diff options
Diffstat (limited to 'source/know/concept/ito-integral')
-rw-r--r-- | source/know/concept/ito-integral/index.md | 268 |
1 files changed, 268 insertions, 0 deletions
diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md new file mode 100644 index 0000000..da3c706 --- /dev/null +++ b/source/know/concept/ito-integral/index.md @@ -0,0 +1,268 @@ +--- +title: "Itō integral" +date: 2021-11-06 +categories: +- Mathematics +- Stochastic analysis +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. +The Itō integral $I_t$ of $G_t$ is defined as follows: + +$$\begin{aligned} + \boxed{ + I_t + \equiv \int_a^b G_t \dd{B_t} + \equiv \lim_{h \to 0} \sum_{t = a}^{t = b} G_t \big(B_{t + h} - B_t\big) + } +\end{aligned}$$ + +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 $B_t$. + + +## Motivation + +Consider the following simple first-order differential equation for $X_t$, +for some function $f$: + +$$\begin{aligned} + \dv{X_t}{t} + = f(X_t) +\end{aligned}$$ + +This can be solved numerically using the explicit Euler scheme +by discretizing it with step size $h$, +which can be applied recursively, leading to: + +$$\begin{aligned} + X_{t+h} + \approx X_{t} + f(X_t) \: h + \quad \implies \quad + X_t + \approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h +\end{aligned}$$ + +In the limit $h \to 0$, this leads to the following unsurprising integral for $X_t$: + +$$\begin{aligned} + \int_0^t f(X_s) \dd{s} + = \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h +\end{aligned}$$ + +In contrast, consider the *stochastic differential equation* below, +where $\xi_t$ represents white noise, +which is informally the $t$-derivative +of the Wiener process $\xi_t = \idv{B_t}{t}$: + +$$\begin{aligned} + \dv{X_t}{t} + = g(X_t) \: \xi_t +\end{aligned}$$ + +Now $X_t$ is not deterministic, +since $\xi_t$ is derived from a random variable $B_t$. +If $g = 1$, we expect $X_t = X_0 + B_t$. +With this in mind, we introduce the **Euler-Maruyama scheme**: + +$$\begin{aligned} + X_{t+h} + &= X_t + g(X_t) \: (\xi_{t+h} - \xi_t) \: h + \\ + &= X_t + g(X_t) \: (B_{t+h} - B_t) +\end{aligned}$$ + +We would like to turn this into an integral for $X_t$, as we did above. +Therefore, we state: + +$$\begin{aligned} + X_t + = X_0 + \int_0^t g(X_s) \dd{B_s} +\end{aligned}$$ + +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. +This is an Itō integral: + +$$\begin{aligned} + \int_0^t g(X_s) \dd{B_s} + \equiv \lim_{h \to 0} \sum_{s = 0}^{s = t} g(X_s) \big(B_{s + h} - B_s\big) +\end{aligned}$$ + +For more information about applying the Itō integral in this way, +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. + +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 $v, w \in \mathbb{R}$: + +$$\begin{aligned} + \int_a^b v G_t + w H_t \dd{B_t} + = v\! \int_a^b G_t \dd{B_t} +\: w\! \int_a^b H_t \dd{B_t} +\end{aligned}$$ + +By adding multiple summations, +the Itō integral clearly satisfies, for $a < b < c$: + +$$\begin{aligned} + \int_a^c G_t \dd{B_t} + = \int_a^b G_t \dd{B_t} + \int_b^c G_t \dd{B_t} +\end{aligned}$$ + +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 exists by the definition of Itō-integrability): + +$$\begin{aligned} + \boxed{ + \mathbf{E} \bigg( \int_a^b G_t \dd{B_t} \bigg)^2 + = \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t} + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-isometry"/> +<label for="proof-isometry">Proof</label> +<div class="hidden" markdown="1"> +<label for="proof-isometry">Proof.</label> +We write out the left-hand side of the Itō isometry, +where eventually $h \to 0$: + +$$\begin{aligned} + \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 + &= \sum_{t = a}^{t = b} \sum_{s = a}^{s = b} \mathbf{E} \bigg[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \bigg] +\end{aligned}$$ + +In the particular case $t \ge s \!+\! h$, +a given term of this summation can be rewritten +as follows using the *law of total expectation* +(see [conditional expectation](/know/concept/conditional-expectation/)): + +$$\begin{aligned} + \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] + = \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big| \mathcal{F}_t \Big] \bigg] +\end{aligned}$$ + +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: + +$$\begin{aligned} + \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] + = \mathbf{E} \bigg[ G_t G_s (B_{s + h} \!-\! B_s) \: \mathbf{E} \Big[ (B_{t + h} \!-\! B_t) \Big| \mathcal{F}_t \Big] \bigg] +\end{aligned}$$ + +However, $\mathcal{F}_t$ says nothing about +the increment $(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$, +meaning that the conditional expectation is zero: + +$$\begin{aligned} + \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] + = 0 + \qquad \mathrm{for}\; t \ge s + h +\end{aligned}$$ + +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: + +$$\begin{aligned} + \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 + &= \sum_{t = a}^{t = b} \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big] + \\ + &= \sum_{t = a}^{t = b} \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big| \mathcal{F}_t \Big] \bigg] +\end{aligned}$$ + +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 $h$: + +$$\begin{aligned} + \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 + &= \sum_{t = a}^{t = b} \mathbf{E} \big[ G_t^2 \big] h + \longrightarrow + \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t} +\end{aligned}$$ +</div> +</div> + +Furthermore, Itō integrals are [martingales](/know/concept/martingale/), +meaning that the average noise contribution is zero, +which makes intuitive sense, +since true white noise cannot be biased. + +<div class="accordion"> +<input type="checkbox" id="proof-martingale"/> +<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 $\mathcal{F}_s$: + +$$\begin{aligned} + \mathbf{E} \big[ I_t \!-\! I_s | \mathcal{F}_s \big] + = \mathbf{E} \bigg[ \int_s^t G_u \dd{B_u} \bigg| \mathcal{F}_s \bigg] + = \lim_{h \to 0} \sum_{u = s}^{u = t} \mathbf{E} \Big[ G_u (B_{u + h} \!-\! B_u) \Big| \mathcal{F}_s \Big] +\end{aligned}$$ + +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: + +$$\begin{aligned} + \mathbf{E} \big[ I_t \!-\! I_s | \mathcal{F}_s \big] + &= \lim_{h \to 0} \sum_{u = s}^{u = t} + \mathbf{E} \bigg[ \mathbf{E} \Big[ G_u (B_{u + h} \!-\! B_u) \Big| \mathcal{F}_u \Big] \bigg| \mathcal{F}_s \bigg] + = 0 +\end{aligned}$$ + +We now have everything we need to calculate $\mathbf{E} [ I_t | \mathcal{F_s} ]$, +giving the martingale property: + +$$\begin{aligned} + \mathbf{E} \big[ I_t | \mathcal{F}_s \big] + = \mathbf{E} \big[ I_s | \mathcal{F}_s \big] + \mathbf{E} \big[ I_t \!-\! I_s | \mathcal{F}_s \big] + = I_s + \mathbf{E} \big[ I_t \!-\! I_s | \mathcal{F}_s \big] + = I_s +\end{aligned}$$ + +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. +</div> +</div> + + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. |