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-rw-r--r--source/know/concept/ito-integral/index.md57
1 files changed, 24 insertions, 33 deletions
diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md
index 4a725e1..9b092d6 100644
--- a/source/know/concept/ito-integral/index.md
+++ b/source/know/concept/ito-integral/index.md
@@ -10,8 +10,7 @@ layout: "concept"
The **Itō integral** offers a way to integrate
a given [stochastic process](/know/concept/stochastic-process/) $$G_t$$
-with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$,
-which is also a stochastic process.
+with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$.
The Itō integral $$I_t$$ of $$G_t$$ is defined as follows:
$$\begin{aligned}
@@ -47,21 +46,21 @@ which can be applied recursively, leading to:
$$\begin{aligned}
X_{t+h}
\approx X_{t} + f(X_t) \: h
- \quad \implies \quad
+ \qquad \implies \qquad
X_t
\approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h
\end{aligned}$$
-In the limit $$h \to 0$$, this leads to the following unsurprising integral for $$X_t$$:
+In the limit $$h \to 0$$, this unsurprisingly leads to the following integral for $$X_t$$:
$$\begin{aligned}
- \int_0^t f(X_s) \dd{s}
- = \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h
+ \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h
+ = \int_0^t f(X_s) \dd{s}
\end{aligned}$$
In contrast, consider the *stochastic differential equation* below,
where $$\xi_t$$ represents white noise,
-which is informally the $$t$$-derivative
+which is informally defined as the $$t$$-derivative
of the Wiener process $$\xi_t = \idv{B_t}{t}$$:
$$\begin{aligned}
@@ -89,9 +88,9 @@ $$\begin{aligned}
= X_0 + \int_0^t g(X_s) \dd{B_s}
\end{aligned}$$
-This integral is *defined* as below,
-analogously to the first, but with $$h$$ replaced by
-the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process.
+The meaning of such an integral is *defined* below.
+It is analogous to the deterministic case,
+but $$h$$ is replaced by the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process.
This is an Itō integral:
$$\begin{aligned}
@@ -100,7 +99,7 @@ $$\begin{aligned}
\end{aligned}$$
For more information about applying the Itō integral in this way,
-see the [Itō calculus](/know/concept/ito-process/).
+see [Itō calculus](/know/concept/ito-process/).
@@ -131,7 +130,7 @@ $$\begin{aligned}
A more interesting property is the **Itō isometry**,
which expresses the expectation of the square of an Itō integral of $$G_t$$
as a simpler "ordinary" integral of the expectation of $$G_t^2$$
-(which exists by the definition of Itō-integrability):
+(which exists due to the definition of Itō-integrability):
$$\begin{aligned}
\boxed{
@@ -172,24 +171,16 @@ $$\begin{aligned}
However, $$\mathcal{F}_t$$ says nothing about
the increment $$(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$$,
-meaning that the conditional expectation is zero:
+meaning that the conditional expectation is zero for $$t \ge s + h$$:
$$\begin{aligned}
\mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big]
= 0
- \qquad \mathrm{for}\; t \ge s + h
\end{aligned}$$
-By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$:
-
-$$\begin{aligned}
- \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big]
- = 0
- \qquad \mathrm{for}\; s \ge t + h
-\end{aligned}$$
-
-This leaves only one case which can be nonzero: $$[t, t\!+\!h] = [s, s\!+\!h]$$.
-Applying the law of total expectation again yields:
+By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$.
+This leaves only one possibly nonzero case: $$[t, t\!+\!h] = [s, s\!+\!h]$$.
+Applying the law of total expectation again:
$$\begin{aligned}
\mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2
@@ -198,15 +189,15 @@ $$\begin{aligned}
&= \sum_{t = a}^{t = b} \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big| \mathcal{F}_t \Big] \bigg]
\end{aligned}$$
-We know $$G_t$$, and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$,
-since the increment is normally distributed, is simply the variance $$h$$:
+We know $$G_t$$,
+and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$ is simply the variance $$h$$:
$$\begin{aligned}
\mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2
&= \sum_{t = a}^{t = b} \mathbf{E} \big[ G_t^2 \big] h
- \longrightarrow
- \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t}
\end{aligned}$$
+
+Taking the limit $$h \to 0$$ then yields the desired result.
{% include proof/end.html id="proof-isometry" %}
@@ -239,7 +230,7 @@ $$\begin{aligned}
\end{aligned}$$
We now have everything we need to calculate $$\mathbf{E} [ I_t | \mathcal{F_s} ]$$,
-giving the martingale property:
+leading to the martingale property:
$$\begin{aligned}
\mathbf{E} \big[ I_t | \mathcal{F}_s \big]
@@ -250,10 +241,10 @@ $$\begin{aligned}
For the existence of $$I_t$$,
we need $$\mathbf{E}[G_t^2]$$ to be integrable over the target interval,
-so from the Itō isometry we have $$\mathbf{E}[I]^2 < \infty$$,
-and therefore $$\mathbf{E}[I] < \infty$$,
-so $$I_t$$ has all the properties of a Martingale,
-since it is trivially $$\mathcal{F}_t$$-adapted.
+which implies via the Itō isometry that $$\mathbf{E}[I]^2$$ is finite.
+Therefore $$\mathbf{E}[I]$$ is also finite,
+so $$I_t$$ has all the properties of a Martingale
+(since it is trivially $$\mathcal{F}_t$$-adapted).
{% include proof/end.html id="proof-martingale" %}