diff options
author | Prefetch | 2022-10-27 20:40:09 +0200 |
---|---|---|
committer | Prefetch | 2022-10-27 20:40:09 +0200 |
commit | 6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch) | |
tree | a8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/dynkins-formula | |
parent | bcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (diff) |
Optimize last images, add proof template, improve CSS
Diffstat (limited to 'source/know/concept/dynkins-formula')
-rw-r--r-- | source/know/concept/dynkins-formula/index.md | 23 |
1 files changed, 8 insertions, 15 deletions
diff --git a/source/know/concept/dynkins-formula/index.md b/source/know/concept/dynkins-formula/index.md index c0d20c5..307f098 100644 --- a/source/know/concept/dynkins-formula/index.md +++ b/source/know/concept/dynkins-formula/index.md @@ -39,11 +39,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-kolmogorov"/> -<label for="proof-kolmogorov">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-kolmogorov">Proof.</label> + +{% include proof/start.html id="proof-kolmogorov" -%} We define a new process $$Y_t \equiv h(X_t)$$, and then apply Itō's lemma, leading to: $$\begin{aligned} @@ -84,9 +81,8 @@ $$\begin{aligned} \hat{L}\{h(X_0)\} \approx \frac{1}{t} \mathbf{E}[Y_t - Y_0| X_0] \end{aligned}$$ +{% include proof/end.html id="proof-kolmogorov" %} -</div> -</div> The general definition of resembles that of a classical derivative, and indeed, the generator $$\hat{A}$$ can be thought of as a differential operator. @@ -104,11 +100,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dynkin"/> -<label for="proof-dynkin">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dynkin">Proof.</label> + +{% include proof/start.html id="proof-dynkin" -%} The proof is similar to the one above. Define $$Y_t = h(X_t)$$ and use Itō’s lemma: @@ -136,9 +129,9 @@ $$\begin{aligned} = \mathbf{E}\bigg[ Y_\tau - Y_0 - \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} \bigg| X_0 \bigg] \end{aligned}$$ -Isolating this equation for $$\mathbf{E}[Y_\tau | X_0]$$ then gives Dynkin's formula. -</div> -</div> +Isolating this equation for $$\mathbf{E}[Y_\tau \!\mid\! X_0]$$ then gives Dynkin's formula. +{% include proof/end.html id="proof-dynkin" %} + A common application of Dynkin's formula is predicting when the stopping time $$\tau$$ occurs, and in what state $$X_\tau$$ this happens. |