Transcript On the Mathematics and Economics Assumptions of Continuous

On the Mathematics and
Economics Assumptions of
Continuous-Time Models
BaoheWang
[email protected]
3.1 Introduction
• This chapter attempts to
(1) bridge the gap by using only elementary
probability theory and calculus to derive the
basic theorems required for continuous time
analysis.
(2) make explicit the economics assumptions
implicitly embedded in the mathematical
assumption.
• The general approach is to keep the assumption
as weak as possible.
• But we need make a choice between the losses
in generality and the reduction in mathematical
complexity.
• The substantive contributions of continuous time
analysis to financial economic theory:
(1) trading take place continuously in time
(2) the underlying stochastic variables follow
diffusion type motions with continuous sample
paths
• The twin assumptions lead to a set of behavioral
equations for intertemporal portfolio selection
that are both simpler and rich than those derived
from the corresponding discrete trading model.
• The continuous trading is an abstraction from
physical reality.
• If the length of time between revisions is very
short, the continuous trading solution will be a
approximation to the discrete trading solution.
• The application of continuous time analysis in
the empirical study of financial economic data is
more recent and less developed.
• In early studies, we assume that the logarithm of
the ratio of successive prices had a Gaussian
distribution.
• But the sample characteristics of the time series
were frequently inconsistent with these assumed
population properties.
• Attempts to resolve these discrepancies
proceeded along two separate paths.
• The first maintains the independent increments
and stationarity assumptions but replaces the
Gaussian with a more general stable (ParatoLevy) distribution.
• The stable family frequently fit the tails of the
empirical distributions better than the Gaussian.
• But there is little empirical evidence to support
adoption of the stable Paretian hypothesis over
that of any leptokurtotic distribution.
• Moreover, the infinite variance property of the
non-Gaussian stable distributions implis :
(1) most of our statistical tools, which are based
upon finite-moment assumptions are useless.
(2) the first-moment or expected value of the
arithmetic price change does not exist.
• The second, Cootner (1964) consider the
alternative path of finite-moment processes
whose distributions are nonstationary .
• The general continuous time framework requires
that the underlying process be a mixture of
diffusion and Poisson-directed processes.
• The general continuous time framework can
accommodate a wide range of specific
hypotheses including the “ reflecting barrier”
model.
• Rosenberg (1972) shows that a Gaussion model
with a changing variance rate appear to explain
the observed fat-tail characteristics of return.
• Rosenberg (1980) has developed statistical
techniques for estimating the parameters of
continuous time processes.
• As discussed by Merton, If the parameters are
slowly varying functions of time, then it is
possible to exploit the different “ time scales” to
identify and estimate these parameters.
• The second distribution still required more
research before a judgment can be made as to
the success of this approach.
• Their finite moment properties make the
development of hypothesis tests considerably
easier for these processes than for the stable
Pareto-Levy processes.
• With this as a background, we development the
assumptions of continuous time models.
• If X (t ) denote the price of a security at time t, the
change in the price of nthe security:
X (T )  X (0)  [ X (k )  X (k  1)]
1
where h denote the trading horizon, T  nh  0 ,
X (k )  X (k  1) denote X (kh)  X [(k 1)h]
• The continuous time trading interval assumption
implies that the trading interval h is equal to the
continuous time infinitesimal dt
• Note: it is unreasonable to assume that the
equilibrium distribution of returns on a security
over a specified time period will be invariant to
the trading interval for that security.
• Because investors’ optimal demand function will
depend upon how frequently they can revise
their portfolio.
• Define the random variables  (k ) by:
 (k )  X (k )  X (k 1)  Ek 1{X (k )  X (k 1)}
•  (k ) is the unanticipated price change in the
security between k  1 and k , conditional on
being at time
.
k 1
• Because Ek  j { (k )}  0 for j  1, , k , hence the
partial sums Sn  1n  (k ) form a martingale.
• The theory of martingales is usually associated
in the financial economics literature with the
“ Efficient Marker Hypothesis”.
• Two economics assumptions:
• Assumption 1: For each finite time interval [0,T]
there exists a number A1  0 , independent of the
number of trading intervals n, such that var(Sn )  A1
n
where var( Sn )  E0{[1  (k )]2 } .
• Assumption 2: For each finite time interval [0,T],
there exists a number A2   , independent of
n, such that var(Sn )  A2 .
• The second assumption rules out the variance
become unbounded such as Pareto-Levy stable
distribution with infinite variance.
• Assumption 3: There exists a number A3 , 1  A3  0
independent of n, such that for V (k ) V  A3 k  1, , n
2
V
(
k
)

E
{

(k )}, k  1, , n and V  max k V (k )
where
0
• This assumption rules out the possibility that all
the uncertainty in the unanticipated price change
over [0,T] is concentrated in a few of the many
trading periods, such as lottery ticket.
• Proposition 3.1: If Assumption 1,2,3 hold, then
V (k ) h, k  1, , n. That is, V (k )  O(h) and
V (k )  o(h) and V (k ) is asymptotically
proportional to h where the proportionality factor
is positive.
Proof:
var( S n )  E0 {1 1  (k ) ( j )}  1 1 E0 { ( k ) ( j )}
n
n
n
n
Suppose k  j, k  j then
E0{ (k ) ( j)}  E0{ ( j) E j { (k )}}  0 k  j
Therefore var( Sn )  1 V (k )
From Assumption 3 and 2
n
nVA3  1 V (k )  A2 
n

A2 h
VA3T
A2 h
VA3T
1
 V  V (k )
where 0  A2 A3   , hence V (k )  O(h)
From Assumption 3 and1
1
1
AAh
V (k )  A3V  A3 var( S n )  A3 A1  1 3
T
n
n
where A1 A3  0 hence V (k )  o(h)
• Suppose that  (k ) can take on any one of m
distinct values denoted by  j (k ), j  1, , m
where m is finite.
• Suppose that there exist a number M   ,
2
independent of n, such than  j  M .
• If p j (k )  prob{ (k )   j | information available
as of time zero}, then from Proposition 3.1 it
m
2
p

follow that: 1 j j  O(h) and because m is
2
p

finite it follows that: j j  O(h) for every j.
• Without lost generally, we assume p j 2j h for
every j .
• Assumption 4: Forj  1, , m, p j and  j are
sufficiently “well-behaved” functions of that
qj
q
there exist numbers j and r j such that p j h
rj
and  j h .
• This assumption is stronger than is necessary.
q j  2 rj
2
• From Assumption 4, we have h p j j h
j  1, , m
So q j  2rj  1
• This say that “ the larger the magnitude of the
outcome, the smaller the likelihood that the even
will occur.
2
p

1

• Because j
and j is bounded , both q j and
must be nonnegative, and therefore, we have:
0  q j  1 and 0  rj  1 2, j  1,
, m.
• We can partition its outcomes into three type:
(1) “type I” outcome is one such that rj  1 2 .
(2) “type II” outcome is one such that 0  rj  1 2
(3) “type III” outcome is one such that rj  0
rj
• Let J denote the set of events j such that the
outcomes  j are of type I.
• For j  J , q j  0, and therefore p j  o(1) .
c
• For j  J , p j  o(1)
.
• So for small trading intervals h, virtually all
observations of  (k ) will be type I outcomes,
c
and therefore an apt name for J might be
“ the set of rare events.”.
3.2 Continuous-Sample-Path
Processes With “No Rare Events”
• In this section, it is assumed that all possible
outcomes for  (k ), k  1, , n are of type I,
and therefore J c is empty.
• Define k  Ek 1{X (k )  X (k 1)} h , denote
the conditional expected dollar return per unit
time on the security.
• Assumption 5: For every h, it is assumed that  k
exists, k  1, , n and that there exists a
number    , independent of h, such that
| k | .
• This assumption ensures that for all securities
with a finite price the expected rate of return per
unit time over the trading horizon is finite.
• From before formula, we have:
X (k )  X (k 1)  k h   (k )
• Proposition 3.2: If, for k  1, , n, all possible
outcomes for  (k ) are type I outcomes, then the
continuous-time sample path for the price of the
security will be continuous.
• Proof: Let Qk ( )  prob{| X (k )  X (k 1) |  | I k 1}
A necessary and sufficient condition for
continuity of the sample path for X is that, for
every   0, Qk ( )  o(h).
12
u

max
|

|
/
h
Define
, so u  O(1)
{ j}
j
For every   0 define function h ( ) as the

 12
solution of    h  u (h ) .
Because  and u are O(1) , h ( )  0 for every
  0 . Therefore, for every h, 0  h  h ( )
| X (k )  X (k  1) |  , Qk ( )  0 , and hence
Q ( )
lim[ k
]  0 as h  0 .
h
• The sample for X(t) is continuous, but it is almost
nowhere differentiable.
X (k )  X (k 1) / h  k   j / h (1 h)1/ 2
• Which diverges as h  0 .
• So we need a generalized calculus and
corresponding theory of stochastic differential
equations.
• Some moment properties for X (k )  X (k  1) .
  Ek 1{ (k )}/ h O(1) k  1, , n
2
k
2
Ek 1{[ X (k )  X (k  1)]2 }  Ek 1{( k h   (k )) 2 }
  k2 h  0(h)
h
E0{|  (k ) | }  1 p j |  j |
m
N
N
 1 p j (u ) h N 2  u N h N / 2  o(h)
m
N
for N  2
E0{| X (k )  X (k  1) |N }  ( h  uh1/ 2 ) N
 u N h N / 2  o( h N 2 )
Hence, the unconditional Nth central and
noncentral absolute moments of X (k )  X (k  1)
are the same.
• Because the unconditional Nth central and
noncentral do not depend on the probabilities of
specific outcomes { p j }. Therefore:
Ek 1{|  (k ) |N }  o(h) for N  2
Ek 1{| X (k )  X (k 1) | }  Ek 1{|  (k ) | }  o(h
N
N
• Define u(k )   (k ) /( h) , where
u j   j /( k2h)1 2  O(1), j  1, , m; so
Ek 1{u(k )}  0 Ek 1{u 2 (k )}  1; and
2
k
Ek 1{| u(k ) |N }  O(1), N  2
1/ 2
N /2
)
• We have: X (k )  X (k 1)  k h   k u(k )h1/ 2
• This form makes explicit an important property
frequently observed in security returns: the
realized return on a security over a short trading
interval will be completely dominated by its
unanticipated.
• However , it does not follow that in choosing an
optimal portfolio the investor should neglect
differences in the expected returns among
stocks, because the first and second moments
of the returns are of the same order.
• Let F (t )  f ( X , t ), where X (t )  X , f is a C 2
function with bounded third partial derivatives.
• Denote by X the known value of X (k  1) , then
1/ 2
X j  X  k h   k u j h
j  1, , m
• We use Taylor’s theorem
f ( X j , k )  f ( X , k  1)  f1 ( X , k  1)( k h   k u j h1/ 2 )
 f 2 ( X , k  1)h  1 f11 ( X , k  1)( k h   k u j h1/ 2 )2  R j
2
R j  1 f 22 ( X , k  1)h 2  f12 ( X , k  1)( k h   k u j h1/ 2 )h
2
 1 f111 ( j ,  j )( X j  X )3  1 f112 ( j ,  j )( X j  X ) 2 h
6
2
 1 f122 ( j ,  j )( X j  X )h 2  1 f 222 ( j ,  j )h3
2
6
• Where  j  X   j ( X j  X ) and  j  (k 1)  v j
3/ 2
|
R
|

O
(
h
)  o(h) j  1, , m
• For each and every j, j
• So we have
f ( X j , k )  f ( X , k  1)  f1 ( X , k  1)( k h   k u j h )
1/ 2
 f 2 ( X , k  1)h  1 f11 ( X , k  1) k2u 2j h  o(h)
2
• We can describe the dynamics for F (k )
F (k )  F (k  1)  { f1[ X (k  1), k  1] k
2 2
1
 f 2 [ X (k  1), k  1]h 
f11[ X (k  1), k  1] k u j }h
2
 f1[ X (k  1), k  1] k u j h1/ 2  o(h)
• Applying the conditional expectation to both side
Ek 1[ F (k )  F (k  1)]  { f1[ X (k  1), k  1] k
 f 2 [ X (k  1), k  1]h  1 f11[ X (k  1), k  1] k2 }h  o(h)
2
• Define k  Ek 1[F (k )  F (k 1)]/ h
k  { f1[ X (k  1), k  1] k  f 2 [ X (k  1), k  1]
 1 f11[ X (k  1), k  1] k2 }  o(1)
2
• So F (k )  F (k  1)   h
k
2
1

f11[ X (k  1), k  1] k [u (k )  1]h
2
1/ 2
 f1[ X (k  1), k  1] k u (k )h  o(h)
• It is clear that the realized change in F over a
very short time interval is completely dominated
1/ 2
by the f1[ X (k 1), k 1] k u(k )h component of
the unanticipated change.
• We have:
Ek 1{[ F (k )  F (k  1)] }
2
 { f1[ X (k  1), k  1] k }2 h  o(h)
Ek 1{[F (k )  F (k 1)] }  O(h
N
N /2
)  o(h)
• So the conditional moment of F (k )  F (k  1) and
X (k )  X (k  1) is same.
Ek 1{[ F (k )  F (k  1)][ X (k )  X (k  1)]}
 { f1[ X (k  1), k  1] k2 }h  o(h)
Ek 1{[ F (k )  F (k  1)] j [ X (k )  X (k  1)]N  j }
 O ( h N / 2 )  o( h)
• The co-moments between F (k )  F (k  1) and
X (k )  X (k  1) is also same
• The conditional correlation coefficient per unit
time between contemporaneous changes in F
and X is
k  1  o(1)
 1  o(1)
if
f1[ X (k  1),(k  1)]  0
if
f1[ X (k  1),(k  1)]  0
• Now we study O(h) contribution to the change in
F over a finite time interval.
• Define
G(k )  G(k  1)  F (k )  F (k  1)  k h
 f1[ X (k  1), k  1] k u (k )h1/ 2
• If we define
2
2
y(k )  f11[ X (k 1), k 1] k [u (k ) 1]/ 2
• We have
G(k )  G(k 1)  y(k )  o(h)
• By construction Ek 1{y(k )}  0, and therefore
Ek  j {y(k )}  0 j  1, , k . Therefore , the partial
sums  y (k ) form a martingale.
n 2
E
{
• Because 0 1 y (k ) / k 2 }   , from the Law of
Large Numbers for martingales that
n
1
lim[h1
n
1 n
y (k )]  T lim[ 1 y (k )]  0 as n  
n
• We have that for fixed T
G (T )  G (0)  h1 y (k )  1 o(h)  h1 y (k )  o(1)
n
n
n
• Taking the limit we have G(T )  G(0)  0
• So, the cumulative error of the approximation
goes to zero with probability one.
• Hence, for T  0 , we have
F (T )  F (0)  1 [ F (k )  F (k  1)]
n
 1 k h  1 f1[ X (k  1), k  1] k u(k )h1/ 2
n
n
• Hence the stochastic term will have a negligible
effect on the change in F over a finite time
interval.
• By the usual limiting arguments for Riemann
integration, we have
lim(1 k h)    (t )dt
T
n
n
0
lim{1 f1[ X (k  1), k  1] k u (k )h }
n
1/ 2
n 
T
  f1[ X (t ), t ] (t )u (t )(dt )
0
1/ 2
• So we have
T
T
0
0
F (T )  F (0)    (t )dt   f1[ X (t ), t ] (t )u(t )(dt )1/ 2
• The stochastic differential for F is
dF (t )   (t )dt  f1[ X (t ), t ] (t )u(t )(dt )
1/ 2
• There is no difference because the contribution
of this O(dt )stochastic term to the moments of dF
over the infinitesimal interval dt is o(dt ), and over
finite intervals it disappears.
• Let f ( X , t )  X , we get the stochastic
differential for X
T
T
0
0
X (T )  X (0)    (t )dt    (t )u(t )(dt )1 2
dX (t )   (t )dt   (t )u(t )(dt )
1/ 2
• Throughout this analysis, the only restrictions on
E
{
u
(
t
)}

0,
u
(
t
)
the distribution for
where (a)
2
(b) E{u(t ) }  1 (c) u(t )  O(1)
(d)the distribution for u(t ) is discrete.
• Assumption 6: The stochastic process for X (t ) is
a Markov process.
E{X (t ) | X t 1, X t 2, X 0}  E{X (t ) | X t 1}
• The Assumption 6 can be weakened to say that
the conditional probabilities for X depend on only
a finite amount of past information.
p( x, t )  p( x, t; X , T )
 prob{ X (T )  X | X (t )  x}, t  T
• Therefore , provide that p is a well behaved
function of x and t, it will satisfy all the properties
previous derived for F(t)
 (t )  Et 1[ p(t )  p(t 1)]/ h  0
1 2
0   ( x, t ) p11 ( x, t )   ( x, t ) p1 ( x, t )  p2 ( x, t )
2
• This is a “Kolmogorov backward equation”.
• Therefore, subject to boundary condition, the
formula completely specifies the transition
probability densities for the two price.
• The only characteristics of the distribution for the {u(t )}
that affect the asymptotic distribution for the
security price are the first and second moments.
• Hence, in the limit of continuous trading, nothing
of economic content is lost by assuming that the {u(t )}
are independent and identical.
• Define Z (t ) is a random variable
Z (T )  Z (0)  1 [Z (k )  Z (k  1)]
n
 1  k h  1  k u(k )h1/ 2
n
n
• When k  0,  k  1
Z (T )  Z (0)  T
 u (k )
n
1/ 2
1
1/ 2
n
• When {u (k )} are independent and identically
distributed with a zero mean and unit variance,
we have Z (T )  Z (0) is normal distribution.
• Then the solution to the “Kolmogorov backward
2
equation” with   1,   0
exp[( X  x)2 / 2(T  t )]
p( x, t; X , T ) 
[2 (T  t )]1/ 2
• We define dZ (t )  u(t )(dt )1/ 2
• When the {u(t )} are independent and distribution
standard normal, the dZ process is called a
Wiener or Brownian.
• Then the dynamics of X (t ) can be wrote as:
T
T
0
0
X (T )  X (0)   [ X (t ), t ]dt    [ X (t ), t ]dZ (t )
• And
dX  [ X (t ), t ]dt   [ X (t ), t ]dZ (t )
• The class of continuous-time Markov processes
whose dynamics can be written in the two form
are call Ito’s processes.
• Ito’s lemma: Let f ( X , t ) be a C 2 function define
on R[0, ] and take the stochastic integral
defined by the two form, then the timedependent random variable F  f ( X , t ) is a
stochastic integral and its stochastic differential
is:
1
dF  f1 ( X , t )dX  f 2 ( X , t )dt  f11 ( X , t )(dX ) 2
2
• Where the product of the differentials is defined
2
by the multiplication rules (dZ )  dt , dZdt  0
and (dt )2  0
• If the economic structure to be analyzed is such
that Assumption 1-6 obtain and can have only
type I outcomes.
• Then in continuous-trading model, securitypricing dynamics can always be described by Ito
processes with no loss of generality.
• Note:
• (1) The normality assumption for the
imposes no further restrictions on the process
beyond those of Assumptions 1-6.
• (2) The distribution for the security price change
over a finite interval [0,T] may not be normally
distribution. Such as [ X (t ), t ]  aX ,  [ X (t ), t ]  bX
dX  [ X (t ), t ]dt   [ X (t ), t ]dZ (t )
X (t )can be shown to have a log-normal
distribution
• The continuous time models with Ito process
have substantive benefits.
• Such as, (1) the analysis of corporate liability
and option pricing is simplified by Ito process.
(2) in solving the intertemporal portfolio selection
problem, the optimal portfolio demand functions
will depend only upon the first two moment of the
security return distributions.
Continuous-Sample-Path Processes
With “ Rare Events”
• In this section, it is assumed that the outcomes
for  (k ), k  1, , n can be either of type I or
type II, but not type III.
• The principal conclusion of this analysis will be
that, in the limit of continuous trading, the
distributional properties of security return are
indistinguishable form those of section 3.2.
• Proposition 3.3: If, for k  1, , n, all possible
outcomes for  (k ) are either type I or type II
outcome, then the continuous-time sample path
for the price of the security will be continuous.
• Proof: Let Qk ( )  prob{| X (k )  X (k 1) |  | I k 1}
A necessary and sufficient condition for
continuity of the sample path for X is that, for
every   0, Qk ( )  o(h).
r
u

max
|

|
h
Define
, so u  O(1)
{ j}
j
For every   0 define function h ( ) as the

 r
solution of    h  u (h )
Because  and u are O(1) , h ( )  0 for every
  0 . Therefore, for every h, 0  h  h ( )
| X (k )  X (k  1) |  , Qk ( )  0 , and hence
Q ( )
lim[ k
]  0 as h  0 .
h
• From Assumption 5: Ek 1{X (k )  X (k 1)} is
asymptotically proportional to h, and therefore so
is E0{X (k )  X (k 1)}.
• From Proposition 3.1 the unconditional variance
of X (k )  X (k  1) is asymptotically proportional
to h.
• The Nth unconditional absolute moment of  (k )
E0{|  (k ) |}  1 p j |  j |  O(1 h
m
N
 O(h
( N  2) rj 1
N
m
( N  2) rj 1
)
)  o(h) for N  2
• The Nth unconditional noncentral moments:
N
r N
E0{[ X (k )  X (k  1)] }  ( h  uh )
 u N h Nr  o(h)
• The second conditional moments is
Ek 1{[ X (k )  X (k  1)]2 }  Ek 1{( k h   (k )) 2 }
  k2 h  0(h)
h
• The Nth ( N  2) conditional moments is
Ek 1{[ X (k )  X (k 1)]N }  Ek 1{(k h   (k )) N }  o(h)
• Hence, the moment relations for X (k )  X (k  1)
are identical with those derived in section 3.2
where only type I outcomes were allowed.
• Define F (t )  f ( X ) if X (t )  X is a C 2
function with a bounded ( K  1)th order
derivative, then from Taylor’s theorem, we have:
(1)
Ek 1[ F (k )  F (k  1)]  { f [ X (k  1)] k
 1 f (2) [ X (k  1)] k2 }h  o(h)
2
Ek 1{[ F (k )  F (k  1)] }
2
 { f1[ X (k  1), k  1] k } h  o(h)
2
Ek 1{[ F (k )  F (k 1)] }  O(h )  o(h)
N
Nr
• Hence, the order relation for the conditional
moments of F (k )  F (k  1) is the same as for
the conditional of X (k )  X (k  1) .
• Over short intervals of time, the unanticipated
part of the change will be dominated by
f (1) [ X (k 1)] (k )
• We now examine stochastic properties for the
change in F over a finite time interval.
• Define random variables
j
j
[
X
(
k
)

X
(
k

1)]

E
{[
X
(
k
)

X
(
k

1)]
}
( j)
k 1
y j (k )  f [ X (k  1)]
j!
G(k )  G(k  1)  F (k )  F (k  1)
 Ek 1{F (k )  F (k  1)}  f [ X (k  1)] (k )
• Which by Taylor’s theorem can be rewritten as
(1)
G (k )  G (k  1)  1 y j (k )  RK 1
K
RK 1  f ( K 1) [ X (k  1)  (1   ) X (k )]
[ X (k )  X (k  1)]K 1  Ek 1{[ X (k )  X (k  1)]K 1}

( K  1)!
• For some  , 0    1. But f ( k 1) is bounded
and [k h   (k )]K 1  O(hr ( K 1) )  o(h) .
• So
RK 1  o(h)
G (k )  G (k  1)  1 y j (k )  o(h)
K
• The unconditional variance of G(k )  G(k 1) is
var[G (k )  G (k  1)]  E0 { 2
K
 2
K

K
2

K
2
yi (k ) y j (k )}  o(h)
M i M j E0 {{[ X (k )  X (k  1)]i
 Ek 1[ X (k )  X (k  1)]i }
{[ X (k )  X (k  1)] j
 Ek 1[ X (k )  X (k  1)] j }}/ i ! j ! o(h)
 O(h
2 r 1
)  o( h)  o( h)
• Where M i is the least upper bound on | f (i ) |
• For the finite tiem interval [0,T], we have
G(T )  G(0)  1 [G(k )  G(k  1)]
n
 1 1 yi (k )  o(1)
n
•

n
K
1
1
K
yi (k ) forms a martingale, so
var[G(T )  G(0)]  O(h )  o(1)  o(1)
2r
• As h  0 , the variance of
zero.
G(T )  G(0)
goes to
• Hence, we have
F (T )  F (0)  1 [ F (k )  F (k  1)]
n
 1 k h  1 f (1) [ X (k  1)] (k )
n
n
• With probability one where
k  f (1) [ X (k  1)] k  1 2 f (2) [ X (k  1)] k2
• So
T
T
o
0
F (T )  F (0)    (t )dt   f (1) [ X (t )] (t )
dF (t )   (t )dt  f (1) [ X (t )] (t )
• Hence, the limit of continuous trading, processes
with type I and type II outcomes are
indistinguishable from processes with type I
outcomes only.
Discontinuous-Sample-Path
Processes With “Rare Events’
• In the concluding section, the general case is
analyzed where the outcomes for  (k ) can be
type I, type II or type III.
• Type III resulting sample path for X will be
discontinuous.
• Proposition 3.4: If, for k  1, , n, at least one
possible outcome for  (k ) is a type III outcome,
then the continuous time sample path for the
price of the security will not be continuous.
• Proof:Let Qk ( )  prob{| X (k )  X (k 1) |  | I k 1}
A necessary and sufficient condition for
continuity of the sample path for X is that, for
every   0, Qk ( )  o(h) . If suppose event j
denote a type III outcome for  (k ) where  (k )   j
and  j is O(1) . If p j is the probability that event j
occurs. Then p j   j h where  j  O(1) .
Define

if  k  j  0
h 
(  1) j /  k if  k  j  0

  |  j |

Note h  0,    0 independent of h.
For all h such that 0  h  h , if  (k )   j
then | X (k )  X (k 1) |   .


0




Hence for 0  h  h , any  such that
| X (k )  X (k  1) |  if  (k )   j , and therefore
Qk ( )   j h  O(h) . So the path is not continuous.
• These fundamental discontinuities in the sample
path manifest themselves in the moment
properties of X (k )  X (k  1)
• The first and second unconditional moments of
X (k )  X (k  1) are asymptotically proportional to h
• The Nth unconditional absolute moments is O(h)
E0{|  (k ) | }  1 p j |  (k ) |
N
 O(1 h
m
( N  2) rj 1
m
N
)  O(h)
• So none of the moments can be neglected.
• Define the conditional random variable u(k )   (k ) / h1/ 2
conditional on  (k ) having type I outcome.
• Define the conditional random variable y(k )   (k )
conditional on  (k ) having type I outcome.
• Then
u(k )h1/ 2 with probability 1   (k )h
 (k ) 
with probability  (k )h
y (k )
• Where u(k ), y(k ),and  (k ) are all O(1)
• Define
y (k )  Eky1{ y (k )}
 Ek 1{ (k ) | type III outcome}
1/ 2
u (k )h
 h E {u (k )}
1/ 2
u
k 1
 Ek 1{ (k ) | type III outcome}
• Because Ek 1{ (k )}  0
 (k ) y (k )h
u (k )  
1   (k )h
1/ 2
  yh
1/ 2
 o(h)
• Define
 h  Ek 1{y (k ) | type III outcome}
2
y
2
 h  Ek 1{u (k ) | type I outcome}
2
u
2
 h  Ek 1{ (k )}
2
k
2
• Then we have:
2
2



k
y
2
u 
  k2   y2  O(h)
1  h
• Where  u ,  y ,  k are all O(1)
• Further for N  2
Ek 1{ (k )}  (k )E {y (k )}h  o(h)
N
y
k 1
N
• Thus, both the type I and type III contribute
significantly to the mean and variance of  (k )
• Let F (t )  f ( X , t ) , X (t )  X , where f is a C 2
function with bounded third partial derivatives.
• By Taylor’s theorem
f [ X (k  1)   k h  y, k ]  f [ X (k  1)  y, k  1]
 f1[ X (k  1)  y, k  1] k h
 f 2 [ X (k  1)  y, k  1]h  o(h)
f [ X (k  1)   k h  uh1/ 2 , k ]  f [ X (k  1), k  1]
 f1[ X (k  1), k  1]( k h  uh )  f 2 [ X (k  1), k  1]h
1/ 2
1
2
 f11[ X (k  1), k  1]u h  o(h)
2
• The properties of conditional expectation:
Ek 1{F (k )  F (k  1)}
  (k )hE {F (k )  F (k  1)}
y
k 1
[1   (k )h]E {F (k )  F (k  1)}
u
k 1
1
2
 ( f11[ X (k  1), k  1] u
2
 f1[ X (k  1), k  1]( k   y )  f 2 [ X (k  1), k  1]
 Eky1{ f [ X (k  1)  y (k ), k  1]
 f [ X (k  1, k  1)]})h  o(h)
• Define k  Ek 1{F (k )  F (k 1)}/ h
• As h  0 , we can write the instantaneous
condition expect change in F per unit time as
1
 (t )  f11[ X (t ), t ] u2 (t )  f1[ X (t ), t ][ (t )   (t ) y (t )]
2
 f 2 [ X (t ), t ]   (t ) Ety { f [ X (t )  y (t ), t ]  f [ X (t ), t ]}
• When  (t )  0 and there are no type III outcomes,
it became into the form of Section 3.2
• The higher conditional moments
Ek 1{[ F (k )  F (k  1)] }
2
 ( E { f [ X (k  1)  y (k ), k  1]
y
k 1
 f [ X (k  1, k  1)]}
2
 f [ X (k  1, k  1)] )h  o(h)
2
1
2
u
• For N  2
Ek 1{[ F (k )  F (k  1)]N }
  E { f [ X (k  1)  y (k ), k  1]
y
k 1
 f [ X (k  1, k  1)]} h  o(h)
N
• Only the type III outcomes contribute significantly
to the moment higher than the second.
• The moment of X (k )  X (k  1) and F (k )  F (k  1)
are same.
• Let
denote the conditional probability
density for X(T)=X, at time T, which is a function
of security price at time t, so we have p satisfy:
1 2
0   u p11 ( x, t )  (   y ) p1 ( x, t )  p2 ( x, t )
2
  [ p( x  y, t )  p( x, t )]g ( y; x, t )dt
 u2   g
• Hence, knowledge of the functions
is sufficient to determine the probability
distribution for the change in X between two data.
• Moreover, this process is identical with that of a
stochastic process driven by a linear
superposition of a continuous sample path
diffusion process and a “Poisson directed”
process.
• Let Q(t  h)  Q(t ) be a Poisson distribution
random variable.
0 with probability 1  [ X (t ), t ]dt  o(dt )
dQ(t )  1 with probability [ X (t ), t ]dt  o(dt )
N with probability
o(dt ), N  2
• Define dX1 (t )  y(t )dQ(t ) , then dX1 is an
example of a “Poisson direction” process.
• Define dX 2   dt   dZ
• Then the limiting process for the change in X will
be identical with the process described by dX1  dX 2
• The stochastic differential equation of X(t) is
dX (t )  { [ X (t ), t ]  [ X (t ), t ] y (t )}dt
 [ X (t ), t ]dZ (t )  y(t )dQ(t )
• Where  is the probability per unit time that the
change in X is a type III outcome.
• When   0 it is mean only type I outcomes
can occur.
• The stochastic differential representation for F
1 2
dF (t )  {  f11[ X (t ), t ]  (   y ) f1[ X (t ), t ]
2
 f 2 [ X (t ), t ]}dt   f1[ X (t ), t ]dZ (t )
{ f [ X (t )  y (t ), t ]  f [ X (t ), t ]}dQ(t )
• In summary, if the economic structure to be
analyzed is such that Assumption 1-5 obtain,
then in continuous trading models security price
dynamics can always be described by a mixture
of continuous sample path diffusion processes
and Poisson directed processes with no loss in
generality.
• The diffusion process describes the frequent
local changes in prices. The Poisson directed
process is used to capture those rare events
• The transition probabilities are completely
specified by only four functions    g
• This make the testing of these model structures
empirically feasible.
The End
Thanks!