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Martingales, Detrending Data, and the Efficient Market Hypothesis

Martingales, Detrending Data, and the Efficient Market Hypothesis
Martingales, Detrending Data, and the Efficient Market Hypothesis

Martingales, Detrending Data, and the Efficient

Market Hypothesis

Joseph L. McCauley+, Kevin E. Bassler++, and Gemunu H.

Gunaratne+++

Physics Department

University of Houston

Houston, Tx. 77204-5005

jmccauley@https://www.wendangku.net/doc/c316620628.html,

+Senior Fellow

COBERA

Department of Economics

J.E.Cairnes Graduate School of Business and Public Policy

NUI Galway, Ireland

++Texas Center for Superconductivity

University of Houston

Houston, Texas 77204-5005

+++Institute of Fundamental Studies

Kandy, Sri Lanka

Key Words: Martingales, Markov processes, detrending, memory, stationary and nonstationary increments, correlations, efficient market hypothesis.

Abstract

We discuss martingales, detrending data, and the efficient market hypothesis for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). Beginning with x-independent drift coefficients R(t) we show that Martingale stochastic

processes generate uncorrelated, generally nonstationary increments. Generally, a test for a martingale is therefore a test for uncorrelated increments. A detrended process with an x-dependent drift coefficient is generally not a martingale, and so we extend our analysis to include the class of (x,t)-dependent drift coefficients of interest in finance. We explain why martingales look Markovian at the level of both simple averages and2-point correlations. And while a Markovian market has no memory to exploit and presumably cannot be beaten systematically, it has never been shown that martingale memory cannot be exploited in 3-point or higher correlations to beat the market. We generalize our Markov scaling solutions presented earlier, and also generalize the martingale formulation of the efficient market hypothesis (EMH) to include (x,t)-dependent drift in log returns. We also use the analysis of this paper to correct a misstatement of the ‘fair game’ condition in terms of serial correlations in Fama’s paper on the EMH. We end with a discussion of Levy’s characterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.

1. Introduction

Recently [1] we focused on the condition for long time correlations like and including fractional Brownian motion (fBm), which is stationarity of the increments in a stochastic process x(t) with variance nonlinear in the time. There, we derived the 2-point and 1-point densities including the transition density for fBm. We will point out below that there are nonMarkov systems where the pair correlations canot be distinguished from those of a Markov process, but time series with stationary increments (like fBm) exhibit long time memory that can be seen at the level of pair

correlations: fBm cannot be mistaken for a Markov process at the 2-point level. We correspondingly emphasized that

neither 1-point averages nor Hurst exponents can be used to identify the presence or absence of history-dependence in a time series, or to identify the underlying stochastic process

(see [2] for the conclusion that an equation of motion for a 1-point density cannot be used to decide if a process is Markovian or not). In the same paper, we pointed out that

the opposite class, systems with no memory at all (Markov processes) and with x-independent drift coefficients generate uncorrelated, typically nonstationary increments. The

conclusions in [1] about Markov processes are more general than we realized at the time. Here, we generalize that work by focusing on martingales.

In applications to finance, by “x” we always mean x(t)=ln(p(t)/p c) where p(t) is a price at time t and p c is a

reference price, the consensus price or ‘value’ [3]. The consensus price p c is simply the price that determines the peak of the 1-point returns density f1(x,t). The reason why

log increments x(t;T)=lnp(t+T)/p(t) and price differences Δp=p(t+T)-p(t) generally cannot be taken as ‘good’ variables describing a stochastic process (neither theoretically nor in

data analysis) is explained below in part 4. It is impossible for a martingale, excepting the special case of a variance linear in the time t, to develop either stochastic dynamics or probability theory based on increments x(t;T) or Δp, because if the increments are nonstationary, as they generally are, then the starting time t matters and consequently histograms derived empirically from time series assuming that the starting time doesn’t matter exhibit ‘significant artifacts’ like fat tails and spurious Hurst exponents [3,4]. In contrast, in a system with long time autocorrelations (like fBm), the stationary increment x(t;T)=x(t+T)-x(t)=x(T), ‘in distribution’, is a perfectly good variable. But real markets [4] are hard to beat and rule out increment autocorrelations.

Stated briefly, if increments are nonstationary then the 1-point density that describes the increments is not independent of t (see part 5 below). Next, we define the required underlying ideas.

2. Conditional expectations with memory

Imagine a collection of time series generated by an unknown stochastic process that we would like to discover via data analysis. Simple averages require only a 1-point density f1(x,t), e.g., =∫x n f1(x,t)dx. No dynamical process can be identified by specifying merely either the 1-point density or a scaling exponent [1]. Both conditioned and unconditioned two-point correlations, e.g. =∫dydxyx f2(y,t+T;x,t), require a two point density f2(y,t+T;x,t) for their description and provide us with limited information about the class of dynamics under consideration.

Two point conditional probability densities p k, or transition probability densities, can then be defined as [5,6]:

!f

2

(x

1

,t

1

;x

1

,t

1

)=p

2

(x

2

,t

2

x

1

,t

1

)f

1

(x

1

,t

1

), (1)

!f

3

(x

3

,t

3

;x

2

,t

2

;x

1

,t

1

)=p

3

(x

3

,t

3

x

2

,t

2

,x

1

,t

1

)p

2

(x

2

,t

2

x

1

,t

1

)f

1

(x

1

,t

1

) ,

(2)

and more generally as

!f

n

(x

n

,t

n

;...;x

1

,t

1

)=p

n

(x

n

,t

n

x

n"1

,t

n"1

; (x)

1

,t

1

)f

n"1

(x

n"1

,t

n"1

;...;x

1

,t

1

)

=p

n

(x

n

,t

n

x

n"1

,t

n"1

; (x)

1

,t

1

)...p

2

(x

2

,t

2

x

1

,t

1

)f

1

(x

1

,t

1

)

, (3)

where p n is the 2-point conditional probability density to find x n at time t n, given the last observed point (x n-1,t n-1) and the previous history (x n-2,t n-2;…;x1,t1). When memory is present in the system then one cannot use the simplest 2-point transition density p2to describe the complete time evolution of the dynamical system that generates x(t).

In a Markov process the picture is much simpler. A Markov process [5,6] remembers only the last observed point in the time series. There, we have

!

f

n

(x

n

,t

n

;...;x

1

,t

1

)=p

2

(x

n

,t

n

x

n"1

,t

n"1

)...p

2

(x

2

,t

2

x

1

,t

1

)f

1

(x

1

,t

1

),

(4)

because all transition rates p n, n>2, are built up as products of p2,

!

p

k

(x

k

,t

k

x

k"1

,t

k"1

;...;x

1

,t

1

)=p

2

(x

k

,t

k

x

k"1

,t

k"1

),

(5)

for k=3,4, …. , and so p2cannot depend on an initial state (x1,t1) or on any previous state other than the last observed point (x k-1,t k-1). Only in the absence of memory does the 2-

point density p2 describe the complete time evolution of the dynamical system. E.g., we can prove that for an arbitrary process with or without memory

!

p

k"1

(x

k

,t

k

x

k"2

,t

k"2

;...;x

1

,t

1

)=dx

k"1

p

k

(x

k

,t

k

x

k"1

,t

k"1

;...;x

1

,t

1

)

#p

k"1

(x

k"1

,t

k"1

x

k"2

,t

k"2

;...;x

1

,t

1

)

(6)

and therefore that

!

p

2

(x

3

,t

3

x

1

,t

1

)=dx

2

p

3

(x

3

,t

3

x

2

,t

2

;x

1

,t

1

)

"p

2

(x

2

,t

2

x

1

,t

1

),

(7)

whereas the Chapman-Kolmogorov (CK) equation for a Markov process follows with p n=p2 for n=2,3,…, from (6) so that

!

p

2

(x

3

,t

3

x

1

,t

1

)=dx

2

p

2

(x

3

,t

3

x

2

,t

2

)

"p

2

(x

2

,t

2

x

1

,t

1

). (8)

The Markov property is expressed by p n=p2 for all n≥3, the complete lack of memory excepting the last observed point. The CK Equation (8) is a necessary but not sufficient condition for a Markov process [7,8,9,10].

A time translationally invariant Markov process defines a 1-parameter semi-group U(t2,t1) of transformations [10], where p2(x n,t n:x n-1,t n-1)=p2(x n,t n-t n-1:x n-1,0)), but time translational invariance is not a property of FX data [4] and will not be assumed here. In any and all cases, the identity element is defined by the equal times transition density

!

p

2

(y,t x,t)="(y#x). (9)

Arbitrary processes with memory do not obey the CK eqn. Instead, the class of path-dependent time evolutions is defined by the entire hierarchy eqns. (3,6), for n=2,3,4,… . That the transition density for fBm, e.g., obeys no CK eqn. is shown implicitly in Appendix B of [11], where the authors show that for general Gaussian processes one obtains the semi-group property iff. the Gaussian describes a Markov process. However, both CK and Fokker-Planck eqns. have been shown to hold for Ito processes with only finitely many states in memory [9].

Memory-dependent processes in statistical physics have been discussed by H?nggi and Thomas [11]. They point our that

!p

2

(x

3

,t

3

x

2

,t

2

)=

dx

1

p

3

(x

3

,t

3

x

2

,t

2

;x

1

,t

1

)p

2

(x

2

,t

2

x

1

,t

1

)f

1

(x

1

,t

1

)dx

1

"

p

2

(x

2

,t

2

x

1

,t

1

)f

1

(x

1

,t

1

)dx

1

"

(10)

is a functionals of the initial state f1(x1,t1) in which the system was prepared at the initial time t1, unless the process is Markovian. In a nonMarkov system one may sometimes be able to mask this dependence on state preparation by choosing the initial condition to be f1(x1,t1)=δ(x1). If, instead, we would or could choose f1(x1,t1)=δ(x1-x’o) at t1=0, e.g., then we obtain p2(x3,t3;x2,t2)= p3(x3,t3;x2,t2,x o’), introducing a dependence on x o’ in both the drift and diffusion coefficients. So in this case, what appears superficially as p2is really a special case of p3. The authors of [11] point out that the origin of memory in statistical physics is often a consequence of averaging over other, rapidly changing, variables. We will mention memory as a consequence of averaging over other variables in the section below on the efficient market hypothesis.

A class of Markov processes with scaling more general than Hurst exponent scaling [1,3,12] is defined as follows: let

!

f

1

(x,t)="

1

#1(t)F(u)(11)

with initial condition f1(x,0)=δ(x), where u=x/σ1(t), with variance1

!

"2(t)=x2(t)="

1

2(t)u2.(12)

1Mathematicians often write x(t)=tH x(1). The variable x(1) is the same as our variable u: the time t is here dimensionless, otherwise the diffusion coefficient D(x,t) must be multiplied by a constant factor with dimension sec-1.

Then with the diffusion coefficient scaling as

!D(x,t)=(d"

1

2/dt)D (u)(13)

where dσ1/dt>0 is required, f1(x,t) satisfies the Fokker-Planck pde

!"f

1

"t

=1

2

"2(Df

1

)

"x2

(14)

and yields the scale invariant part of the solution

!F(u)=

C

(u)

e"udu/D (u)

#. (15)

An example is given by Hurst exponent scaling σ1(t)=t H, 0

The Green function g(x,t;x o,t o) of (14) for an arbitrary initial condition (x o,t o)≠(0,0) does not scale [12], but then the 2-point transition density p2(x2,t2; x1,t1) for fBm does not scale either1. In all cases scaling, when it occurs, can only be seen in the special choice of conditional density f1(x,t)=p2(x,t;0,t o) with t o=0 for a Markov process.

The same 1-point density f1(x,t) may describe nonMarkovian processes because a 1-point density taken alone, without the 1

In [1] it is wrongly asserted that x(-∞)=0 for fBm. Instead, x(0)=0 and the variance diverges at t=-∞. I.e., every path is ‘filtered’ through the point x(0)=0. In different ‘runs’ of the Gedankenexperiment, initial conditions x(t o) are drawn from a Gaussian, with the limits t o=-∞, σ2(-∞)=∞.

information provided by the transition densities, defines no specific stochastic process and may be generated by many different completely unrelated processses, including systems with long time increment autocorrelations like fBm [1]. We will show below that the 2-point transition density delineates fBM from a martingale, but that pair correlations cannot be used to distinguish an arbitrary martingale from a drift-free Markov process.

3. Absence of trend and martingales

By a trend, we mean that d/dt≠0, conversely, by lack of trend we mean that d/dt=0. If a stochastic process can be detrended, then d/dt=0 is possible via a transformation of variables but one must generally specify which average is used to define . If the drift coefficient R(x,t) depends on x, then detrending with respect to a specific average generally will not produce a detrended series if a different average is then used (e.g., one can choose different conditional averages, or an absolute average). We next discuss processes that can be detrended once and for all by a simple subtraction. I.e., we assume for the time being a trivial drift coefficient but allow for nontrivial diffusion coefficients. This case is of interest both theoretically and for FX data analysis.

A trivial drift coefficient R(t) is a function of time alone. A nontrivial drift coefficient R(x,t) depends on x, on (x,t), or on (x,t) plus memory {x}, and is defined for Ito/Langevin processes by [5,6]

!R(x,t,{x})"

1

T

dy

#$

$

%(y#x)p n(y,t+T;x,t,{x})(16)

as T vanishes, where {x} denotes the history dependence in

p n, e.g. with y=x n and x=x n-1(y,t+T,x,t;{x}) denotes (x n,t n;x n-1,t n-1,x n-2,t n-2,…,x1,t1) with y=x n and x=x n-1. If R(x,t)=0 then

!

dy

"#

#

$yp n(y,t+T;x,t,{x})=x,(17)

so that the conditional average over x at a later time is given by the last observed point in the time series, cond=x(t). This is the notion of a fair game: there is no systematic change in x on the average as t increases, dcond/dt=0. The process x(t) is generally nonstationary, and the condition (17) is called a local martingale[13]. The possibility of vanishing trend, d/dt=0, implies a local martingale x(t), and vice-versa.

This is essentially the content of the Martingale Representation Theorem [14], which states that an arbitrary martingale can be built up from a Wiener process B(t), the most fundamental martingale, via stochastic integration ala Ito,

!

x(t)=b(x(s),s;{x})dB(s)

". (18)

There is no drift term in (18). If a stochastic differential equation (sde)

!

dx(t)=b(x(t),t;{x})dB(t), (19)

follows from (18) then the diffusion coefficient is defined by [5,6]

!D(x,t,{x})"

1

T

dy(y#x)2p

n

(y,t+T x,t,{x})

$(20)

as T vanishes, so that D=b2. In a Markov system the drift and diffusion coefficients depend on (x,t) alone, have no history dependence. Ito calculus based on martingales has been developed systematically by Durrett, including the derivation of Girsanov’s Theorem for arbitrary diffusion coefficients D(x,t) [13]. Many discussions of Girsanov’s Theorem [14,15] implicitly rule out the general case (19) where D(x,t) may depend on x as well as t. In this paper we do not appeal to Girsanov’s theorem because the emphasis is on application to data analysis, to detecting martingales in empirical data. A new and simplified proof of Girsanov’s theorem for variable diffusion coefficients has been presented elsewhere [16].

There are stochastic processes that are inherently biased, and fBm provides an example. There, although the absolute average vanishes =∫c f1(x,t)dx=0, the conditional average yields dc/dt≠0 where the time dependence arises from long time correlations rather from a drift term: in fBm one obtains [1]

!

x(t)

cond

=dyyp

2

(y,s x,t)=C(t,s)x

"(21)

instead of the martingale condition (19). Here, dc/dt=xdC/dt≠0 because the factor C(t,s)≠1 is proportional to the autocorrelation function where the stationarity of increments guaranteeing long time memory was built in [1]. Such processes cannot be ‘detrended’ (R(x,t)=0 by construction in fBm [1,17]) because what appears locally to be a trend in a conditional average is simply the strongly correlated behavior of the entire time series.

Note next that subtracting an average drift ∫dt from a process x(t) defined by x-dependent drift term plus a Martingale,

!

x(t)=x(t"T)+R(x(s),s;{x})ds+b(x(s),s;{x}dB(s)

#

t"T

t

#, (22)

does not produce a martingale. Here, if we replace x(t) by x(t)-∫dt where the average drift term defined conditionally from some initial condition (x1,t1),

!

R=dxR(x,t)p

2

"(x,t x1,t1)(23)

depends on t alone we do not get drift free motion, and choosing absolute or other averages of R will not change this. In financial analysis, e.g., may represent an average from the opening return x1at opening time t1up to some arbitrary intraday return x at time t. The subtraction yields

!x(t)=x(t"T)+R(x(s),s;{x})ds"R

t"T

t

#ds+b(x(s),s;{x}dB(s)

#

t"T

t

#

(24)

and is not a martingale unless R is independent of x: we obtain c=x iff. x=x o. Constructing martingales for the special case of an (x,t) dependent drift R(x,t) in financial applications is carried out in part 6 below. Until part 6, we assume a trivial drift R(t) that has been subtracted, so that by x(t) we really mean x(t)-∫R(t)dt.

With that assumption we can, for our present purposes, divide stochastic processes into those that satisfy the martingale condition

!

x(t)

cond

=x(t

o

), (25)

where <..>cond denotes the conditional average (19), and those that do not. Those that do not satisfy (19) can be classified further into processes that consist of a nontrivial (i.e., (x,t)-dependent) drift plus a martingale (20), and those (including fBm) that are not defined by an underlying martingale.

Finally, an Ito sde with or without drift included can be used to derive a Fokker-Planck pde for a stochastic process with memory. The Fokker-Planck pde is usually derived from the CK eqn. for a Markov process as an approximation, but this is not necessary. The derivation of the Fokker-Planck pde from the Ito sde, without assuming a C-K eqn. apriori, is provided in [9,19] goes through even if the drift and diffusion coefficients R and D are memory dependent. In that case one has a pde for a 2-point conditional probability p n depending on a history of n-2 earlier states.

4. Stationary vs. nonstationary increments

Let us preface this section with a comment: in contrast with what is assumed in the econophysics and finance literature, we know of only two stochastic processes with both finite variance and stationary increments: the Wiener process and fractional Brownian motion. Furthermore, we know of no finance data with stationary increments. Furthermore, stationary increments is a very different condition than either time or space translational invariance. Nonstationary increments are ubiquitous in both theory and data analysis. In this section we generalize an argument in [1] that assumed Markov processes with trivially removable drift R(t). In fact, that argument was based on no specifically Markovian assumption and applies quite generally to

nonMarkovian martingales. In the analysis that follows, we assume a drift-free nonstationary process x(t) with the initial condition x(t o)=0, so that the variance is given by σ2==∫x2f1(x,t). By the increments of the process we

mean x(t;T) = x(t+T)-x(t) and x(t;-T)=x(t)-x(t-T).

We state in advance that we assume that [-∞

Stationary increments are defined by

!x(t+T)"x(t)=x(T), (27)

‘in distribution’, and by nonstationary increments [1,3,4,5] we mean that

!x(t+T)"x(t)#x(T). (28)

in distribution. When (27) holds, then given the density of ‘positions’ f1(x,t), we also know the density f1(x(T),T)=f1(x(t+T)-x(t),T) of increments independently of the starting time t. Whenever the increments are nonstationary then any analysis of the increments inherently requires the two-point density, f2(x(t+T),t+T;x(t),t). From the standpoint of theory there exists no 1-point density of increments f(x;T),T) depending on T alone, independent of t, and spurious 1-point histograms of increments are typically constructed empirically by assuming that the converse is possible [4]. Next, we place an important restriction on the class of stochastic processes under consideration. According to Mandelbrot, so-called ‘efficient market’ has no memory that can be easily exploited in trading [18]. Beginning with that idea we can assert the necessary but not

sufficient condition, the absence of increment autocorrelations,

!(x(t

1

)"x(t

1

"T

1

))(x(t

2

+T

2

)"x(t

2

))=0,(29)

when there is no time interval overlap, t10. This is a much weaker condition and far more interesting than asserting that the increments are statistically independent. We will see that this condition leaves the question of the dynamics of x(t) open, except to rule out processes with increment autocorrelations, specifically stationary increment processes like fBm [1,20], but also processes with correlated nonstationary increments like the time translationally invariant Gaussian transition densities described in [2].

Consider a stochastic process x(t) where the increments (29) are uncorrelated. From this condition we easily obtain the autocorrelation function for positions (returns), sometimes called ‘serial autocorrelations’. If t>s then

!

x(t)x(s)=(x(t)"x(s))x(s)+x2(s)=x2(s)>0, (30) since with x(t o)=0 x(s)-x(t o)=x(s), so that = is simply the variance in x. Given a history (x(t), ….,x(s),…,x(0)), or (x(t n),…x(t k),…,x(t1)), (30) reflects a martingale property:

!

x(t

n

)x(t

k

)=dx

n

...dx

1

x

n

"x k p n(x n,t n x n,t n,...,x n,t n,...)p n#1(...)...p k+1(...)f k(...)

=x

k

2

"f k(x k,t k;...;x1,t1)dx k...dx1=x2

"f1(x,t)dx=x k2(t k)

(31)

where

!

x

m

dx

m

p

m

(x

m

,t

m

x

m"1

,t

m"1

;...;x

1

,t

1

)=x

m"1

#. (32)

Every martingale generates uncorrelated increments and conversely, and so for a Martingale = if s

In a martingale process, the history dependence cannot be detected at the level of 2-point correlations, memory effects can at best first appear at the level 3-point correlations requiring the study of a transition density p3. Here, we have not postulated a martingale, instead we’ve deduced that property from the lack of pair wise increment correlations. But this is only part of the story. What follows next is crucial for avoiding mistakes in data analysis [4].

Combining

!

(x(t+T)"x(t))2=+(x2(t+T)+x2(t)"2x(t+T)x(t)

(33)

with (34), we get

!

(x(t+T)"x(t))2=x2(t+T)"x2(t)(34)

which depends on both t and T, excepting the case where is linear in t. Uncorrelated increments are generally nonstationary. Therefore, martingales generate uncorrelated, typically nonstationary increments. So, at the level of pair correlations a martingale with memory cannot be

2 Note that (30,31) hold for time translationally invariant martingales, where p

2

(x,t:y,s)=p2(x,t-s:y,0). One can easily check this for a drift-free Gaussian Markov process. I.e., time translational invariance does not imply that is

a function of t-s alone. Time translational invariance of p n, n≥2, does not imply that a statistical equilibrium density

f1(x) exists and is approached asymptotically by f1(x,t) [21]. I.e., a time translationally invariant martingale on [-∞,∞] cannot yield a stationary process, cannot lead to statistical equilibrium.

distinguished empirically from a drift-free Markov process. To see the memory in a martingale one must study at the very least the 3-point correlations.

Summarizing, we’ve shown explicitly that fBm is not a martingale [1], while every Markov process with trivial drift R(t) can be transformed into a (local) Martingale via the substitution of x(t)-∫Rdt for x(t): Ito sdes with vanishing drift describe local martingales [13]. A martingale may have memory, and we’ve provided a model diffusion coefficient to illustrate the appearance of memory (any drift or diffusion coefficient depending on a state (x’,t’) other than the present state (x,t) exhibits memory, so diffusive models with memory are quite easy to construct). We’ve shown that uncorrelated increments are nonstationary unless the variance is linear in t. This means that looking for memory in two point correlations is useless: at that level of description a martingale with memory will look Markovian. To find the memory in a martingale one must study the transition densities p n and correlations for n≥3. This has not been discussed in the literature, so far as we know.

5. The Efficient Market Hypothesis

We begin by sumarizing our viewpoint for the reader. Real finance markets are hard to beat, arbitrage posibilites are hard to find and, once found, tend to disappear fast. In our opinion the EMH is simply an attempt to mathematize the idea that the market is very hard to beat. If there is no useful information in market prices, then those prices can be counted as noise, the product of ‘noise trading’. A martingale formulation of the EMH embodies the idea that the market is hard to beat, is overwhelmingly noise, but leaves open the question of hard to find correlations that might be exploited for exceptional profit.

A strict interpretation of the EMH is that there are no correlations, no patterns of any kind, that can be employed systematically to beat the average return reflecting the market itself: if one wants a higher return, then one must take on more risk. A Markov market is unbeatable, it has no systematically repeated patterns, no memory to exploit. We will argue below that the stipulation should be added that in discussing the EMH we should consider only normal, liquid markets, meaning very liquid markets with small enough transactions that approximately reversible trading is possible on a time scale of seconds [3]. Otherwise, ‘Brownian’ market models do not apply. Liquidity, ‘the money bath’ created by the noise traders whose behavior is reflected in the diffusion coefficient [3], is somewhat qualitatively analogous to the idea of the heat bath in thermodynamics [24]: the second by second fluctuations in x(t) are created by the continual noise trading.

Mandelbrot [18] proposed a less strict and very attractive definition of the EMH, one that directly reflects the fact that financial markets are hard to beat but leaves open the question whether the market can be beaten in principle at some high level of insight. He suggested that a martingale condition on returns realistically reflects the notion of the EMH. A martingale may contain memory, but that memory can’t be easily exploited to beat the market precisely because the expectation of a martingale process x(t) at any later time is simply the last observed return. In addition, as we’ve shown above, pair correlations in increments cannot be exploited to beat the market either. The idea that memory may arise (in commodities, e,g.) from other variables (like the weather) [18] correponds in statistical physics [11] to the appearance of memory as a consequence of averaging over other, more rapidly changing, variables in the larger dynamical system.

The martingale (as opposed to Markov) version of the EMH

is also interesting because technical traders assume that certain price sequences give signals either to sell or buy. In principle, that is permitted in a martingale. A particular

price sequence (p(t n), ….,p(t1)), were it quasi-systematically

to repeat, can be encoded as returns (x n,…,x1) so that a conditional probability density p n(x n;x n-1,…,x1) could be interpreted as a providing a risk measure to buy or sell. By

‘quasi-repetition’ of the sequence we mean that p n(x n;x n-1,…,x1) is significantly greater than a Markovian prediction. Typically, technical traders make the mistake of

trying to interpret random price sequences quasi-deterministically, which differs from our interpretation of

‘technical trading’ based on conditional probabilities (see Lo

et al [25] for a discussion of technical trading claims, but

based on a non-martingale, non-empirically based model of prices). With only a conditional probability for ‘signaling’ a specific price sequence, an agent with a large debt to equity

ratio can easily suffer the Gamblers’ Ruin. In any case, we

can offer no advice about technical trading, because the existence of market memory has not been firmly established

(the question is left open by the analysis of ref. [25]), liquid finance markets look pretty Markovian so far as we’ve been

able to understand the data [4], but one can go systematically beyond the level of pair correlations to try to

find memory. Apparently, this remains to be done, or at

least to be published.

Memory could reflect heavy trading around a particular

price and can, of course, be lost in the course of time. The

writer remembers well the period of a few months ca. 1999

when CPQ sold for around $22, and was traded often in the

range $18-$25 before crashing further. Whether that provides an example is purely speculation at this point.

Fama [26] took Mandelbrot’s proposal seriously and tried to test finance data at the simplest level for a fair game condition. We continue our discussion by first correcting a mathematical mistake made by Fama (see the first two of three unnumbered equations at the bottom of pg. 391 in [26]), who wrongly concluded in his discussion of martingales as a fair game condition that =0. Here’s his argument, rewritten partly in our notation. Let x(t) denote a ‘fair game’. With the initial condition chosen as x(t o)=0, then we have the unconditioned expectation =∫xdxf1(x,t)=0 (there is no drift). Then the so-called ‘serial covariance’ is given by

!

x(t+T)x(t)=xdx

cond(x)

f

1

(x,t)

". (40)

Fama states that this vanishes because cond=0. This is impossible: by a fair game we mean a Martingale, the conditional expectation is cond=∫ydyp2(y,t+T;x,t)=x=x(t)≠0, and so Fama should have concluded instead that = as we showed in the last section. Vanishing of (40) would be true of statistically independent variables but is violated by a ‘fair game’. Can Fama’s argument be salvaged? Suppose that instead of x(t) we would try to use the increment x(t,T)=x(t+T)-x(t) as variable. Then =0 for a Martingale, as we showed in part 4. However, Fama’s argument still would not be generally correct because x(t,T) canno t be taken as a ‘fair game’ variable unless the variance is linear in t, and in financial markets the variance is not linear in t [3,4]. Fama’s mislabeling of time dependent averages (typical in economics and finance literature) as ‘market equilibrium’ has been corrected elsewhere [24].

In our discussion of the EMH we shall not follow the economists’ tradition and discuss three separate forms (weak, semi-strong, and strong [27]) of the EMH, where a

免疫学的临床应用

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常用免疫学检验技术的基本原理

常用免疫学检验技术的基本原理 免疫学检测即是根据抗原、抗体反应的原理,利用已知的抗原检测未知的抗体或利用已知的抗体检测未知的抗原。由于外源性和内源性抗原均可通过不同的抗原递呈途径诱导生物机体的免疫应答,在生物体内产生特异性和非特异性T 细胞的克隆扩增,并分泌特异性的免疫球蛋白(抗体)。由于抗体-抗原的结合具有特异性和专一性的特点,这种检测可以定性、定位和定量地检测某一特异的蛋白(抗原或抗体)。免疫学检测技术的用途非常广泛,它们可用于各种疾病的诊断、疗效评价及发病机制的研究。 最初的免疫检测方法是将抗原或抗体的一方或双方在某种介质中进行扩散,通过观察抗原-抗体相遇时产生的沉淀反应,检测抗原或抗体,最终达到诊断的目的。这种扩散可以是蛋白的自然扩散,例如环状沉淀试验、单向免疫扩散试验、双向免疫扩散实验。单向免疫扩散试验就是在凝胶中混入抗体,制成含有抗体的凝胶板,而将抗原加入凝胶板预先打好的小孔内,让抗原从小孔向四周的凝胶自然扩散,当一定浓度的抗原和凝胶中的抗体相遇时便能形成免疫复合物,出现以小孔为中心的圆形沉淀圈,沉淀圈的直径与加入的抗原浓度成正比。 利用蛋白在不同酸碱度下带不同电荷的特性,可以利用人为的电场将抗原、抗体扩散,例如免疫电泳试验和双向免疫电泳。免疫电泳首先将抗原加入凝胶中电泳,将抗原各成分依次分散开。然后沿电泳方向平行挖一直线形槽,于槽内加入含有针对各种抗原的混合抗体,让各抗原成分与相应抗体进行自然扩散,形成沉淀线。然后利用标准的抗原-抗体沉淀线进行抗原蛋白(或抗体)的鉴别。上述的方法都是利用肉眼观察抗原-抗体反应产生的沉淀,因此灵敏度有很大的局限。比浊法引入沉淀检测产生的免疫比浊法就是利用浊度计测量液体中抗原-抗体反应产生的浊度,根据标准曲线来计算抗原(或抗体)的含量。该方法不但大大提高了检测的灵敏度,且可对抗原、抗体进行定量的检测。

免疫学发展简史

免疫学发展简史 分三个时期:①经验免疫学时期(公元前400年~18世纪末); ②免疫学科建立时期(19世纪~1975年);③现代免疫学时期(1975年至今)。 一、经验免疫学时期(公元前400年~18世纪末) (一)天花的危害 天花是一种古老的、世界流行的烈性传染病,死亡率可高达25%~40%,我国民间早有“生了孩子算一半,得了天花才算全”的说法。患天花痊愈后留下永久的疤痕,但可获得终身免疫。 16世纪由于西班牙殖民者侵略,将天花传播到美洲,墨西哥土著人从16世纪初(1518年)的2000~3000万人到16世纪末减少到100万人,阿茨特克帝国消亡。16世纪中期之后向南进发,在美洲中部毁灭了玛雅和印加文明,随后又毁灭了秘鲁。 (二)人痘苗接种 1.人痘苗接种实践: 中医称天花为“痘疮”,据史书记载人痘苗接种预防天花的方法是在公元前约400年由我们中华民族的祖先建立的。Zinsser微生物学(1988):发明于中国2000多年之前。 明庆隆年间(1567~1572);16~17世纪人痘苗接种预防天花已在全国普遍展开。清康熙27年(1688)俄国曾派医生到北京学习种痘技术。并经丝绸之路东传至朝鲜、日本和东南亚国家,西传至欧亚、北非及北美各国。 1700年传入英国/Momtagu夫人在英国积极推广人痘苗接种中起了重要的作用。 1721~1722年天花在英国爆发流行期间,英国皇家学会在国王的特许下,主持进行了用犯人和孤儿做人痘苗接种的试验,均获得了成功,试验者无一人死于天花。在此基础上,1722年给英国威尔士王子的两个女儿(一个9岁,一个11岁)也进行了人痘苗接种,也都获得成功。 2.人痘苗接种意义:有三个方面: ①能有效预防天花。 ②在接种方法、痘苗的制备和保存建立了一整套完整的科学方法,为以后疫苗的发展提供了丰富的经验和借鉴。 清代吴谦所著的《医宗金鉴·幼科种痘心法要旨》(1742年)中介绍了四种接种法:痘衣法-痘浆法-旱苗法-水苗法。并指出这些方法的优劣:“水苗为上,旱苗次之,痘衣多不应验,痘浆太涉残忍。” 对痘苗保存指出:“若遇热则气泄,日久则气薄,触污秽则气不清,藏不洁则气不正,此蓄苗之法。”“须贮新磁瓶内,上以物密覆

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免疫球蛋白的类、亚类、型、亚型 根据CH不同分为五类: CH:γαμδε Ig: IgG IgA IgM IgD IgE 亚类:IgG1~IgG4; IgA1、IgA2。 根据CL不同分为两型:κ型、λ型 亚型:λ1~λ4 (二)Ig的功能区 L链:VL、CL。 H链: VH、CH1、CH2、CH3(IgA、IgG、IgD)VH、CH1、CH2、CH3、CH4(IgM、IgE)铰链区:位于CH1与CH2之间,易弯曲。各功能区主要功能: VH、VL:与抗原特异性结合部位 CH和CL:遗传标志所在处 IgG的CH2和IgM的CH3:激活补体 IgG的CH2/CH3和IgE的CH4:结合细胞

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免疫学研究报告现状及发展前景

XX学院 Hefei University 医学免疫学 题目:医学免疫学述 系别:生物与环境工程系 专业:_ 12级生物技术 学号: 1202021037 XX:戎晓娜 指导教师:甤 2015年4月10日

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免疫学研究的发展趋势现状

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免疫学发展简史

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天花。在此基础上,1722年给英国威尔士王子的两个女儿(一个9岁,一个11 岁)也进行了人痘苗接种,也都获得成功。 羅2.人痘苗接种意义:有三个方面: 袄①能有效预防天花。 蚁②在接种方法、痘苗的制备和保存建立了一整套完整的科学方法,为以后疫 苗的发展提供了丰富的经验和借鉴。 羆清代吴谦所著的《医宗金鉴?幼科种痘心法要旨》(1742年)中介绍了四种接种法:痘衣法-痘浆法-旱苗法-水苗法。并指出这些方法的优劣:“水苗为上,旱苗次之,痘衣多不应验,痘浆太涉残忍。” 蚇对痘苗保存指出:“若遇热则气泄,日久则气薄,触污秽则气不清,藏不洁则气不正,此蓄苗之法。” “须贮新磁瓶内,上以物密覆之,置之洁净之所,清凉之处。” 薃痘苗有“时苗”和“熟苗”之分,开始采用的痘痂叫时苗,经人体接种传代后制备的叫熟苗。清代朱奕梁编著的《种痘心法》中写道:“其苗传种愈久,则药力之提拔愈清。人工之选炼愈熟,火毒汰尽,精气独存,所以万全而无害也。若‘时苗’ 能连种七次,精加选炼,则为‘熟苗',不可不知。” 蚁③“以毒攻毒”的思想对防治疾病意义深远。 莇首届诺贝尔医学奖获得者贝林(Emil von Behring)深受“以毒攻毒”这种观念的影响,开创了抗毒素免疫治疗的方法。他说:“中国人远在两千年前即知’以毒攻毒’的医理,这是合乎现代科学的一句古训!” 肅(三)牛痘苗接种 莂英国乡村医生琴纳(Edward Jenne)1798发明牛痘苗接种,1804年传入中国。牛痘接种预防天花既安全又有效,是一划时代的发明。 螀他于1796年9月17日给一个8岁男孩的右臂划痕接种了牛痘,两天后男孩感到有些不适,可是很快就好了。6周后再接种天花患者的痘浆,未发生天花。以后又继续试验,证实了牛痘苗接种预防天花的作用。于1798年公布了他的研 究论文。

免疫学在食品科学中的应用-

免疫学在食品科学中的应用 吉林大学生物与农业工程学院 09级食品质量与安全 (45090729)

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免疫学综述

免疫学对生理学发展的重 要性 贵州大学生命科学学院 2008级生物科学 张萌 学号:080704110118

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合肥学院 Hefei University 医学免疫学 : 生物与环境工程系 专业:_ 12级生物技术 学号: 1202021037 姓名: 戎晓娜 指导教师: 李甤 2015年 4月 10日

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