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Diffstat (limited to 'content/know/concept/ito-calculus')
| -rw-r--r-- | content/know/concept/ito-calculus/index.pdc | 180 |
1 files changed, 164 insertions, 16 deletions
diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-calculus/index.pdc index 576e09a..3527b1d 100644 --- a/content/know/concept/ito-calculus/index.pdc +++ b/content/know/concept/ito-calculus/index.pdc @@ -12,10 +12,10 @@ markup: pandoc # Itō calculus -Given two time-indexed [random variables](/know/concept/random-variable/) -(i.e. stochastic processes) $F_t$ and $G_t$, -then consider the following random variable $X_t$, -where $B_t$ is the [Wiener process](/know/concept/wiener-process/): +Given two [stochastic processes](/know/concept/stochastic-process/) +$F_t$ and $G_t$, consider the following random variable $X_t$, +where $B_t$ is the [Wiener process](/know/concept/wiener-process/), +i.e. Brownian motion: $$\begin{aligned} X_t @@ -27,7 +27,7 @@ assuming $G_t$ is Itō-integrable. We call $X_t$ an **Itō process** if $F_t$ is locally integrable, and the initial condition $X_0$ is known, i.e. $X_0$ is $\mathcal{F}_0$-measurable, -where $\mathcal{F}_t$ is the [filtration](/know/concept/sigma-algebra/) +where $\mathcal{F}_t$ is the filtration to which $F_t$, $G_t$ and $B_t$ are adapted. The above definition of $X_t$ is often abbreviated as follows, where $X_0$ is implicit: @@ -39,8 +39,18 @@ $$\begin{aligned} Typically, $F_t$ is referred to as the **drift** of $X_t$, and $G_t$ as its **intensity**. +Because the Itō integral of $G_t$ is a +[martingale](/know/concept/martingale/), +it does not contribute to the mean of $X_t$: + +$$\begin{aligned} + \mathbf{E}[X_t] + = \int_0^t \mathbf{E}[F_s] \dd{s} +\end{aligned}$$ + Now, consider the following **Itō stochastic differential equation** (SDE), -where $\xi_t = \dv*{B_t}{t}$ is white noise: +where $\xi_t = \dv*{B_t}{t}$ is white noise, +informally treated as the $t$-derivative of $B_t$: $$\begin{aligned} \dv{X_t}{t} @@ -51,15 +61,6 @@ An Itō process $X_t$ is said to satisfy this equation if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$, in which case $X_t$ is also called an **Itō diffusion**. -Because the Itō integral of $G_t$ is a -[martingale](/know/concept/martingale/), -it does not contribute to the mean of $X_t$: - -$$\begin{aligned} - \mathbf{E}[X_t] - = \int_0^t \mathbf{E}[F_s] \dd{s} -\end{aligned}$$ - ## Itō's lemma @@ -204,9 +205,156 @@ $$\begin{aligned} 0 &= f(x) \: h'(x) + \frac{1}{2} g^2(x) \: h''(x) \\ - &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg) + &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g^2(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg) +\end{aligned}$$ + + +## Existence and uniqueness + +It is worth knowing under what condition a solution to a given SDE exists, +in the sense that it is finite on the entire time axis. +Suppose the drift $f$ and intensity $g$ satisfy these inequalities, +for some known constant $K$ and for all $x$: + +$$\begin{aligned} + x f(x) \le K (1 + x^2) + \qquad \quad + g^2(x) \le K (1 + x^2) +\end{aligned}$$ + +When this is satisfied, we can find the following upper bound +on an Itō process $X_t$, +which clearly implies that $X_t$ is finite for all $t$: + +$$\begin{aligned} + \boxed{ + \mathbf{E}[X_t^2] + \le \big(X_0^2 + 3 K t\big) \exp\!\big(3 K t\big) + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-existence"/> +<label for="proof-existence">Proof</label> +<div class="hidden"> +<label for="proof-existence">Proof.</label> +If we define $Y_t \equiv X_t^2$, +then Itō's lemma tells us that the following holds: + +$$\begin{aligned} + \dd{Y_t} + = \big( 2 X_t \: f(X_t) + g^2(X_t) \big) \dd{t} + 2 X_t \: g(X_t) \dd{B_t} \end{aligned}$$ +Integrating and taking the expectation value +removes the Wiener term, leaving: + +$$\begin{aligned} + \mathbf{E}[Y_t] + = Y_0 + \mathbf{E}\! \int_0^t 2 X_s f(X_s) + g^2(X_s) \dd{s} +\end{aligned}$$ + +Given that $K (1 \!+\! x^2)$ is an upper bound of $x f(x)$ and $g^2(x)$, +we get an inequality: + +$$\begin{aligned} + \mathbf{E}[Y_t] + &\le Y_0 + \mathbf{E}\! \int_0^t 2 K (1 \!+\! X_s^2) + K (1 \!+\! X_s^2) \dd{s} + \\ + &\le Y_0 + \int_0^t 3 K (1 + \mathbf{E}[Y_s]) \dd{s} + \\ + &\le Y_0 + 3 K t + \int_0^t 3 K \big( \mathbf{E}[Y_s] \big) \dd{s} +\end{aligned}$$ + +We then apply the +[Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/), +noting that $(Y_0 \!+\! 3 K t)$ does not decrease with time, leading us to: + +$$\begin{aligned} + \mathbf{E}[Y_t] + &\le (Y_0 + 3 K t) \exp\!\bigg( \int_0^t 3 K \dd{s} \bigg) + \\ + &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big) +\end{aligned}$$ +</div> +</div> + +If a solution exists, it is also worth knowing whether it is unique. +Suppose that $f$ and $g$ satisfy the following inequalities, +for some constant $K$ and for all $x$ and $y$: + +$$\begin{aligned} + \big| f(x) - f(y) \big| \le K \big| x - y \big| + \qquad \quad + \big| g(x) - g(y) \big| \le K \big| x - y \big| +\end{aligned}$$ + +Let $X_t$ and $Y_t$ both be solutions to a given SDE, +but the initial conditions need not be the same, +such that the difference is initially $X_0 \!-\! Y_0$. +Then the difference $X_t \!-\! Y_t$ is bounded by: + +$$\begin{aligned} + \boxed{ + \mathbf{E}\big[ (X_t - Y_t)^2 \big] + \le (X_0 - Y_0)^2 \exp\!\Big( \big(2 K \!+\! K^2 \big) t \Big) + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-uniqueness"/> +<label for="proof-uniqueness">Proof</label> +<div class="hidden"> +<label for="proof-uniqueness">Proof.</label> +We define $D_t \equiv X_t \!-\! Y_t$ and $Z_t \equiv D_t^2 \ge 0$, +together with $F_t \equiv f(X_t) \!-\! f(Y_t)$ and $G_t \equiv g(X_t) \!-\! g(Y_t)$, +such that Itō's lemma states: + +$$\begin{aligned} + \dd{Z_t} + = \big( 2 D_t F_t + G_t^2 \big) \dd{t} + 2 D_t G_t \dd{B_t} +\end{aligned}$$ + +Integrating and taking the expectation value +removes the Wiener term, leaving: + +$$\begin{aligned} + \mathbf{E}[Z_t] + = Z_0 + \mathbf{E}\! \int_0^t 2 D_s F_s + G_s^2 \dd{s} +\end{aligned}$$ + +The *Cauchy-Schwarz inequality* states that $|D_s F_s| \le |D_s| |F_s|$, +and then the given fact that $F_s$ and $G_s$ satisfy +$|F_s| \le K |D_s|$ and $|G_s| \le K |D_s|$ gives: + +$$\begin{aligned} + \mathbf{E}[Z_t] + &\le Z_0 + \mathbf{E}\! \int_0^t 2 K D_s^2 + K^2 D_s^2 \dd{s} + \\ + &\le Z_0 + \int_0^t (2 K \!+\! K^2) \: \mathbf{E}[Z_s] \dd{s} +\end{aligned}$$ + +Where we have implicitly used that $D_s F_s = |D_s F_s|$ +because $Z_t$ is positive for all $G_s^2$, +and that $|D_s|^2 = D_s^2$ because $D_s$ is real. +We then apply the +[Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/), +recognizing that $Z_0$ does not decrease with time (since it is constant): + +$$\begin{aligned} + \mathbf{E}[Z_t] + &\le Z_0 \exp\!\bigg( \int_0^t 2 K \!+\! K^2 \dd{s} \bigg) + \\ + &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big) +\end{aligned}$$ +</div> +</div> + +Using these properties, it can then be shown +that if all of the above conditions are satisfied, +then the SDE has a unique solution, +which is $\mathcal{F}_t$-adapted, continuous, and exists for all times. + ## References |