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diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md index 3f17a9a..f087f97 100644 --- a/source/know/concept/ito-integral/index.md +++ b/source/know/concept/ito-integral/index.md @@ -9,10 +9,10 @@ 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/) $B_t$, +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. -The Itō integral $I_t$ of $G_t$ is defined as follows: +The Itō integral $$I_t$$ of $$G_t$$ is defined as follows: $$\begin{aligned} \boxed{ @@ -22,17 +22,17 @@ $$\begin{aligned} } \end{aligned}$$ -Where have partitioned the time interval $[a, b]$ into steps of size $h$. -The above integral exists if $G_t$ and $B_t$ are adapted -to a common filtration $\mathcal{F}_t$, -and $\mathbf{E}[G_t^2]$ is integrable for $t \in [a, b]$. -If $I_t$ exists, $G_t$ is said to be **Itō-integrable** with respect to $B_t$. +Where have partitioned the time interval $$[a, b]$$ into steps of size $$h$$. +The above integral exists if $$G_t$$ and $$B_t$$ are adapted +to a common filtration $$\mathcal{F}_t$$, +and $$\mathbf{E}[G_t^2]$$ is integrable for $$t \in [a, b]$$. +If $$I_t$$ exists, $$G_t$$ is said to be **Itō-integrable** with respect to $$B_t$$. ## Motivation -Consider the following simple first-order differential equation for $X_t$, -for some function $f$: +Consider the following simple first-order differential equation for $$X_t$$, +for some function $$f$$: $$\begin{aligned} \dv{X_t}{t} @@ -40,7 +40,7 @@ $$\begin{aligned} \end{aligned}$$ This can be solved numerically using the explicit Euler scheme -by discretizing it with step size $h$, +by discretizing it with step size $$h$$, which can be applied recursively, leading to: $$\begin{aligned} @@ -51,7 +51,7 @@ $$\begin{aligned} \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$$, this leads to the following unsurprising integral for $$X_t$$: $$\begin{aligned} \int_0^t f(X_s) \dd{s} @@ -59,18 +59,18 @@ $$\begin{aligned} \end{aligned}$$ In contrast, consider the *stochastic differential equation* below, -where $\xi_t$ represents white noise, -which is informally the $t$-derivative -of the Wiener process $\xi_t = \idv{B_t}{t}$: +where $$\xi_t$$ represents white noise, +which is informally the $$t$$-derivative +of the Wiener process $$\xi_t = \idv{B_t}{t}$$: $$\begin{aligned} \dv{X_t}{t} = g(X_t) \: \xi_t \end{aligned}$$ -Now $X_t$ is not deterministic, -since $\xi_t$ is derived from a random variable $B_t$. -If $g = 1$, we expect $X_t = X_0 + B_t$. +Now $$X_t$$ is not deterministic, +since $$\xi_t$$ is derived from a random variable $$B_t$$. +If $$g = 1$$, we expect $$X_t = X_0 + B_t$$. With this in mind, we introduce the **Euler-Maruyama scheme**: $$\begin{aligned} @@ -80,7 +80,7 @@ $$\begin{aligned} &= X_t + g(X_t) \: (B_{t+h} - B_t) \end{aligned}$$ -We would like to turn this into an integral for $X_t$, as we did above. +We would like to turn this into an integral for $$X_t$$, as we did above. Therefore, we state: $$\begin{aligned} @@ -89,8 +89,8 @@ $$\begin{aligned} \end{aligned}$$ 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. +analogously to the first, but with $$h$$ replaced by +the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process. This is an Itō integral: $$\begin{aligned} @@ -104,14 +104,14 @@ see the [Itō calculus](/know/concept/ito-calculus/). ## Properties -Since $G_t$ and $B_t$ must be known (i.e. $\mathcal{F}_t$-adapted) -in order to evaluate the Itō integral $I_t$ at any given $t$, -it logically follows that $I_t$ is also $\mathcal{F}_t$-adapted. +Since $$G_t$$ and $$B_t$$ must be known (i.e. $$\mathcal{F}_t$$-adapted) +in order to evaluate the Itō integral $$I_t$$ at any given $$t$$, +it logically follows that $$I_t$$ is also $$\mathcal{F}_t$$-adapted. Because the Itō integral is defined as the limit of a sum of linear terms, it inherits this linearity. -Consider two Itō-integrable processes $G_t$ and $H_t$, -and two constants $v, w \in \mathbb{R}$: +Consider two Itō-integrable processes $$G_t$$ and $$H_t$$, +and two constants $$v, w \in \mathbb{R}$$: $$\begin{aligned} \int_a^b v G_t + w H_t \dd{B_t} @@ -119,7 +119,7 @@ $$\begin{aligned} \end{aligned}$$ By adding multiple summations, -the Itō integral clearly satisfies, for $a < b < c$: +the Itō integral clearly satisfies, for $$a < b < c$$: $$\begin{aligned} \int_a^c G_t \dd{B_t} @@ -127,8 +127,8 @@ $$\begin{aligned} \end{aligned}$$ A more interesting property is the **Itō isometry**, -which expresses the expectation of the square of an Itō integral of $G_t$ -as a simpler "ordinary" integral of the expectation of $G_t^2$ +which expresses the expectation of the square of an Itō integral of $$G_t$$ +as a simpler "ordinary" integral of the expectation of $$G_t^2$$ (which exists by the definition of Itō-integrability): $$\begin{aligned} @@ -144,14 +144,14 @@ $$\begin{aligned} <div class="hidden" markdown="1"> <label for="proof-isometry">Proof.</label> We write out the left-hand side of the Itō isometry, -where eventually $h \to 0$: +where eventually $$h \to 0$$: $$\begin{aligned} \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 &= \sum_{t = a}^{t = b} \sum_{s = a}^{s = b} \mathbf{E} \bigg[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \bigg] \end{aligned}$$ -In the particular case $t \ge s \!+\! h$, +In the particular case $$t \ge s \!+\! h$$, a given term of this summation can be rewritten as follows using the *law of total expectation* (see [conditional expectation](/know/concept/conditional-expectation/)): @@ -161,18 +161,18 @@ $$\begin{aligned} = \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big| \mathcal{F}_t \Big] \bigg] \end{aligned}$$ -Recall that $G_t$ and $B_t$ are adapted to $\mathcal{F}_t$: -at time $t$, we have information $\mathcal{F}_t$, -which includes knowledge of the realized values $G_t$ and $B_t$. -Since $t \ge s \!+\! h$ by assumption, we can simply factor out the known quantities: +Recall that $$G_t$$ and $$B_t$$ are adapted to $$\mathcal{F}_t$$: +at time $$t$$, we have information $$\mathcal{F}_t$$, +which includes knowledge of the realized values $$G_t$$ and $$B_t$$. +Since $$t \ge s \!+\! h$$ by assumption, we can simply factor out the known quantities: $$\begin{aligned} \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] = \mathbf{E} \bigg[ G_t G_s (B_{s + h} \!-\! B_s) \: \mathbf{E} \Big[ (B_{t + h} \!-\! B_t) \Big| \mathcal{F}_t \Big] \bigg] \end{aligned}$$ -However, $\mathcal{F}_t$ says nothing about -the increment $(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$, +However, $$\mathcal{F}_t$$ says nothing about +the increment $$(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$$, meaning that the conditional expectation is zero: $$\begin{aligned} @@ -181,7 +181,7 @@ $$\begin{aligned} \qquad \mathrm{for}\; t \ge s + h \end{aligned}$$ -By swapping $s$ and $t$, the exact same result can be obtained for $s \ge t \!+\! h$: +By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$: $$\begin{aligned} \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] @@ -189,7 +189,7 @@ $$\begin{aligned} \qquad \mathrm{for}\; s \ge t + h \end{aligned}$$ -This leaves only one case which can be nonzero: $[t, t\!+\!h] = [s, s\!+\!h]$. +This leaves only one case which can be nonzero: $$[t, t\!+\!h] = [s, s\!+\!h]$$. Applying the law of total expectation again yields: $$\begin{aligned} @@ -199,8 +199,8 @@ $$\begin{aligned} &= \sum_{t = a}^{t = b} \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big| \mathcal{F}_t \Big] \bigg] \end{aligned}$$ -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 $$(B_{t+h} \!-\! B_t)^2$$, +since the increment is normally distributed, is simply the variance $$h$$: $$\begin{aligned} \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 @@ -208,6 +208,7 @@ $$\begin{aligned} \longrightarrow \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t} \end{aligned}$$ + </div> </div> @@ -221,9 +222,9 @@ since true white noise cannot be biased. <label for="proof-martingale">Proof</label> <div class="hidden" markdown="1"> <label for="proof-martingale">Proof.</label> -We will prove that an arbitrary Itō integral $I_t$ is a martingale. -Using additivity, we know that the increment $I_t \!-\! I_s$ -is as follows, given information $\mathcal{F}_s$: +We will prove that an arbitrary Itō integral $$I_t$$ is a martingale. +Using additivity, we know that the increment $$I_t \!-\! I_s$$ +is as follows, given information $$\mathcal{F}_s$$: $$\begin{aligned} \mathbf{E} \big[ I_t \!-\! I_s | \mathcal{F}_s \big] @@ -232,8 +233,8 @@ $$\begin{aligned} \end{aligned}$$ We rewrite this [conditional expectation](/know/concept/conditional-expectation/) -using the *tower property* for some $\mathcal{F}_u \supset \mathcal{F}_s$, -such that $G_u$ and $B_u$ are known, but $B_{u+h} \!-\! B_u$ is not: +using the *tower property* for some $$\mathcal{F}_u \supset \mathcal{F}_s$$, +such that $$G_u$$ and $$B_u$$ are known, but $$B_{u+h} \!-\! B_u$$ is not: $$\begin{aligned} \mathbf{E} \big[ I_t \!-\! I_s | \mathcal{F}_s \big] @@ -242,7 +243,7 @@ $$\begin{aligned} = 0 \end{aligned}$$ -We now have everything we need to calculate $\mathbf{E} [ I_t | \mathcal{F_s} ]$, +We now have everything we need to calculate $$\mathbf{E} [ I_t | \mathcal{F_s} ]$$, giving the martingale property: $$\begin{aligned} @@ -252,12 +253,12 @@ $$\begin{aligned} = I_s \end{aligned}$$ -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. +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. </div> </div> |