Almost Sure

26 September 16

Do Convex and Decreasing Functions Preserve the Semimartingale Property?

Some years ago, I spent considerable effort trying to prove the hypothesis below. After failing at this, I spent time trying to find a counterexample, but also with no success. I did post this as a question on mathoverflow, but it has so far received no conclusive answers. So, as far as I am aware, the following statement remains unproven either way.

Hypothesis H1 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and right-continuous and decreasing in t. Then, for any semimartingale X, {f(t,X_t)} is a semimartingale.

It is well known that convex functions of semimartingales are themselves semimartingales. See, for example, the Ito-Tanaka formula. More generally, if {f(t,x)} was increasing in t rather than decreasing, then it can be shown without much difficulty that {f(t,X_t)} is a semimartingale. Consider decomposing {f(t,X_t)} as

\displaystyle  f(t,X_t)=\int_0^tf_x(s,X_{s-})\,dX_s+V_t, (1)

for some process V. By convexity, the right hand derivative of {f(t,x)} with respect to x always exists, and I am denoting this by {f_x}. In the case where f is twice continuously differentiable then the process V is given by Ito’s formula which, in particular, shows that it is a finite variation process. If {f(t,x)} is convex in x and increasing in t, then the terms in Ito’s formula for V are all increasing and, so, it is an increasing process. By taking limits of smooth functions, it follows that V is increasing even when the differentiability constraints are dropped, so {f(t,X_t)} is a semimartingale. Now, returning to the case where {f(t,x)} is decreasing in t, Ito’s formula is only able to say that V is of finite variation, and is generally not monotonic. As limits of finite variation processes need not be of finite variation themselves, this does not say anything about the case when f is not assumed to be differentiable, and does not help us to determine whether or not {f(t,X_t)} is a semimartingale.

Hypothesis H1 can be weakened by restricting to continuous functions of continuous martingales.

Hypothesis H2 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and continuous and decreasing in t. Then, for any continuous martingale X, {f(t,X_t)} is a semimartingale.

As continuous martingales are special cases of semimartingales, hypothesis H1 implies H2. In fact, the reverse implication also holds so that hypotheses H1 and H2 are equivalent.

Hypotheses H1 and H2 can also be recast as a simple real analysis statement which makes no reference to stochastic processes.

Hypothesis H3 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be convex in x and decreasing in t. Then, {f=g-h} where {g(t,x)} and {h(t,x)} are convex in x and increasing in t.

(more…)

14 September 16

Failure of the Martingale Property For Stochastic Integration

If X is a cadlag martingale and {\xi} is a uniformly bounded predictable process, then is the integral

\displaystyle  Y=\int\xi\,dX (1)

a martingale? If {\xi} is elementary this is one of most basic properties of martingales. If X is a square integrable martingale, then so is Y. More generally, if X is an {L^p}-integrable martingale, any {p > 1}, then so is Y. Furthermore, integrability of the maximum {\sup_{s\le t}\lvert X_s\rvert} is enough to guarantee that Y is a martingale. Also, it is a fundamental result of stochastic integration that Y is at least a local martingale and, for this to be true, it is only necessary for X to be a local martingale and {\xi} to be locally bounded. In the general situation for cadlag martingales X and bounded predictable {\xi}, it need not be the case that Y is a martingale. In this post I will construct an example showing that Y can fail to be a martingale. (more…)

12 September 16

Martingales with Non-Integrable Maximum

Filed under: Examples and Counterexamples — George Lowther @ 12:01 PM
Tags: , ,

It is a consequence of Doob’s maximal inequality that any {L^p}-integrable martingale has a maximum, up to a finite time, which is also {L^p}-integrable for any {p > 1}. Using {X^*_t\equiv\sup_{s\le t}\lvert X_s\rvert} to denote the running absolute maximum of a cadlag martingale X, then {X^*} is {L^p}-integrable whenever {X} is. It is natural to ask whether this also holds for {p=1}. As martingales are integrable by definition, this is just asking whether cadlag martingales necessarily have an integrable maximum. Integrability of the maximum process does have some important consequences in the theory of martingales. By the Burkholder-Davis-Gundy inequality, it is equivalent to the square-root of the quadratic variation, {[X]^{1/2}}, being integrable. Stochastic integration over bounded integrands preserves the martingale property, so long as the martingale has integrable maximal process. The continuous and purely discontinuous parts of a martingale X are themselves local martingales, but are not guaranteed to be proper martingales unless X has integrable maximum process.

The aim of this post is to show, by means of some examples, that a cadlag martingale need not have an integrable maximum. (more…)

11 September 16

The Optimality of Doob’s Maximal Inequality

One of the most fundamental and useful results in the theory of martingales is Doob’s maximal inequality. Use {X^*_t\equiv\sup_{s\le t}\lvert X_s\rvert} to denote the running (absolute) maximum of a process X. Then, Doob’s {L^p} maximal inequality states that, for any cadlag martingale or nonnegative submartingale X and real {p > 1},

\displaystyle  \lVert X^*_t\rVert_p\le c_p \lVert X_t\rVert_p (1)

with {c_p=p/(p-1)}. Here, {\lVert\cdot\rVert_p} denotes the standard Lp-norm, {\lVert U\rVert_p\equiv{\mathbb E}[U^p]^{1/p}}.

An obvious question to ask is whether it is possible to do any better. That is, can the constant {c_p} in (1) be replaced by a smaller number. This is especially pertinent in the case of small p, since {c_p} diverges to infinity as p approaches 1. The purpose of this post is to show, by means of an example, that the answer is no. The constant {c_p} in Doob’s inequality is optimal. We will construct an example as follows.

Example 1 For any {p > 1} and constant {1 \le c < c_p} there exists a strictly positive cadlag {L^p}-integrable martingale {\{X_t\}_{t\in[0,1]}} with {X^*_1=cX_1}.

For X as in the example, we have {\lVert X^*_1\rVert_p=c\lVert X_1\rVert_p}. So, supposing that (1) holds with any other constant {\tilde c_p} in place of {c_p}, we must have {\tilde c_p\ge c}. By choosing {c} as close to {c_p} as we like, this means that {\tilde c_p\ge c_p} and {c_p} is indeed optimal in (1). (more…)

6 September 16

The Maximum Maximum of Martingales with Known Terminal Distribution

In this post I will be concerned with the following problem — given a martingale X for which we know the distribution at a fixed time, and we are given nothing else, what is the best bound we can obtain for the maximum of X up until that time? This is a question with a long history, starting with Doob’s inequalities which bound the maximum in the {L^p} norms and in probability. Later, Blackwell and Dubins (3), Dubins and Gilat (5) and Azema and Yor (1,2) showed that the maximum is bounded above, in stochastic order, by the Hardy-Littlewood transform of the terminal distribution. Furthermore, this bound is the best possible in the sense that there do exists martingales for which it can be attained, for any permissible terminal distribution. Hobson (7,8) considered the case where the starting law is also known, and this was further generalized to the case with a specified distribution at an intermediate time by Brown, Hobson and Rogers (4). Finally, Henry-Labordère, Obłój, Spoida and Touzi (6) considered the case where the distribution of the martingale is specified at an arbitrary set of times. In this post, I will look at the case where only the terminal distribution is specified. This leads to interesting constructions of martingales and, in particular, of continuous martingales with specified terminal distributions, with close connections to the Skorokhod embedding problem.

I will be concerned with the maximum process of a cadlag martingale X,

\displaystyle  X^*_t=\sup_{s\le t}X_s,

which is increasing and adapted. We can state and prove the bound on {X^*} relatively easily, although showing that it is optimal is more difficult. As the result holds more generally for submartingales, I state it in this case, although I am more concerned with martingales here.

Theorem 1 If X is a cadlag submartingale then, for each {t\ge0} and {x\in{\mathbb R}},

\displaystyle  {\mathbb P}\left(X^*_t\ge x\right)\le\inf_{y < x}\frac{{\mathbb E}\left[(X_t-y)_+\right]}{x-y}. (1)

Proof: We just need to show that the inequality holds for each {y < x}, and then it immediately follows for the infimum. Choosing {y < x^\prime < x}, consider the stopping time

\displaystyle  \tau=\inf\{s\ge0\colon X_s\ge x^\prime\}.

Then, {\tau \le t} and {X_\tau\ge x^\prime} whenever {X^*_t \ge x}. As {f(z)\equiv(z-y)_+} is nonnegative and increasing in z, this means that {1_{\{X^*_t\ge x\}}} is bounded above by {f(X_{\tau\wedge t})/f(x^\prime)}. Taking expectations,

\displaystyle  {\mathbb P}\left(X^*_t\ge x\right)\le{\mathbb E}\left[f(X_{\tau\wedge t})\right]/f(x^\prime).

Since f is convex and increasing, {f(X)} is a submartingale so, using optional sampling,

\displaystyle  {\mathbb P}\left(X^*_t\ge x\right)\le{\mathbb E}\left[f(X_t)\right]/f(x^\prime).

Letting {x^\prime} increase to {x} gives the result. ⬜

The bound stated in Theorem 1 is also optimal, and can be achieved by a continuous martingale. In this post, all measures on {{\mathbb R}} are defined with respect to the Borel sigma-algebra.

Theorem 2 If {\mu} is a probability measure on {{\mathbb R}} with {\int\lvert x\rvert\,d\mu(x) < \infty} and {t > 0} then there exists a continuous martingale X (defined on some filtered probability space) such that {X_t} has distribution {\mu} and (1) is an equality for all {x\in{\mathbb R}}.

(more…)

14 August 16

Purely Discontinuous Semimartingales

As stated by the Bichteler-Dellacherie theorem, all semimartingales can be decomposed as the sum of a local martingale and an FV process. However, as the terms are only determined up to the addition of an FV local martingale, this decomposition is not unique. In the case of continuous semimartingales, we do obtain uniqueness, by requiring the terms in the decomposition to also be continuous. Furthermore, the decomposition into continuous terms is preserved by stochastic integration. Looking at non-continuous processes, there does exist a unique decomposition into local martingale and predictable FV processes, so long as we impose the slight restriction that the semimartingale is locally integrable.

In this post, I look at another decomposition which holds for all semimartingales and, moreover, is uniquely determined. This is the decomposition into continuous local martingale and purely discontinuous terms which, as we will see, is preserved by the stochastic integral. This is distinct from each of the decompositions mentioned above, except for the case of continuous semimartingales, in which case it coincides with the sum of continuous local martingale and FV components. Before proving the decomposition, I will start by describing the class of purely discontinuous semimartingales which, although they need not have finite variation, do have many of the properties of FV processes. In fact, they comprise precisely of the closure of the set of FV processes under the semimartingale topology. The terminology can be a bit confusing, and it should be noted that purely discontinuous processes need not actually have any discontinuities. For example, all continuous FV processes are purely discontinuous. For this reason, the term `quadratic pure jump semimartingale’ is sometimes used instead, referring to the fact that their quadratic variation is a pure jump process. Recall that quadratic variations and covariations can be written as the sum of continuous and pure jump parts,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t&\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t&\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array} (1)

The statement that the quadratic variation is a pure jump process is equivalent to saying that its continuous part, {[X]^c}, is zero. As the only difference between the generalized Ito formula for semimartingales and for FV processes is in the terms involving continuous parts of the quadratic variations and covariations, purely discontinuous semimartingales behave much like FV processes under changes of variables and integration by parts. Yet another characterisation of purely discontinuous semimartingales is as sums of purely discontinuous local martingales — which were studied in the previous post — and of FV processes.

Rather than starting by choosing one specific property to use as the definition, I prove the equivalence of various statements, any of which can be taken to define the purely discontinuous semimartingales.

Theorem 1 For a semimartingale X, the following are equivalent.

  1. {[X]^c=0}.
  2. {[X,Y]^c=0} for all semimartingales Y.
  3. {[X,Y]=0} for all continuous semimartingales Y.
  4. {[X,M]=0} for all continuous local martingales M.
  5. {X=M+V} for a purely discontinuous local martingale M and FV process V.
  6. there exists a sequence {\{X^n\}_{n=1,2,\ldots}} of FV processes such that {X^n\rightarrow X} in the semimartingale topology.

(more…)

8 August 16

Purely Discontinuous Local Martingales

The previous post introduced the idea of a purely discontinuous local martingale. In the context of that post, such processes were used to construct local martingales with prescribed jumps, and enabled us to obtain uniqueness in the constructions given there. However, purely discontinuous local martingales are a very useful concept more generally in martingale and semimartingale theory, so I will go into more detail about such processes now. To start, we restate the definition from the previous post.

Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.

We can show that every local martingale decomposes uniquely into continuous and purely discontinuous parts. Continuous local martingales are well understood — for instance, they can always be realized as time-changed Brownian motions. On the other hand, as we will see in a moment, purely discontinuous local martingales can be realized as limits of FV processes, and arguments involving FV local martingales can often to be extended to the purely discontinuous case. So, decomposition (1) below is useful as it allows arguments involving continuous-time local martingales to be broken down into different approaches involving their continuous and purely discontinuous parts. As always, two processes are considered to be equal if they are equivalent up to evanescence.

Theorem 2 Every local martingale X decomposes uniquely as

\displaystyle  X = X^{\rm c} + X^{\rm d} (1)

where {X^{\rm c}} is a continuous local martingale with {X^{\rm c}_0=0} and {X^{\rm d}} is a purely discontinuous local martingale.

Proof: As the process {H=\Delta X} is, by definition, equal to the jump process of a local martingale then it satisfies the hypothesis of Theorem 5 of the previous post. So, there exists a purely discontinuous local martingale {X^{\rm d}} with {\Delta X^{\rm d}=H=\Delta X}. We can take {X^{\rm d}_0=X_0} so that {X^{\rm c}=X-X^{\rm d}} is a continuous local martingale starting from 0.

If {X=\tilde X^{\rm c}+\tilde X^{\rm d}} is another such decomposition, then {\tilde X^{\rm d}} and {X^{\rm d}} have the same jumps and initial value so, by Lemma 3 of the previous post, {\tilde X^{\rm d}=X^{\rm d}}. ⬜

Throughout the remainder of this post, the notation {X^{\rm c}} and {X^{\rm d}} will be used to denote the continuous and purely discontinuous parts of a local martingale X, as given by decomposition (1). Using the notation {\mathcal{M}_{\rm loc}}, {\mathcal{M}_{{\rm loc},0}^{\rm c}} and {\mathcal{M}_{\rm loc}^{\rm d} } respectively for the spaces of local martingales, continuous local martingales starting from zero and the purely discontinuous local martingales, Theorem 2 can be expressed succinctly as

\displaystyle  \mathcal{M}_{\rm loc} = \mathcal{M}_{{\rm loc},0}^{\rm c} \oplus \mathcal{M}_{\rm loc}^{\rm d}. (2)

That is, {\mathcal{M}_{\rm loc}} is the direct sum of {\mathcal{M}_{{\rm loc},0}^{\rm c}} and {\mathcal{M}_{\rm loc}^{\rm d}}. Definition 2 identifies the purely discontinuous local martingales to be, in a sense, orthogonal to the continuous local martingales. Then, (2) can be understood as the decomposition of {\mathcal{M}_{\rm loc}} into the direct sum of the closed subspace {\mathcal{M}_{{\rm loc},0}^{\rm c}} and its orthogonal complement. This does in fact give an alternative, elementary, and commonly used, method of proving decomposition (1). As we have already shown the rather strong result of Theorem 5 from the previous post, the quickest way of proving the decomposition was to simply apply this result. I’ll give more details on the more elementary approach further below.

Definition 1 used above for the class of purely discontinuous local martingales was very convenient for our purposes, as it leads immediately to the proof of Theorem 2. However, there are many alternative characterizations of such processes. For example, they are precisely the processes which are limits of FV local martingales in a strong enough sense. They can also be characterized in terms of their quadratic variations and covariations. Recall that the quadratic variation and covariation are FV processes with jumps {\Delta[X]=(\Delta X)^2} and {\Delta[X,Y]=\Delta X\Delta Y}, so that they can be decomposed into continuous and pure jump components,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t &\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t &\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array} (3)

The following theorem gives several alternative characterizations of the class of purely discontinuous local martingales.

Theorem 3 For a local martingale X, the following are equivalent.

  1. X is purely discontinuous.
  2. {[X,Y]=0} for all continuous local martingales Y.
  3. {[X,Y]^c=0} for all local martingales Y.
  4. {[X]^c=0}.
  5. there exists a sequence {\{X^n\}_{n=1,2,\ldots}} of FV local martingales such that

    \displaystyle  {\mathbb E}\left[\sup_{t\ge0}(X^n_t-X_t)^2\right]\rightarrow0.

(more…)

25 July 16

Constructing Martingales with Prescribed Jumps

In this post we will describe precisely which processes can be realized as the jumps of a local martingale. This leads to very useful decomposition results for processes — see Theorem 10 below, where we give a decomposition of a process X into martingale and predictable components. As I will explore further in future posts, this enables us to construct particularly useful decompositions for local martingales and semimartingales.

Before going any further, we start by defining the class of local martingales which will be used to match prescribed jump processes. The purely discontinuous local martingales are, in a sense, the orthogonal complement to the class of continuous local martingales.

Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.

The class of purely discontinuous local martingales is often denoted as {\mathcal{M}_{\rm loc}^{\rm d}}. Clearly, any linear combination of purely discontinuous local martingales is purely discontinuous. I will investigate {\mathcal{M}_{\rm loc}^{\rm d}} in more detail later but, in order that we do have plenty of examples of such processes, we show that all FV local martingales are purely discontinuous.

Lemma 2 Every FV local martingale is purely discontinuous.

Proof: If X is an FV local martingale and M is a continuous local martingale then we can compute the quadratic covariation,

\displaystyle  [X,M]_t=\sum_{s\le t}\Delta X_s\Delta M_s=0.

The first equality follows because X is an FV process, and the second because M is continuous. So, {XM=XM-[X,M]} is a local martingale and X is purely discontinuous. ⬜

Next, an important property of purely discontinuous local martingales is that they are determined uniquely by their jumps. Throughout these notes, I am considering two processes to be equal whenever they are equal up to evanescence.

Lemma 3 Purely discontinuous local martingales are uniquely determined by their initial value and jumps. That is, if X and Y are purely discontinuous local martingales with {X_0=Y_0} and {\Delta X = \Delta Y}, then {X=Y}.

Proof: Setting {M=X-Y} we have {M_0=0} and {\Delta M = 0}. So, M is a continuous local martingale and {M^2= MX-MY} is a local martingale starting from zero. Hence, it is a supermartingale and we have

\displaystyle  {\mathbb E}[M_t^2]\le{\mathbb E}[M_0^2]=0.

So {M_t=0} almost surely and, by right-continuity, {M=0} up to evanescence. ⬜

Note that if X is a continuous local martingale, then the constant process {Y_t=X_0} has the same initial value and jumps as X. So Lemma 3 has the immediate corollary.

Corollary 4 Any local martingale with is both continuous and purely discontinuous is almost surely constant.

Recalling that the jump process, {\Delta X}, of a cadlag adapted process X is thin, we now state the main theorem of this post and describe precisely those processes which occur as the jumps of a local martingale.

Theorem 5 Let H be a thin process. Then, {H=\Delta X} for a local martingale X if and only if

  1. {\sqrt{\sum_{s\le t}H_s^2}} is locally integrable.
  2. {{\mathbb E}[1_{\{\tau < \infty\}}H_\tau\;\vert\mathcal{F}_{\tau-}]=0} (a.s.) for all predictable stopping times {\tau}.

Furthermore, X can be chosen to be purely discontinuous with {X_0=0}, in which case it is unique.

(more…)

18 July 16

The Doob-Meyer Decomposition for Quasimartingales

As previously discussed, for discrete-time processes the Doob decomposition is a simple, but very useful, technique which allows us to decompose any integrable process into the sum of a martingale and a predictable process. If {\{X_n\}_{n=0,1,2,\ldots}} is an integrable discrete-time process adapted to a filtration {\{\mathcal{F}_n\}_{n=0,1,2,\ldots}}, then the Doob decomposition expresses X as

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle X_n&\displaystyle=M_n+A_n,\smallskip\\ \displaystyle A_n&\displaystyle=\sum_{k=1}^n{\mathbb E}\left[X_k-X_{k-1}\;\vert\mathcal{F}_{k-1}\right]. \end{array} (1)

Then, M is then a martingale and A is an integrable process which is also predictable, in the sense that {A_n} is {\mathcal{F}_{n-1}}-measurable for each {n > 0}. The expected value of the variation of A can be computed in terms of X,

\displaystyle  {\mathbb E}\left[\sum_{k=1}^n\lvert A_k-A_{k-1}\rvert\right] ={\mathbb E}\left[\sum_{k=1}^n\left\lvert {\mathbb E}[X_k-X_{k-1}\vert\;\mathcal{F}_{k-1}]\right\rvert\right].

This is the mean variation of X.

In continuous time, the situation is rather more complex, and will require constraints on the process X other than just integrability. We have already discussed the case for submartingales — the Doob-Meyer decomposition. This decomposes a submartingale into a local martingale and a predictable increasing process.

A natural setting for further generalising the Doob-Meyer decomposition is that of quasimartingales. In continuous time, the appropriate class of processes to use for the component A of the decomposition is the predictable FV processes. Decomposition (2) below is the same as that in the previous post on special semimartingales. This is not surprising, as we have already seen that the class of special semimartingales is identical to the class of local quasimartingales. The difference with the current setting is that we can express the expected variation of A in terms of the mean variation of X, and obtain a necessary and sufficient condition for the local martingale component to be a proper martingale.

As was noted in an earlier post, historically, decomposition (2) for quasimartingales played an important part in the development of stochastic calculus and, in particular, in the proof of the Bichteler-Dellacherie theorem. That is not the case in these notes, however, as we have already proven the main results without requiring quasimartingales. As always, any two processes are identified whenever they are equivalent up to evanescence.

Theorem 1 Every cadlag quasimartingale X uniquely decomposes as

\displaystyle  X=M+A (2)

where M is a local martingale and A is a predictable FV process with {A_0=0}. Then, A has integrable variation over each finite time interval {[0,t]} satisfying

\displaystyle  {\rm Var}_t(X)={\rm Var}_t(M)+{\mathbb E}\left[\int_0^t\,\vert dA\vert\right]. (3)

so that, in particular,

\displaystyle  {\mathbb E}\left[\int_0^t\,\vert dA\vert\right]\le{\rm Var}_t(X). (4)

Furthermore, the following are equivalent,

  1. X is of class (DL).
  2. M is a proper martingale.
  3. inequality (4) is an equality for all times t.

(more…)

2 July 16

Properties of Quasimartingales

The previous two posts introduced the concept of quasimartingales, and noted that they can be considered as a generalization of submartingales and supermartingales. In this post we prove various basic properties of quasimartingales and of the mean variation, extending results of martingale theory to this situation.

We start with a version of optional stopping which applies for quasimartingales. For now, we just consider simple stopping times, which are stopping times taking values in a finite subset of the nonnegative extended reals {\bar{\mathbb R}_+=[0,\infty]}. Stopping a process can only decrease its mean variation (recall the alternative definitions {{\rm Var}} and {{\rm Var}^*} for the mean variation). For example, a process X is a martingale if and only if {{\rm Var}(X)=0}, so in this case the following result says that stopped martingales are martingales.

Lemma 1 Let X be an adapted process and {\tau} be a simple stopping time. Then

\displaystyle  {\rm Var}^*(X^\tau)\le{\rm Var}^*(X). (1)

Assuming, furthermore, that X is integrable,

\displaystyle  {\rm Var}(X^\tau)\le{\rm Var}(X). (2)

and, more precisely,

\displaystyle  {\rm Var}(X)={\rm Var}(X^\tau)+{\rm Var}(X-X^\tau) (3)

(more…)

Next Page »

Create a free website or blog at WordPress.com.