ution equation
forused
p,that
we
consider
Here,
⇤ dB]
0, 0,
which
follows
from
thethe
non-anticipatory
Here,we
wehave
have
used
thatE[g(X(t))
E[g(X(t))
⇤ dB]= =
which
follows
from
non-anticipatory
⇥
⇤
definition
byby
recalling
that
definitionofofthe
theIto
Itointegral.
integral.Furthermore,
Furthermore,
recalling
that 2
E[d[ (X x)]] = E (@X (X x)) dX + D(X) @X (X(t) x) dt
⇥d @X (X x) = @x (X x),
⇤
(1.40)
2
@X
x)x)].
=u(X)
@x (X
x),
(1.40)(1.39)
@t p =
= E[E (@X(X(t)
(X(X x))
+ D(X)
@X (X(t) x) dt.
we
wemay
maywrite
write
dt
we have used that E[g(X(t)) ⇥⇤ dB] = 0, which follows from ⇤ the non-anticipato
E[d[⇥ (X
] = E (@X
⇥
= E (@X
2
⇤ + D(X) @X
x)]] = E (@X2⇥ (X x)) dX
(X(t) ⇤x) dt
2
(XE[d[x))
+ D(X)
x)
⇥d= @EX(@(X(t)
⇤
(XdX x)]]
x))dtdX
+ D(X) @X (X(t)
x) dt
X (X
2
⇤
⇥ (X
⇤ x) dt. (1.39)
2
=
(X(t)
x)].
= E[
ED(X)
(@
x))
u(X)
+
D(X)
2 @X (X(t)
X
t p u(X)
(X @x))
+
@
(X(t)
x)
dt.
= E (@
x)) u(X) + D(X) @ (X(t) x) dt.
X X (X
dt
X
t E[g(X(t))
⇤ dB]
=used
0,E[g(X(t))
which
follows
from
the
non-anticipatory
Here,used
we
have
that E[g(X(t))
⇤ dB]
0,which
which follows
from
the non-anticipatory
we
have
that
⇤ dB]
= =0,
follows
from
the non-anticipato
s., we
apply
Ito’s
formula
to the di↵erential
x)]] and find
gral.
Furthermore,
by Ito
recalling
that
definition
of the
integral.
Furthermore,
by recalling thatd[ (X(t)
on of the Ito integral. Furthermore, by recalling that
⇤ (1.40)
(1.40)
x),
2
x)]] = E (@X @(X
x))x)dX
+@xD(X)
@x),
x) dt
X (X(t)
(X
=
(X
(1.4
X
⇥
⇤
we may write
2
= E (@X (X ⇥ x)) u(X) + D(X) @X (X(t) ⇤x) dt.
⇥
⇤
write
2
E[d[ x))
(X u(X)
x)]]+ =
E (@x@x(X(t)
(X x))
= E ( @x (X
D(X)
x) u(X)
dt + D(X) @x2 (X(t) x) dt
⇤
=@x2 E[D(X)
@=
(X(X(t)
x) u(X)]
dt
(X(t) x)] dt.
= that
@x E[ E[g(X(t))
(X x) u(X)] ⇤
dt⇥+
x)]
dt.+ @x2 E[D(X)
x E[0,
ed
dB]
which
follows
from
2 the non-anticipatory
@X (X
x) = ⇥ @x (X
x), @X (X
x) =
@x (X
E[d[ (X
x)]] = E ( @x (X x)) u(X) + D(X) @x (X(t) x) dt
o the
integral.
Furthermore,
recalling that 2
Using
another
property of the by
-function
of
-function
=
@x E[ (X x) u(X)] dt + @x E[D(X) (X(t) x)] dt.
f (y) (y
x) = f (x) (y
x) f (y) (y
x) = f (x) (y
@
(X
x)
=
@
(X
X
x
nother property of the -function
x),
x)
(1.41)
(1.41)
(1.40)
we obtain
2x)
f
(y)
x)
=
f
(x)
(y
2 (y
E[d[
(X
x)]]
=
@
E[
(X
x)
u(x)]
dt
+
@
x)] dt
x
x E[D(x) (X(t)
=
@x E[ (X x) u(x)] dt + @x E[D(x) (X(t) x)] dt
2
2 @x {u(x) E[ (X
=
x)]}
dt
+
@
⇥
x
=
@x {u(x) E[ (X x)]} dt + @x {D(x) E[ (X(t) x)]} dt 2{D(x) E[ (X(t) ⇤x)]} dt
ain
x)]]@x {u(x)
= pE @(x[D(x)p]}
@x (X
x)) u(X) + D(X) @ (X(t) x) dt
=
=
dt. @x {u(x) p @x [D(x)p]} dt. x
2
=
@
E[
(X
x)
u(X)]
dt
+
@
E[D(X)
(X(t)
x)]
dt.
2 (or Smoluchowski)
x
Combining
this
with
Eq.
(1.39)
yields
the
Fokker-Planck
equation
x
E[d[
(X
x)]]
=
@
E[
(X
x)
u(x)]
dt
+
@
E[D(x)
(X(t)
x)]
dt
. (1.39) yields the Fokker-Planck
(or Smoluchowski) equation
x
x
2
@
p
=
@
{u(x)
p
@
[D(x)p]}
. {D(x) E[ (X(t)
=
@
{u(x)
E[
(X
x)]}
dt
+
@
t .
x
x
x
x
@
p
=
@
{u(x)
p
@
[D(x)p]}
(1.42)
t
x
perty
of xthe -function
=
@x {u(x) p @x [D(x)p]} dt.
(1.42)dt
x)]}
(1.4
E[d[⇥ (X
] = E (@X
⇥
= E (@X
2
⇤ + D(X) @X
x)]] = E (@X2⇥ (X x)) dX
(X(t) ⇤x) dt
2
(XE[d[x))
+ D(X)
x)
⇥d= @EX(@(X(t)
⇤
(XdX x)]]
x))dtdX
+ D(X) @X (X(t)
x) dt
X (X
2
⇤
⇥ (X
⇤ x) dt. (1.39)
2
=
(X(t)
x)].
= E[
ED(X)
(@
x))
u(X)
+
D(X)
2 @X (X(t)
X
t p u(X)
(X @x))
+
@
(X(t)
x)
dt.
= E (@
x)) u(X) + D(X) @ (X(t) x) dt.
X X (X
dt
X
t E[g(X(t))
⇤ dB]
=used
0,E[g(X(t))
which
follows
from
the
non-anticipatory
Here,used
we
have
that E[g(X(t))
⇤ dB]
0,which
which follows
from
the non-anticipatory
we
have
that
⇤ dB]
= =0,
follows
from
the non-anticipato
s., we
apply
Ito’s
formula
to the di↵erential
x)]] and find
gral.
Furthermore,
by Ito
recalling
that
definition
of the
integral.
Furthermore,
by recalling thatd[ (X(t)
on of the Ito integral. Furthermore, by recalling that
⇤ (1.40)
(1.40)
x),
2
x)]] = E (@X @(X
x))x)dX
+@xD(X)
@x),
x) dt
X (X(t)
(X
=
(X
(1.4
X
⇥
⇤
we may write
2
= E (@X (X ⇥ x)) u(X) + D(X) @X (X(t) ⇤x) dt.
⇥
⇤
write
2
E[d[ x))
(X u(X)
x)]]+ =
E (@x@x(X(t)
(X x))
= E ( @x (X
D(X)
x) u(X)
dt + D(X) @x2 (X(t) x) dt
⇤
=@x2 E[D(X)
@=
(X(X(t)
x) u(X)]
dt
(X(t) x)] dt.
= that
@x E[ E[g(X(t))
(X x) u(X)] ⇤
dt⇥+
x)]
dt.+ @x2 E[D(X)
x E[0,
ed
dB]
which
follows
from
2 the non-anticipatory
@X (X
x) = ⇥ @x (X
x), @X (X
x) =
@x (X
E[d[ (X
x)]] = E ( @x (X x)) u(X) + D(X) @x (X(t) x) dt
o the
integral.
Furthermore,
recalling that 2
Using
another
property of the by
-function
of
-function
=
@x E[ (X x) u(X)] dt + @x E[D(X) (X(t) x)] dt.
f (y) (y
x) = f (x) (y
x) f (y) (y
x) = f (x) (y
@
(X
x)
=
@
(X
X
x
nother property of the -function
x),
x)
(1.41)
(1.41)
(1.40)
we obtain
2x)
f
(y)
x)
=
f
(x)
(y
2 (y
E[d[
(X
x)]]
=
@
E[
(X
x)
u(x)]
dt
+
@
x)] dt
x
x E[D(x) (X(t)
=
@x E[ (X x) u(x)] dt + @x E[D(x) (X(t) x)] dt
2
2 @x {u(x) E[ (X
=
x)]}
dt
+
@
⇥
x
=
@x {u(x) E[ (X x)]} dt + @x {D(x) E[ (X(t) x)]} dt 2{D(x) E[ (X(t) ⇤x)]} dt
ain
x)]]@x {u(x)
= pE @(x[D(x)p]}
@x (X
x)) u(X) + D(X) @ (X(t) x) dt
=
=
dt. @x {u(x) p @x [D(x)p]} dt. x
2
=
@
E[
(X
x)
u(X)]
dt
+
@
E[D(X)
(X(t)
x)]
dt.
2 (or Smoluchowski)
x
Combining
this
with
Eq.
(1.39)
yields
the
Fokker-Planck
equation
x
E[d[
(X
x)]]
=
@
E[
(X
x)
u(x)]
dt
+
@
E[D(x)
(X(t)
x)]
dt
. (1.39) yields the Fokker-Planck
(or Smoluchowski) equation
x
x
2
@
p
=
@
{u(x)
p
@
[D(x)p]}
. {D(x) E[ (X(t)
=
@
{u(x)
E[
(X
x)]}
dt
+
@
t .
x
x
x
x
@
p
=
@
{u(x)
p
@
[D(x)p]}
(1.42)
t
x
perty
of xthe -function
=
@x {u(x) p @x [D(x)p]} dt.
(1.42)dt
x)]}
(1.4
E[d[⇥ (X
] = E (@X
⇥
= E (@X
2
⇤ + D(X) @X
x)]] = E (@X2⇥ (X x)) dX
(X(t) ⇤x) dt
2
(XE[d[x))
+ D(X)
x)
⇥d= @EX(@(X(t)
⇤
(XdX x)]]
x))dtdX
+ D(X) @X (X(t)
x) dt
X (X
2
⇤
⇥ (X
⇤ x) dt. (1.39)
2
=
(X(t)
x)].
= E[
ED(X)
(@
x))
u(X)
+
D(X)
2 @X (X(t)
X
t p u(X)
(X @x))
+
@
(X(t)
x)
dt.
= E (@
x)) u(X) + D(X) @ (X(t) x) dt.
X X (X
dt
X
t E[g(X(t))
⇤ dB]
=used
0,E[g(X(t))
which
follows
from
the
non-anticipatory
Here,used
we
have
that E[g(X(t))
⇤ dB]
0,which
which follows
from
the non-anticipatory
we
have
that
⇤ dB]
= =0,
follows
from
the non-anticipato
s., we
apply
Ito’s
formula
to the di↵erential
x)]] and find
gral.
Furthermore,
by Ito
recalling
that
definition
of the
integral.
Furthermore,
by recalling thatd[ (X(t)
on of the Ito integral. Furthermore, by recalling that
⇤ (1.40)
(1.40)
x),
2
x)]] = E (@X @(X
x))x)dX
+@xD(X)
@x),
x) dt
X (X(t)
(X
=
(X
(1.4
X
⇥
⇤
we may write
2
= E (@X (X ⇥ x)) u(X) + D(X) @X (X(t) ⇤x) dt.
⇥
⇤
write
2
E[d[ x))
(X u(X)
x)]]+ =
E (@x@x(X(t)
(X x))
= E ( @x (X
D(X)
x) u(X)
dt + D(X) @x2 (X(t) x) dt
⇤
=@x2 E[D(X)
@=
(X(X(t)
x) u(X)]
dt
(X(t) x)] dt.
= that
@x E[ E[g(X(t))
(X x) u(X)] ⇤
dt⇥+
x)]
dt.+ @x2 E[D(X)
x E[0,
ed
dB]
which
follows
from
2 the non-anticipatory
@X (X
x) = ⇥ @x (X
x), @X (X
x) =
@x (X
E[d[ (X
x)]] = E ( @x (X x)) u(X) + D(X) @x (X(t) x) dt
o the
integral.
Furthermore,
recalling that 2
Using
another
property of the by
-function
of
-function
=
@x E[ (X x) u(X)] dt + @x E[D(X) (X(t) x)] dt.
f (y) (y
x) = f (x) (y
x) f (y) (y
x) = f (x) (y
@
(X
x)
=
@
(X
X
x
nother property of the -function
x),
x)
(1.41)
(1.41)
(1.40)
we obtain
2x)
f
(y)
x)
=
f
(x)
(y
2 (y
E[d[
(X
x)]]
=
@
E[
(X
x)
u(x)]
dt
+
@
x)] dt
x
x E[D(x) (X(t)
=
@x E[ (X x) u(x)] dt + @x E[D(x) (X(t) x)] dt
2
2 @x {u(x) E[ (X
=
x)]}
dt
+
@
⇥
x
=
@x {u(x) E[ (X x)]} dt + @x {D(x) E[ (X(t) x)]} dt 2{D(x) E[ (X(t) ⇤x)]} dt
ain
x)]]@x {u(x)
= pE @(x[D(x)p]}
@x (X
x)) u(X) + D(X) @ (X(t) x) dt
=
=
dt. @x {u(x) p @x [D(x)p]} dt. x
2
=
@
E[
(X
x)
u(X)]
dt
+
@
E[D(X)
(X(t)
x)]
dt.
2 (or Smoluchowski)
x
Combining
this
with
Eq.
(1.39)
yields
the
Fokker-Planck
equation
x
E[d[
(X
x)]]
=
@
E[
(X
x)
u(x)]
dt
+
@
E[D(x)
(X(t)
x)]
dt
. (1.39) yields the Fokker-Planck
(or Smoluchowski) equation
x
x
2
@
p
=
@
{u(x)
p
@
[D(x)p]}
. {D(x) E[ (X(t)
=
@
{u(x)
E[
(X
x)]}
dt
+
@
t .
x
x
x
x
@
p
=
@
{u(x)
p
@
[D(x)p]}
(1.42)
t
x
perty
of xthe -function
=
@x {u(x) p @x [D(x)p]} dt.
(1.42)dt
x)]}
(1.4
Ito-FPE
=
=
@x {u(x) E[ (X x)]} dt + @x {D(x) E[ (X(t)
@x {u(x) p @x [D(x)p]} dt.
x)]} dt
Combining this with Eq. (1.39) yields the Fokker-Planck (or Smoluchowski) equation
@t p =
Backward-Ito FPE
@x {u(x) p
@x [D(x)p]} .
(1.42)
10
For comparison, an analogous calculation for the backward-Ito SDE
p
dX(t) = uB (X) dt + 2DB (X) • dB(t),
(1.43)
gives
@t p = @functions,
@x p] .
x [uB (x) p itDis
B (x)
For sufficiently smooth coefficient
straightforward
to transform(1.44)
back and
forth between
di↵erent types of SDEs (see Appendix A). That is, a given backward Ito
Compared with the Ito FPE (1.42), the di↵usion coefficient DB now enters in front of the
SDE with
coefficients
(uB ,however,
DB ) canthat
be transformed
intoFPEs
a stochastically
equivalent
gradient
@x p. Note,
the two di↵erent
coincide if one
identifiesIto
theSDE
by adapting
the coe↵ficients
(u,: D) accordingly. More precisely, by fixing
coefficients
as in Eq. (1.33).
A summary of Fokker-Planck equations for the three di↵erent stochastic integral conu = uB +and
@x D
D =inDarbitrary
(1.33)
ventions (Ito, Strantonovich-Fisk
backward-Ito)
space dimensions can be
B,
B
found in Appendix A.
one obtains an Ito SDE that is stochastically equivalent to Eqs. (1.31).
Another discretization convention, that is popular in the physics literature is the
Stratonovich-Fisk discretization, denoted by