Almost Sure

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)

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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}}.

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23 December 18

Projection in Discrete Time

It has been some time since my last post, but I am continuing now with the stochastic calculus notes on optional and predictable projection. In this post, I will go through the ideas in the discrete-time situation. All of the main concepts involved in optional and predictable projection are still present in discrete time, but the theory is much simpler. It is only really in continuous time that the projection theorems really show their power, so the aim of this post is to motivate the concepts in a simple setting before generalising to the full, continuous-time situation. Ideally, this would have been published before the posts on optional and predictable projection in continuous time, so it is a bit out of sequence.

We consider time running through the discrete index set {{\mathbb Z}^+=\{0,1,2,\ldots\}}, and work with respect to a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_n\}_{n=0,1,\ldots},{\mathbb P})}. Then, {\mathcal{F}_n} is used to represent the collection of events observable up to and including time n. Stochastic processes will all be real-valued and defined up to almost-sure equivalence. That is, processes X and Y are considered to be the same if {X_n=Y_n} almost surely for each {n\in{\mathbb Z}^+}. The projections of a process X are defined as follows.

Definition 1 Let X be a measurable process. Then,

  1. the optional projection, {{}^{\rm o}\!X}, exists if and only if {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_n]} is almost surely finite for each n, in which case
    \displaystyle  {}^{\rm o}\!X_n={\mathbb E}[X_n\,\vert\mathcal{F}_n]. (1)
  2. the predictable projection, {{}^{\rm p}\!X}, exists if and only if {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_{n-1}]} is almost surely finite for each n, in which case
    \displaystyle  {}^{\rm p}\!X_n={\mathbb E}[X_n\,\vert\mathcal{F}_{n-1}]. (2)

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