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Diffstat (limited to 'content/know/concept/ito-calculus/index.pdc')
| -rw-r--r-- | content/know/concept/ito-calculus/index.pdc | 5 |
1 files changed, 4 insertions, 1 deletions
diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-calculus/index.pdc index 3527b1d..7a80e2f 100644 --- a/content/know/concept/ito-calculus/index.pdc +++ b/content/know/concept/ito-calculus/index.pdc @@ -60,6 +60,9 @@ $$\begin{aligned} An Itō process $X_t$ is said to satisfy this equation if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$, in which case $X_t$ is also called an **Itō diffusion**. +All Itō diffusions are [Markov processes](/know/concept/markov-process/), +since only the current value of $X_t$ determines the future, +and $B_t$ is also a Markov process. ## Itō's lemma @@ -80,7 +83,7 @@ known as **Itō's lemma**: $$\begin{aligned} \boxed{ \dd{Y_t} - = \pdv{h}{t} \dd{t} + \bigg( \pdv{h}{x} F_t + \frac{1}{2} G_t^2 \pdv[2]{h}{x} \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} + = \bigg( \pdv{h}{t} + \pdv{h}{x} F_t + \frac{1}{2} \pdv[2]{h}{x} G_t^2 \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} } \end{aligned}$$ |