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author | Prefetch | 2024-07-17 10:01:43 +0200 |
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committer | Prefetch | 2024-07-17 10:01:43 +0200 |
commit | 075683cdf4588fe16f41d9f7b46b9720b42b2553 (patch) | |
tree | f200b341b35e9c89aa1030a2f6da94cd0e1e958b /source/know/concept/ito-integral | |
parent | c4d597e8d695eb145755464cffbf88a68fd0c88a (diff) |
Improve knowledge base
Diffstat (limited to 'source/know/concept/ito-integral')
-rw-r--r-- | source/know/concept/ito-integral/index.md | 57 |
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" %} |