Almost Sure

23 November 19

Algebraic Probability: Quantum Theory

We continue the investigation of representing probability spaces as states on algebras. Whereas, previously, I focused attention on the commutative case and on classical probabilities, in the current post I will look at non-commutative quantum probability.

Quantum theory is concerned with computing probabilities of outcomes of measurements of a physical system, as conducted by an observer. The standard approach is to start with a Hilbert space {\mathcal H}, which is used to represent the states of the system. This is a vector space over the complex numbers, together with an inner product {\langle\cdot,\cdot\rangle}. By definition, this is linear in one argument and anti-linear in the other,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\langle\phi,\lambda\psi+\mu\chi\rangle=\lambda\langle\phi,\psi\rangle+\mu\langle\phi,\chi\rangle,\smallskip\\ &\displaystyle\langle\lambda\phi+\mu\psi,\chi\rangle=\bar\lambda\langle\phi,\chi\rangle+\bar\mu\langle\psi,\chi\rangle,\smallskip\\ &\displaystyle\langle\psi,\phi\rangle=\overline{\langle\phi,\psi\rangle}, \end{array}

for {\phi,\psi,\chi\in\mathcal H} and {\lambda,\mu\in{\mathbb C}}. Positive definiteness is required, so that {\langle\psi,\psi\rangle > 0} for {\psi\not=0}. I am using the physicists’ convention, where the inner product is linear in the second argument and anti-linear in the first. Furthermore, physicists often use the bra–ket notation {\langle\phi\vert\psi\rangle}, which can be split up into the `bra’ {\langle\phi\vert} and `ket’ {\vert\psi\rangle} considered as elements of the dual space of {\mathcal H} and of {\mathcal H} respectively. For a linear operator {A\colon\mathcal H\rightarrow\mathcal H}, the expression {\langle\phi,A\psi\rangle} is often expressed as {\langle\phi\vert A\vert\psi\rangle} in the physicists’ language. By the Hilbert space definition, {\mathcal H} is complete with respect to the norm {\lVert\psi\rVert=\sqrt{\langle\psi,\psi\rangle}}. (more…)

17 November 19

Algebraic Probability (continued)

Filed under: Probability Theory — George Lowther @ 8:19 PM
Tags: , , ,

Continuing on from the previous post, I look at cases where the abstract concept of states on algebras correspond to classical probability measures. Up until now, we have considered commutative real algebras but, before going further, it will help to look instead at algebras over the complex numbers {{\mathbb C}}. In the commutative case, we will see that this is equivalent to using real algebras, but can be more convenient, and in the non-commutative case it is essential. When using complex algebras, we will require the existence of an involution, which can be thought of as a generalisation of complex conjugation.

Recall that, by an algebra {\mathcal A} over a field {K}, we mean that {\mathcal A} is a {K}-vector space together with a binary product operation satisfying associativity, distributivity over addition, compatibility with scalars, and which has a multiplicative identity.

Definition 1 A *-algebra {\mathcal A} is an algebra over {{\mathbb C}} together with an involution, which is a unary operator {\mathcal A\rightarrow\mathcal A}, {a\mapsto a^*}, satisfying,

  1. Anti-linearity: {(\lambda a+\mu b)^*=\bar\lambda a^*+\bar\mu b^*}.
  2. {(ab)^*=b^*a^*}.
  3. {a^{**}=a}

for all {a,b\in\mathcal A} and {\lambda,\mu\in{\mathbb C}}.

(more…)

10 November 19

Algebraic Probability

Filed under: Probability Theory — George Lowther @ 2:16 PM
Tags: , ,

The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities.

In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space {(\Omega,\mathcal F,{\mathbb P})}. This consists of a state space {\Omega}, an event space {\mathcal F}, which is a sigma-algebra of subsets of {\Omega}, and a probability measure {{\mathbb P}}. The measure {{\mathbb P}} is defined as a map {{\mathbb P}\colon\mathcal F\rightarrow{\mathbb R}^+} satisfying countable additivity and normalised as {{\mathbb P}(\Omega)=1}.

A measure space allows us to define integrals of real-valued measurable functions or, in the language of probability, expectations of random variables. We construct the set {L^\infty(\Omega,\mathcal F)} of all bounded measurable functions {X\colon\Omega\rightarrow{\mathbb R}}. This is a real vector space and, as it is closed under multiplication, is an algebra. Expectation, by definition, is the unique linear map {L^\infty\rightarrow{\mathbb R}}, {X\mapsto{\mathbb E}[X]} satisfying {{\mathbb E}[1_A]={\mathbb P}(A)} for {A\in\mathcal F} and monotone convergence: if {X_n\in L^\infty} is a nonnegative sequence increasing to a bounded limit {X}, then {{\mathbb E}[X_n]} tends to {{\mathbb E}[X]}.

In the opposite direction, any nonnegative linear map {p\colon L^\infty(\Omega,\mathcal F)\rightarrow{\mathbb R}} satisfying monotone convergence and {p(1)=1} defines a probability measure by {{\mathbb P}(A)=p(1_A)}. This is the unique measure with respect to which expectation agrees with the linear map, {{\mathbb E}=p}. So, probability measures are in one-to-one correspondence with such linear maps, and they can be viewed as one and the same thing. The Kolmogorov definition of a probability space can be thought of as representing the expectation on the subset of {L^\infty} consisting of indicator functions {1_A}. In practise, it is often more convenient to start with a different subset of {L^\infty}. For example, probability measures on {{\mathbb R}^+} can be defined via their Laplace transform, {\mathcal L_{{\mathbb P}}(a)=\int e^{-ax}d{\mathbb P}(x)}, which represents the expectation on exponential functions {x\mapsto e^{-ax}}. Generalising to complex-valued random variables, probability measures on {{\mathbb R}} are often represented by their characteristic function {\varphi(a)=\int e^{iax}d{\mathbb P}(x)}, which is just the expectation of the complex exponentials {x\mapsto e^{iax}}. In fact, by the monotone class theorem, we can uniquely represent probability measures on {(\Omega,\mathcal F)} by the expectations on any subset {\mathcal K\subseteq L^\infty} which is closed under taking products and generates the sigma-algebra {\mathcal F}. (more…)

27 October 19

The Functional Monotone Class Theorem

Filed under: Probability Theory — George Lowther @ 8:29 PM
Tags: , ,

The monotone class theorem is a very helpful and frequently used tool in measure theory. As measurable functions are a rather general construct, and can be difficult to describe explicitly, it is common to prove results by initially considering just a very simple class of functions. For example, we would start by looking at continuous or piecewise constant functions. Then, the monotone class theorem is used to extend to arbitrary measurable functions. There are different, but related, `monotone class theorems’ which apply, respectively, to sets and to functions. As the theorem for sets was covered in a previous post, this entry will be concerned with the functional version. In fact, even for the functional version, there are various similar, but slightly different, statements of the monotone class theorem. In practice, it is beneficial to use the version which most directly applies to the specific application. So, I will state and prove several different versions in this post. (more…)

6 October 19

The Monotone Class Theorem

Filed under: Probability Theory — George Lowther @ 11:00 AM
Tags: , , , ,

The monotone class theorem, and closely related {\pi}-system lemma, are simple but fundamental theorems in measure theory, and form an essential step in the proofs of many results. General measurable sets are difficult to describe explicitly so, when proving results in measure theory, it is often necessary to start by considering much simpler sets. The monotone class theorem is then used to extend to arbitrary measurable sets. For example, when proving a result about Borel subsets of {{\mathbb R}}, we may start by considering compact intervals and then apply the monotone class theorem. I include this post on the monotone class theorem for reference. (more…)

24 February 19

Properties of the Dual Projections

In the previous post I introduced the definitions of the dual optional and predictable projections, firstly for processes of integrable variation and, then, generalised to processes which are only required to be locally (or prelocally) of integrable variation. We did not look at the properties of these dual projections beyond the fact that they exist and are uniquely defined, which are significant and important statements in their own right.

To recap, recall that an IV process, A, is right-continuous and such that its variation

\displaystyle  V_t\equiv \lvert A_0\rvert+\int_0^t\,\lvert dA\rvert (1)

is integrable at time {t=\infty}, so that {{\mathbb E}[V_\infty] < \infty}. The dual optional projection is defined for processes which are prelocally IV. That is, A has a dual optional projection {A^{\rm o}} if it is right-continuous and its variation process is prelocally integrable, so that there exist a sequence {\tau_n} of stopping times increasing to infinity with {1_{\{\tau_n > 0\}}V_{\tau_n-}} integrable. More generally, A is a raw FV process if it is right-continuous with almost-surely finite variation over finite time intervals, so {V_t < \infty} (a.s.) for all {t\in{\mathbb R}^+}. Then, if a jointly measurable process {\xi} is A-integrable on finite time intervals, we use

\displaystyle  \xi\cdot A_t\equiv\xi_0A_0+\int_0^t\xi\,dA

to denote the integral of {\xi} with respect to A over the interval {[0,t]}, which takes into account the value of {\xi} at time 0 (unlike the integral {\int_0^t\xi\,dA} which, implicitly, is defined on the interval {(0,t]}). In what follows, whenever we state that {\xi\cdot A} has any properties, such as being IV or prelocally IV, we are also including the statement that {\xi} is A-integrable so that {\xi\cdot A} is a well-defined process. Also, whenever we state that a process has a dual optional projection, then we are also implicitly stating that it is prelocally IV.

From theorem 3 of the previous post, the dual optional projection {A^{\rm o}} is the unique prelocally IV process satisfying

\displaystyle  {\mathbb E}[\xi\cdot A^{\rm o}_\infty]={\mathbb E}[{}^{\rm o}\xi\cdot A_\infty]

for all measurable processes {\xi} with optional projection {{}^{\rm o}\xi} such that {\xi\cdot A^{\rm o}} and {{}^{\rm o}\xi\cdot A} are IV. Equivalently, {A^{\rm o}} is the unique optional FV process such that

\displaystyle  {\mathbb E}[\xi\cdot A^{\rm o}_\infty]={\mathbb E}[\xi\cdot A_\infty]

for all optional {\xi} such that {\xi\cdot A} is IV, in which case {\xi\cdot A^{\rm o}} is also IV so that the expectations in this identity are well-defined.

I now look at the elementary properties of dual optional projections, as well as the corresponding properties of dual predictable projections. The most important property is that, according to the definition just stated, the dual projection exists and is uniquely defined. By comparison, the properties considered in this post are elementary and relatively easy to prove. So, I will simply state a theorem consisting of a list of all the properties under consideration, and will then run through their proofs. Starting with the dual optional projection, the main properties are listed below as Theorem 1.

Note that the first three statements are saying that the dual projection is indeed a linear projection from the prelocally IV processes onto the linear subspace of optional FV processes. As explained in the previous post, by comparison with the discrete-time setting, the dual optional projection can be expressed, in a non-rigorous sense, as taking the optional projection of the infinitesimal increments,

\displaystyle  dA^{\rm o}={}^{\rm o}dA. (2)

As {dA} is interpreted via the Lebesgue-Stieltjes integral {\int\cdot\,dA}, it is a random measure rather than a real-valued process. So, the optional projection of {dA} appearing in (2) does not really make sense. However, Theorem 1 does allow us to make sense of (2) in certain restricted cases. For example, if A is differentiable so that {dA=\xi\,dt} for a process {\xi}, then (9) below gives {dA={}^{\rm o}\xi\,dt}. This agrees with (2) so long as {{}^{\rm o}(\xi\,dt)} is interpreted to mean {{}^{\rm o}\xi\,dt}. Also, restricting to the jump component of the increments, {\Delta A=A-A_-}, (2) reduces to (11) below.

We defined the dual projection via expectations of integrals {\xi\cdot A} with the restriction that this is IV. An alternative approach is to first define the dual projections for IV processes, as was done in theorems 1 and 2 of the previous post, and then extend to (pre)locally IV processes by localisation of the projection. That this is consistent with our definitions follows from the fact that (pre)localisation commutes with the dual projection, as stated in (10) below.

Theorem 1

  1. A raw FV process A is optional if and only if {A^{\rm o}} exists and is equal to A.
  2. If the dual optional projection of A exists then,
    \displaystyle  (A^{\rm o})^{\rm o}=A^{\rm o}. (3)
  3. If the dual optional projections of A and B exist, and {\lambda}, {\mu} are {\mathcal F_0}-measurable random variables then,
    \displaystyle  (\lambda A+\mu B)^{\rm o}=\lambda A^{\rm o}+\mu B^{\rm o}. (4)
  4. If the dual optional projection {A^{\rm o}} exists then {{\mathbb E}[\lvert A_0\rvert\,\vert\mathcal F_0]} is almost-surely finite and
    \displaystyle  A^{\rm o}_0={\mathbb E}[A_0\,\vert\mathcal F_0]. (5)
  5. If U is a random variable and {\tau} is a stopping time, then {U1_{[\tau,\infty)}} is prelocally IV if and only if {{\mathbb E}[1_{\{\tau < \infty\}}\lvert U\rvert\,\vert\mathcal F_\tau]} is almost surely finite, in which case
    \displaystyle  \left(U1_{[\tau,\infty)}\right)^{\rm o}={\mathbb E}[1_{\{\tau < \infty\}}U\,\vert\mathcal F_\tau]1_{[\tau,\infty)}. (6)
  6. If the prelocally IV process A is nonnegative and increasing then so is {A^{\rm o}} and,
    \displaystyle  {\mathbb E}[\xi\cdot A^{\rm o}_\infty]={\mathbb E}[{}^{\rm o}\xi\cdot A_\infty] (7)

    for all nonnegative measurable {\xi} with optional projection {{}^{\rm o}\xi}. If A is merely increasing then so is {A^{\rm o}} and (7) holds for nonnegative measurable {\xi} with {\xi_0=0}.

  7. If A has dual optional projection {A^{\rm o}} and {\xi} is an optional process such that {\xi\cdot A} is prelocally IV then, {\xi} is {A^{\rm o}}-integrable and,
    \displaystyle  (\xi\cdot A)^{\rm o}=\xi\cdot A^{\rm o}. (8)
  8. If A is an optional FV process and {\xi} is a measurable process with optional projection {{}^{\rm o}\xi} such that {\xi\cdot A} is prelocally IV then, {{}^{\rm o}\xi} is A-integrable and,
    \displaystyle  (\xi\cdot A)^{\rm o}={}^{\rm o}\xi\cdot A. (9)
  9. If A has dual optional projection {A^{\rm o}} and {\tau} is a stopping time then,
    \displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle(A^{\tau})^{\rm o}=(A^{\rm o})^{\tau},\smallskip\\ &\displaystyle(A^{\tau-})^{\rm o}=(A^{\rm o})^{\tau-}. \end{array} (10)
  10. If the dual optional projection {A^{\rm o}} exists, then its jump process is the optional projection of the jump process of A,
    \displaystyle  \Delta A^{\rm o}={}^{\rm o}\!\Delta A. (11)
  11. If A has dual optional projection {A^{\rm o}} then
    \displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle{\mathbb E}\left[\xi_0\lvert A^{\rm o}_0\rvert + \int_0^\infty\xi\,\lvert dA^{\rm o}\rvert\right]\le{\mathbb E}\left[{}^{\rm o}\xi_0\lvert A_0\rvert + \int_0^\infty{}^{\rm o}\xi\,\lvert dA\rvert\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\xi_0(A^{\rm o}_0)_+ + \int_0^\infty\xi\,(dA^{\rm o})_+\right]\le{\mathbb E}\left[{}^{\rm o}\xi_0(A_0)_+ + \int_0^\infty{}^{\rm o}\xi\,(dA)_+\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\xi_0(A^{\rm o}_0)_- + \int_0^\infty\xi\,(dA^{\rm o})_-\right]\le{\mathbb E}\left[{}^{\rm o}\xi_0(A_0)_- + \int_0^\infty{}^{\rm o}\xi\,(dA)_-\right], \end{array} (12)

    for all nonnegative measurable {\xi} with optional projection {{}^{\rm o}\xi}.

  12. Let {\{A^n\}_{n=1,2,\ldots}} be a sequence of right-continuous processes with variation

    \displaystyle  V^n_t=\lvert A^n_0\rvert + \int_0^t\lvert dA^n\rvert.

    If {\sum_n V^n} is prelocally IV then,

    \displaystyle  \left(\sum\nolimits_n A^n\right)^{\rm o}=\sum\nolimits_n\left(A^n\right)^{\rm o}. (13)

(more…)

8 February 19

Dual Projections

The optional and predictable projections of stochastic processes have corresponding dual projections, which are the subject of this post. I will be concerned with their initial construction here, and show that they are well-defined. The study of their properties will be left until later. In the discrete time setting, the dual projections are relatively straightforward, and can be constructed by applying the optional and predictable projection to the increments of the process. In continuous time, we no longer have discrete time increments along which we can define the dual projections. In some sense, they can still be thought of as projections of the infinitesimal increments so that, for a process A, the increments of the dual projections {A^{\rm o}} and {A^{\rm p}} are determined from the increments {dA} of A as

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dA^{\rm o}={}^{\rm o}(dA),\smallskip\\ &\displaystyle dA^{\rm p}={}^{\rm p}(dA). \end{array} (1)

Unfortunately, these expressions are difficult to make sense of in general. In specific cases, (1) can be interpreted in a simple way. For example, when A is differentiable with derivative {\xi}, so that {dA=\xi dt}, then the dual projections are given by {dA^{\rm o}={}^{\rm o}\xi dt} and {dA^{\rm p}={}^{\rm p}\xi dt}. More generally, if A is right-continuous with finite variation, then the infinitesimal increments {dA} can be interpreted in terms of Lebesgue-Stieltjes integrals. However, as the optional and predictable projections are defined for real valued processes, and {dA} is viewed as a stochastic measure, the right-hand-side of (1) is still problematic. This can be rectified by multiplying by an arbitrary process {\xi}, and making use of the transitivity property {{\mathbb E}[\xi\,{}^{\rm o}(dA)]={\mathbb E}[({}^{\rm o}\xi)dA]}. Integrating over time gives the more meaningful expressions

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle {\mathbb E}\left[\int_0^\infty \xi\,dA^{\rm o}\right]={\mathbb E}\left[\int_0^\infty{}^{\rm o}\xi\,dA\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\int_0^\infty \xi\,dA^{\rm p}\right]={\mathbb E}\left[\int_0^\infty{}^{\rm p}\xi\,dA\right]. \end{array}

In contrast to (1), these equalities can be used to give mathematically rigorous definitions of the dual projections. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}, and processes are identified whenever they are equal up to evanescence. The terminology `raw IV process‘ will be used to refer to any right-continuous integrable process whose variation on the whole of {{\mathbb R}^+} has finite expectation. The use of the word `raw’ here is just to signify that we are not requiring the process to be adapted. Next, to simplify the expressions, I will use the notation {\xi\cdot A} for the integral of a process {\xi} with respect to another process A,

\displaystyle  \xi\cdot A_t\equiv\xi_0A_0+\int_0^t\xi\,dA.

Note that, whereas the integral {\int_0^t\xi\,dA} is implicitly taken over the range {(0,t]} and does not involve the time-zero value of {\xi}, I have included the time-zero values of the processes in the definition of {\xi\cdot A}. This is not essential, and could be excluded, so long as we were to restrict to processes starting from zero. The existence and uniqueness (up to evanescence) of the dual projections is given by the following result.

Theorem 1 (Dual Projections) Let A be a raw IV process. Then,

  • There exists a unique raw IV process {A^{\rm o}} satisfying
    \displaystyle  {\mathbb E}\left[\xi\cdot A^{\rm o}_\infty\right]={\mathbb E}\left[{}^{\rm o}\xi\cdot A_\infty\right] (2)

    for all bounded measurable processes {\xi}. We refer to {A^{\rm o}} as the dual optional projection of A.

  • There exists a unique raw IV process {A^{\rm p}} satisfying
    \displaystyle  {\mathbb E}\left[\xi\cdot A^{\rm p}_\infty\right]={\mathbb E}\left[{}^{\rm p}\xi\cdot A_\infty\right] (3)

    for all bounded measurable processes {\xi}. We refer to {A^{\rm p}} as the dual predictable projection of A.

Furthermore, if A is nonnegative and increasing then so are {A^{\rm o}} and {A^{\rm p}}.

(more…)

21 January 19

Pathwise Properties of Optional and Predictable Projections

Recall that the the optional and predictable projections of a process are defined, firstly, by a measurability property and, secondly, by their values at stopping times. Namely, the optional projection is measurable with respect to the optional sigma-algebra, and its value is defined at each stopping time by a conditional expectation of the original process. Similarly, the predictable projection is measurable with respect to the predictable sigma-algebra and its value at each predictable stopping time is given by a conditional expectation. While these definitions can be powerful, and many properties of the projections follow immediately, they say very little about the sample paths. Given a stochastic process X defined on a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})} with optional projection {{}^{\rm o}\!X} then, for each {\omega\in\Omega}, we may be interested in the sample path {t\mapsto{}^{\rm o}\!X_t(\omega)}. For example, is it continuous, right-continuous, cadlag, etc? Answering these questions requires looking at {{}^{\rm o}\!X_t(\omega)} simultaneously at the uncountable set of times {t\in{\mathbb R}^+}, so the definition of the projection given by specifying its values at each individual stopping time, up to almost-sure equivalence, is not easy to work with. I did establish some of the basic properties of the projections in the previous post, but these do not say much regarding sample paths.

I will now establish the basic properties of the sample paths of the projections. Although these results are quite advanced, most of the work has already been done in these notes when we established some pathwise properties of optional and predictable processes in terms of their behaviour along sequences of stopping times, and of predictable stopping times. So, the proofs in this post are relatively simple and will consist of applications of these earlier results.

Before proceeding, let us consider what kind of properties it is reasonable to expect of the projections. Firstly, it does not seem reasonable to expect the optional projection {{}^{\rm o}\!X} or the predictable projection {{}^{\rm p}\!X} to satisfy properties not held by the original process X. Therefore, in this post, we will be concerned with the sample path properties which are preserved by the projections. Consider a process with constant paths. That is, {X_t=U} at all times t, for some bounded random variable U. This has about as simple sample paths as possible, so any properties preserved by the projections should hold for the optional and predictable projections of X. However, we know what the projections of this process are. Letting M be the martingale defined by {M_t={\mathbb E}[U\,\vert\mathcal F_t]} then, assuming that the underlying filtration is right-continuous, M has a cadlag modification and, furthermore, this modification is the optional projection of X. So, assuming that the filtration is right-continuous, the optional projection of X is cadlag, meaning that it is right-continuous and has left limits everywhere. So, we can hope that the optional projection preserves these properties. If the filtration is not right-continuous, then M need not have a cadlag modification, so we cannot expect optional projection to preserve right-continuity in this case. However, M does still have a version with left and right limits everywhere, which is the optional projection of X. So, without assuming right-continuity of the filtration, we may still hope that the optional projection preserves the existence of left and right limits of a process. Next, the predictable projection is equal to the left limits, {{}^{\rm p}\!X_t=M_{t-}}, which is left-continuous with left and right limits everywhere. Therefore, we can hope that predictable projections preserve left-continuity and the existence of left and right limits. The existence of cadlag martingales which are not continuous, such as the compensated Poisson process, imply that optional projections do not generally preserve left-continuity and the predictable projection does not preserve right-continuity.

Recall that I previously constructed a version of the optional projection and the predictable projection for processes which are, respectively, right-continuous and left-continuous. This was done by defining the projection at each deterministic time and, then, enforcing the respective properties of the sample paths. We can use the results in those posts to infer that the projections do indeed preserve these properties, although I will now more direct proofs in greater generality, and using the more general definition of the optional and predictable projections.

We work with respect to a complete filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}. As usual, we say that the sample paths of a process satisfy any stated property if they satisfy it up to evanescence. Since integrability conditions will be required, I mention those now. Recall that a process X is of class (D) if the set of random variables {X_\tau}, over stopping times {\tau}, is uniformly integrable. It will be said to be locally of class (D) if there is a sequence {\tau_n} of stopping times increasing to infinity and such that {1_{\{\tau_n > 0\}}1_{[0,\tau_n]}X} is of class (D) for each n. Similarly, it will be said to be prelocally of class (D) if there is a sequence {\tau_n} of stopping times increasing to infinity and such that {1_{[0,\tau_n)}X} is of class (D) for each n.

Theorem 1 Let X be pre-locally of class (D), with optional projection {{}^{\rm o}\!X}. Then,

  • if X has left limits, so does {{}^{\rm o}\!X}.
  • if X has right limits, so does {{}^{\rm o}\!X}.

Furthermore, if the underlying filtration is right-continuous then,

  • if X is right-continuous, so is {{}^{\rm o}\!X}.
  • if X is cadlag, so is {{}^{\rm o}\!X}.

(more…)

20 January 19

Properties of Optional and Predictable Projections

Having defined optional and predictable projections in an earlier post, I now look at their basic properties. The first nontrivial property is that they are well-defined in the first place. Recall that existence of the projections made use of the existence of cadlag modifications of martingales, and uniqueness relied on the section theorems. By contrast, once we accept that optional and predictable projections are well-defined, everything in this post follows easily. Nothing here requires any further advanced results of stochastic process theory.

Optional and predictable projections are similar in nature to conditional expectations. Given a probability space {(\Omega,\mathcal F,{\mathbb P})} and a sub-sigma-algebra {\mathcal G\subseteq\mathcal F}, the conditional expectation of an ({\mathcal F}-measurable) random variable X is a {\mathcal G}-measurable random variable {Y={\mathbb E}[X\,\vert\mathcal G]}. This is defined whenever the integrability condition {{\mathbb E}[\lvert X\rvert\,\vert\mathcal G] < \infty} (a.s.) is satisfied, only depends on X up to almost-sure equivalence, and Y is defined up to almost-sure equivalence. That is, a random variable {X^\prime} almost surely equal to X has the same conditional expectation as X. Similarly, a random variable {Y^\prime} almost-surely equal to Y is also a version of the conditional expectation {{\mathbb E}[X\,\vert\mathcal G]}.

The setup with projections of stochastic processes is similar. We start with a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}, and a (real-valued) stochastic process is a map

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle X\colon{\mathbb R}^+\times\Omega\rightarrow{\mathbb R},\smallskip\\ &\displaystyle (t,\omega)\mapsto X_t(\omega) \end{array}

which we assume to be jointly-measurable. That is, it is measurable with respect to the Borel sigma-algebra {\mathcal B({\mathbb R})} on the image, and the product sigma-algebra {\mathcal B({\mathbb R})\otimes\mathcal F} on the domain. The optional and predictable sigma-algebras are contained in the product,

\displaystyle  \mathcal P\subseteq\mathcal O\subseteq \mathcal B({\mathbb R})\otimes\mathcal F.

We do not have a reference measure on {({\mathbb R}^+\times\Omega,\mathcal B({\mathbb R})\otimes\mathcal F)} in order to define conditional expectations with respect to {\mathcal O} and {\mathcal P}. However, the optional projection {{}^{\rm o}\!X} and predictable projection {{}^{\rm p}\!X} play similar roles. Assuming that the necessary integrability properties are satisfied, then the projections exist. Furthermore, the projection only depends on the process X up to evanescence (i.e., up to a zero probability set), and {{}^{\rm o}\!X} and {{}^{\rm p}\!X} are uniquely defined up to evanescence.

In what follows, we work with respect to a complete filtered probability space. Processes are always only considered up to evanescence, so statements involving equalities, inequalities, and limits of processes are only required to hold outside of a zero probability set. When we say that the optional projection of a process exists, we mean that the integrability condition in the definition of the projection is satisfied. Specifically, that {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal F_\tau]} is almost surely finite. Similarly for the predictable projection.

The following lemma gives a list of initial properties of the optional projection. Other than the statement involving stopping times, they all correspond to properties of conditional expectations.

Lemma 1

  1. X is optional if and only if {{}^{\rm o}\!X} exists and is equal to X.
  2. If the optional projection of X exists then,
    \displaystyle  {}^{\rm o}({}^{\rm o}\!X)={}^{\rm o}\!X. (1)
  3. If the optional projections of X and Y exist, and {\lambda,\mu} are {\mathcal{F}_0}-measurable random variables, then,
    \displaystyle  {}^{\rm o}(\lambda X+\mu Y) = \lambda\,^{\rm o}\!X + \mu\,^{\rm o}Y. (2)
  4. If the optional projection of X exists and U is an optional process then,
    \displaystyle  {}^{\rm o}(UX) = U\,^{\rm o}\!X (3)
  5. If the optional projection of X exists and {\tau} is a stopping time then, the optional projection of the stopped process {X^\tau} exists and,
    \displaystyle  1_{[0,\tau]}{}^{\rm o}(X^\tau)=1_{[0,\tau]}{}^{\rm o}\!X. (4)
  6. If {X\le Y} and the optional projections of X and Y exist then, {{}^{\rm o}\!X\le{}^{\rm o}Y}.

(more…)

10 January 19

Proof of the Measurable Projection and Section Theorems

The aim of this post is to give a direct proof of the theorems of measurable projection and measurable section. These are generally regarded as rather difficult results, and proofs often use ideas from descriptive set theory such as analytic sets. I did previously post a proof along those lines on this blog. However, the results can be obtained in a more direct way, which is the purpose of this post. Here, I present relatively self-contained proofs which do not require knowledge of any advanced topics beyond basic probability theory.

The projection theorem states that if {(\Omega,\mathcal F,{\mathbb P})} is a complete probability space, then the projection of a measurable subset of {{\mathbb R}\times\Omega} onto {\Omega} is measurable. To be precise, the condition is that S is in the product sigma-algebra {\mathcal B({\mathbb R})\otimes\mathcal F}, where {\mathcal B({\mathbb R})} denotes the Borel sets in {{\mathbb R}}, and the projection map is denoted

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\pi_\Omega\colon{\mathbb R}\times\Omega\rightarrow\Omega,\smallskip\\ &\displaystyle\pi_\Omega(t,\omega)=\omega. \end{array}

Then, measurable projection states that {\pi_\Omega(S)\in\mathcal{F}}. Although it looks like a very basic property of measurable sets, maybe even obvious, measurable projection is a surprisingly difficult result to prove. In fact, the requirement that the probability space is complete is necessary and, if it is dropped, then {\pi_\Omega(S)} need not be measurable. Counterexamples exist for commonly used measurable spaces such as {\Omega= {\mathbb R}} and {\mathcal F=\mathcal B({\mathbb R})}. This suggests that there is something deeper going on here than basic manipulations of measurable sets.

By definition, if {S\subseteq{\mathbb R}\times\Omega} then, for every {\omega\in\pi_\Omega(S)}, there exists a {t\in{\mathbb R}} such that {(t,\omega)\in S}. The measurable section theorem — also known as measurable selection — says that this choice can be made in a measurable way. That is, if S is in {\mathcal B({\mathbb R})\otimes\mathcal F} then there is a measurable section,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\tau\colon\pi_\Omega(S)\rightarrow{\mathbb R},\smallskip\\ &\displaystyle(\tau(\omega),\omega)\in S. \end{array}

It is convenient to extend {\tau} to the whole of {\Omega} by setting {\tau=\infty} outside of {\pi_\Omega(S)}.

[caption_id=”sectionpic” align=”aligncenter” width=”450″] measurable sectionFigure 1: A section of a measurable set[/caption] The graph of {\tau} is

\displaystyle  [\tau]=\left\{(t,\omega)\in{\mathbb R}\times\Omega\colon t=\tau(\omega)\right\}.

The condition that {(\tau(\omega),\omega)\in S} whenever {\tau < \infty} can alternatively be expressed by stating that {[\tau]\subseteq S}. This also ensures that {\{\tau < \infty\}} is a subset of {\pi_\Omega(S)}, and {\tau} is a section of S on the whole of {\pi_\Omega(S)} if and only if {\{\tau < \infty\}=\pi_\Omega(S)}.

The results described here can also be used to prove the optional and predictable section theorems which, at first appearances, also seem to be quite basic statements. The section theorems are fundamental to the powerful and interesting theory of optional and predictable projection which is, consequently, generally considered to be a hard part of stochastic calculus. In fact, the projection and section theorems are really not that hard to prove.

Let us consider how one might try and approach a proof of the projection theorem. As with many statements regarding measurable sets, we could try and prove the result first for certain simple sets, and then generalise to measurable sets by use of the monotone class theorem or similar. For example, let {\mathcal S} denote the collection of all {S\subseteq{\mathbb R}\times\Omega} for which {\pi_\Omega(S)\in\mathcal F}. It is straightforward to show that any finite union of sets of the form {A\times B} for {A\in\mathcal B({\mathbb R})} and {B\in\mathcal F} are in {\mathcal S}. If it could be shown that {\mathcal S} is closed under taking limits of increasing and decreasing sequences of sets, then the result would follow from the monotone class theorem. Increasing sequences are easily handled — if {S_n} is a sequence of subsets of {{\mathbb R}\times\Omega} then from the definition of the projection map,

\displaystyle  \pi_\Omega\left(\bigcup\nolimits_n S_n\right)=\bigcup\nolimits_n\pi_\Omega\left(S_n\right).

If {S_n\in\mathcal S} for each n, this shows that the union {\bigcup_nS_n} is again in {\mathcal S}. Unfortunately, decreasing sequences are much more problematic. If {S_n\subseteq S_m} for all {n\ge m} then we would like to use something like

\displaystyle  \pi_\Omega\left(\bigcap\nolimits_n S_n\right)=\bigcap\nolimits_n\pi_\Omega\left(S_n\right). (1)

However, this identity does not hold in general. For example, consider the decreasing sequence {S_n=(n,\infty)\times\Omega}. Then, {\pi_\Omega(S_n)=\Omega} for all n, but {\bigcap_nS_n} is empty, contradicting (1). There is some interesting history involved here. In a paper published in 1905, Henri Lebesgue claimed that the projection of a Borel subset of {{\mathbb R}^2} onto {{\mathbb R}} is itself measurable. This was based upon mistakenly applying (1). The error was spotted in around 1917 by Mikhail Suslin, who realised that the projection need not be Borel, and lead him to develop the theory of analytic sets.

Actually, there is at least one situation where (1) can be shown to hold. Suppose that for each {\omega\in\Omega}, the slices

\displaystyle  S_n(\omega)\equiv\left\{t\in{\mathbb R}\colon(t,\omega)\in S_n\right\} (2)

are compact. For each {\omega\in\bigcap_n\pi_\Omega(S_n)}, the slices {S_n(\omega)} give a decreasing sequence of nonempty compact sets, so has nonempty intersection. So, letting S be the intersection {\bigcap_nS_n}, the slice {S(\omega)=\bigcap_nS_n(\omega)} is nonempty. Hence, {\omega\in\pi_\Omega(S)}, and (1) follows.

The starting point for our proof of the projection and section theorems is to consider certain special subsets of {{\mathbb R}\times\Omega} where the compactness argument, as just described, can be used. The notation {\mathcal A_\delta} is used to represent the collection of countable intersections, {\bigcap_{n=1}^\infty A_n}, of sets {A_n} in {\mathcal A}.

Lemma 1 Let {(\Omega,\mathcal F)} be a measurable space, and {\mathcal A} be the collection of subsets of {{\mathbb R}\times\Omega} which are finite unions {\bigcup_kC_k\times E_k} over compact intervals {C_k\subseteq{\mathbb R}} and {E_k\in\mathcal F}. Then, for any {S\in\mathcal A_\delta}, we have {\pi_\Omega(S)\in\mathcal F}, and the debut

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \tau\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\},\smallskip\\ &\displaystyle \omega\mapsto\inf\left\{t\in{\mathbb R}\colon (t,\omega)\in S\right\}. \end{array}

is a measurable map with {[\tau]\subseteq S} and {\{\tau < \infty\}=\pi_\Omega(S)}.

(more…)

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