Definition of elementary stochastic integral

Stream: Brownian motion

[Thomas Zhu]

When defining the elementary stochastic integral $V \bullet X$ (Def. 10.9), a continuous $\mathbb{R}$-bilinear map $B : E \to F \to G$ was proposed in the blueprint where $F$ is the codomain of the simple process $V$, $E$ is the codomain of the stochastic process $X$, and $G$ is the codomain of the resulting process. Here we assume the more general case than the specific case of scalar multiplication $B(x, r) = r \cdot x$ and $E = G$. I am roughly aware this is in line with some things that we want to do with vector integral. However, I am not sure how much more general this is. We require the following properties of $B$ and $F$:

• $B$ is linear in both arguments.

• Lemma 10.11, in the most general case, would be something like: $X$ is $E$-valued, $W$ is $F$-valued, and $B_1 : E \to F \to G$ so $W \bullet X$ is $G$-valued. Then if $V$ is $H$-valued, and there is $B_2 : G \to H \to I$, the final $V \bullet (W \bullet X)$ is $I$-valued. Then we need a “multiplication” $C : H \to F \to J$, another $B_3 : E \to J \to I$, and a condition $B_2(B_1(x, w), v) = B_3(x, C(v, w))$. I think this is too complicated. For example, if we simply require that $B_1 = B_2$, then to make things type-check we already have $E=G=I$ and $F=H=J$, and $B(B(x, w), v) = B(x, C(v, w))$ written normally is just $v \cdot (w \cdot x) = vw \cdot x$.

• In many cases we need to construct an indicator simple process of an elementary predictable set (in fact, all explicitly defined simple processes are defined as this). This requires $1 : F$. In all proofs such as Lemma 10.14 or the upcrossing lemmas where an explicit simple process is constructed, we use some sum of $\mathbb{1}_A\mathbb{1}_{(s,t]}$ where $A \in \mathcal{F}_{s}$. Integrating this indicator $\mathbb{1}_A\mathbb{1}_{(s,t]} \bullet X = \mathbb{1}_A(X^t - X^s)$ = A.indicator (X^t - X^s) then requires $B(x, 1) = x$.

In light of the above, the relationship between $E$ and $F$ is much more like vector/scalar fields rather than two vector fields, and $B$ and $\bullet$ seem much more like scalar multiplications than bilinear maps (in fact, we would have Module (SimpleProcess F 𝓕) (ι → Ω → E) with $\bullet$ as smul if not for the fact that 1 is not a simple process for infinite ι). Observe that the above bullet points are precisely the module axioms for Module F E, if we interpret $B$ to be smul and $C$ to be mul. So I think it is more appropriate to do this: $E = G$ must be a module over $F$. $F$ can be a normed algebra over $\mathbb{R}$. We probably also need IsScalarTower ℝ F E if it is not inferred. I tried this at https://github.com/RemyDegenne/brownian-motion/pull/307 and it seems to work well.

I discussed this in DM with @Rémy Degenne and he agreed with the change, but suggested to ask publicly in this channel. I would like to see what everyone thinks before I proceed.

[Etienne Marion]

I agree that a general bilinear map seems overkill. However I think we still want to take into account something more general, your bullet points seem to match the case where F is a vector space and E is the space of operators over F I think? I think it would be nice to at least cover the case where F is $\mathbb R^n$ and E is the space of $n \times n$ matrices, as this often appears in stochastic calculus. We should be able to deduce it from the $E = F = \mathbb R$ case but this would surely involve a lot of duplication.

[Thomas Zhu]

Sorry, do you mean the other way around, with F space of operators over E? Because otherwise what is a definition of $B : E \to F \to E$?

[Etienne Marion]

Yes probably I mixed up letters within your message. I mean $B : F \to \mathcal L(F, F) \to F$ defined by $B(x, f) = f(x)$.

[Thomas Zhu]

I think mathematically that is covered by my proposed definition, because (in your notation) $\mathcal{L}(F,F)$ is a normed algebra over $\mathbb{R}$, and $F$ is a $\mathcal{L}(F,F)$-module, which is the generality I propose. (And in my above notation that would be $E$ is a $F=\mathcal{L}(E,E)$-module, just to avoid confusion.)

[Thomas Zhu]

I just thought about it more, and I think maybe this could also make sense: in $V \bullet X$, $V$ is $\mathcal{L}(E, G)$-valued (where $G$ is another space) and $X$ is $E$-valued. This is more general because it allows for the case where $V$ is $m\times n$ matrix where $m \ne n$ and $X$ is $\mathbb{R}^n$ (which is considered in e.g. in Pascucci). The formula that was written $V\bullet(W\bullet X)=(V\cdot W)\bullet X$ is now actually $(V\circ W)\bullet X$ where $\circ$ is composition. Where indicators are needed, we require $E=G$.

[Rémy Degenne]

Oh right we definitely want to cover the case of V being linear-map valued (which was my first def before Etienne told me we could generalize). I forgot about it when replying to your DM about restricting to the scalar multiplication.

[Rémy Degenne]

(these slides use that definition with a linear map for example https://www.stoch-ana-hermite-sobolev.uni-freiburg.de/Tappe-Introduction-to-Stochastic-20210625.pdf, sorry for not having a better ref)

[Etienne Marion]

Yes we definitely want the case where the matrix is not a square too!

[Thomas Zhu]

The slides make total sense, I regret not seeing them before!

[Thomas Zhu]

I have only seen the matrix stochastic integral definition as a wrapper around the case where everything is scalar, which would be derived from elementary stochastic integral where $V$ is scalar, but it seems we can go directly to the general stochastic integral, from a general elementary stochastic integral?

[Rémy Degenne]

Yes, that's the way Tappe does it in those slides: start from a general elementary stochastic integral with $V$ taking values in $L(E, G)$, and prove the Itô isometry in that setting (I found those slides when looking for the "right generality" for the isometry).

[Rémy Degenne]

Sorry again for being misleading with my answer in DM earlier. It's the week-end and I don't have a lot of time to think about those questions.

[Thomas Zhu]

No worries! I or someone else should also modify the quadratic (co)variation and Ito isometry parts of the blueprint to generalize to something more like Tappe's slides.

[Anatole Dedecker]

Thomas Zhu said:

• Lemma 10.11, in the most general case, would be something like: $X$ is $E$-valued, $W$ is $F$-valued, and $B_1 : E \to F \to G$ so $W \bullet X$ is $G$-valued. Then if $V$ is $H$-valued, and there is $B_2 : G \to H \to I$, the final $V \bullet (W \bullet X)$ is $I$-valued. Then we need a “multiplication” $C : H \to F \to J$, another $B_3 : E \to J \to I$, and a condition $B_2(B_1(x, w), v) = B_3(x, C(v, w))$. I think this is too complicated. For example, if we simply require that $B_1 = B_2$, then to make things type-check we already have $E=G=I$ and $F=H=J$, and $B(B(x, w), v) = B(x, C(v, w))$ written normally is just $v \cdot (w \cdot x) = vw \cdot x$.

Just wanted to point out that this is what we do in docs#MeasureTheory.convolution_assoc. So it is a bit annoying to write and you obviously want to specialize it right away, but it's not that unreasonable.

[Thomas Zhu]

Update: my PR 340 (now ready for review) still uses the general bilinear map $B$ (as in the blueprint), and the exact most general form of associativity suggested above. I have specialized it for $L(E,F)$-valued simple processes but specializing to $\mathbb{R}$-valued simple process is a TODO.

/-- The most general case of associativity of the elementary stochastic integral. -/
theorem integral_assoc {B₁ : E →L[ℝ] H →L[ℝ] I} {B₂ : F →L[ℝ] G →L[ℝ] H} {B₃ : E →L[ℝ] F →L[ℝ] J}
    {B₄ : J →L[ℝ] G →L[ℝ] I} (hB : ∀ x y z, B₁ x (B₂ y z) = B₄ (B₃ x y) z)
    (V : SimpleProcess E 𝓕) (W : SimpleProcess F 𝓕) (X : ι → Ω → G) :
    integral B₁ V (integral B₂ W X ∘ WithTop.some) = integral B₄ (map₂ B₃ V W) X

[Rémy Degenne]

I merged the PR.
I have a question about an aspect of the definition of the elementary stochastic integral that predates this PR. It is currently defined as a stochastic process indexed by WithTop ι, while the original processes are indexed by ι. It makes it easy to talk about the integral over the whole index by using the value at top, but I worry that the fact that we are adding a WithTop will become inconvenient later. I don't have any example of where it could be bad for now: I just know that ι vs WithTop ι is a recurring design question in files about stochastic processes and that we should keep this choice in mind and be ready to reconsider if we encounter a difficulty.

[Thomas Zhu]

Currently, for the properties of the elementary stochastic integral, it seems to be easy to prove whether using WithTop or not. So if there is something in the future that shows defining without WithTop makes more sense, there is no reason not to switch.