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diff --git a/source/know/concept/ito-process/index.md b/source/know/concept/ito-process/index.md index b82835d..f192e28 100644 --- a/source/know/concept/ito-process/index.md +++ b/source/know/concept/ito-process/index.md @@ -9,8 +9,8 @@ layout: "concept" --- 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/), +$$F_t$$ 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: $$\begin{aligned} @@ -19,25 +19,25 @@ $$\begin{aligned} \end{aligned}$$ Where the latter is an [Itō integral](/know/concept/ito-integral/), -assuming $G_t$ is Itō-integrable. -We call $X_t$ an **Itō process** if $F_t$ is locally integrable, -and the initial condition $X_0$ is known, -i.e. $X_0$ is $\mathcal{F}_0$-measurable, -where $\mathcal{F}_t$ is the filtration -to which $F_t$, $G_t$ and $B_t$ are adapted. -The above definition of $X_t$ is often abbreviated as follows, -where $X_0$ is implicit: +assuming $$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$$ is the filtration +to which $$F_t$$, $$G_t$$ and $$B_t$$ are adapted. +The above definition of $$X_t$$ is often abbreviated as follows, +where $$X_0$$ is implicit: $$\begin{aligned} \dd{X_t} = F_t \dd{t} + G_t \dd{B_t} \end{aligned}$$ -Typically, $F_t$ is referred to as the **drift** of $X_t$, -and $G_t$ as its **intensity**. -Because the Itō integral of $G_t$ is a +Typically, $$F_t$$ is referred to as the **drift** of $$X_t$$, +and $$G_t$$ as its **intensity**. +Because the Itō integral of $$G_t$$ is a [martingale](/know/concept/martingale/), -it does not contribute to the mean of $X_t$: +it does not contribute to the mean of $$X_t$$: $$\begin{aligned} \mathbf{E}[X_t] @@ -45,25 +45,25 @@ $$\begin{aligned} \end{aligned}$$ Now, consider the following **Itō stochastic differential equation** (SDE), -where $\xi_t = \idv{B_t}{t}$ is white noise, -informally treated as the $t$-derivative of $B_t$: +where $$\xi_t = \idv{B_t}{t}$$ is white noise, +informally treated as the $$t$$-derivative of $$B_t$$: $$\begin{aligned} \dv{X_t}{t} = f(X_t, t) + g(X_t, t) \: \xi_t \end{aligned}$$ -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**. +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. +since only the current value of $$X_t$$ determines the future, +and $$B_t$$ is also a Markov process. ## Itō's lemma -Classically, given $y \equiv h(x(t), t)$, +Classically, given $$y \equiv h(x(t), t)$$, the chain rule of differentiation states that: $$\begin{aligned} @@ -71,8 +71,8 @@ $$\begin{aligned} = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{x} \end{aligned}$$ -However, for a stochastic process $Y_t \equiv h(X_t, t)$, -where $X_t$ is an Itō process, +However, for a stochastic process $$Y_t \equiv h(X_t, t)$$, +where $$X_t$$ is an Itō process, the chain rule is modified to the following, known as **Itō's lemma**: @@ -89,7 +89,7 @@ $$\begin{aligned} <div class="hidden" markdown="1"> <label for="proof-lemma">Proof.</label> We start by applying the classical chain rule, -but we go to second order in $x$. +but we go to second order in $$x$$. This is also valid classically, but there we would neglect all higher-order infinitesimals: @@ -98,7 +98,7 @@ $$\begin{aligned} = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{X_t} + \frac{1}{2} \pdvn{2}{h}{x} \dd{X_t}^2 \end{aligned}$$ -But here we cannot neglect $\dd{X_t}^2$. +But here we cannot neglect $$\dd{X_t}^2$$. We insert the definition of an Itō process: $$\begin{aligned} @@ -109,8 +109,8 @@ $$\begin{aligned} + \frac{1}{2} \pdvn{2}{h}{x} \Big( F_t^2 \dd{t}^2 + 2 F_t G_t \dd{t} \dd{B_t} + G_t^2 \dd{B_t}^2 \Big) \end{aligned}$$ -In the limit of small $\dd{t}$, we can neglect $\dd{t}^2$, -and as it turns out, $\dd{t} \dd{B_t}$ too: +In the limit of small $$\dd{t}$$, we can neglect $$\dd{t}^2$$, +and as it turns out, $$\dd{t} \dd{B_t}$$ too: $$\begin{aligned} \dd{t} \dd{B_t} @@ -120,8 +120,8 @@ $$\begin{aligned} \longrightarrow 0 \end{aligned}$$ -However, due to the scaling property of $B_t$, -we cannot ignore $\dd{B_t}^2$, which has order $\dd{t}$: +However, due to the scaling property of $$B_t$$, +we cannot ignore $$\dd{B_t}^2$$, which has order $$\dd{t}$$: $$\begin{aligned} \dd{B_t}^2 @@ -131,8 +131,8 @@ $$\begin{aligned} \longrightarrow \dd{t} \end{aligned}$$ -Where $\chi_1^2(\dd{t})$ is the generalized chi-squared distribution -with one term of variance $\dd{t}$. +Where $$\chi_1^2(\dd{t})$$ is the generalized chi-squared distribution +with one term of variance $$\dd{t}$$. </div> </div> @@ -144,9 +144,9 @@ to make the solution of a given Itō SDE easier. ## Coordinate transformations The simplest coordinate transformation is a scaling of the time axis. -Defining $s \equiv \alpha t$, the goal is to keep the Itō process. -We know how to scale $B_t$, be setting $W_s \equiv \sqrt{\alpha} B_{s / \alpha}$. -Let $Y_s \equiv X_t$ be the new variable on the rescaled axis, then: +Defining $$s \equiv \alpha t$$, the goal is to keep the Itō process. +We know how to scale $$B_t$$, be setting $$W_s \equiv \sqrt{\alpha} B_{s / \alpha}$$. +Let $$Y_s \equiv X_t$$ be the new variable on the rescaled axis, then: $$\begin{aligned} \dd{Y_s} @@ -156,14 +156,14 @@ $$\begin{aligned} &= \frac{1}{\alpha} f(Y_s) \dd{s} + \frac{1}{\sqrt{\alpha}} g(Y_s) \dd{W_s} \end{aligned}$$ -$W_s$ is a valid Wiener process, +$$W_s$$ is a valid Wiener process, and the other changes are small, so this is still an Itō process. To solve SDEs analytically, it is usually best -to have additive noise, i.e. $g = 1$. +to have additive noise, i.e. $$g = 1$$. This can be achieved using the **Lamperti transform**: -define $Y_t \equiv h(X_t)$, where $h$ is given by: +define $$Y_t \equiv h(X_t)$$, where $$h$$ is given by: $$\begin{aligned} \boxed{ @@ -173,8 +173,8 @@ $$\begin{aligned} \end{aligned}$$ Then, using Itō's lemma, it is straightforward -to show that the intensity becomes $1$. -Note that the lower integration limit $x_0$ does not enter: +to show that the intensity becomes $$1$$. +Note that the lower integration limit $$x_0$$ does not enter: $$\begin{aligned} \dd{Y_t} @@ -185,9 +185,9 @@ $$\begin{aligned} &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g'(X_t) \bigg) \dd{t} + \dd{B_t} \end{aligned}$$ -Similarly, we can eliminate the drift $f = 0$, +Similarly, we can eliminate the drift $$f = 0$$, thereby making the Itō process a martingale. -This is done by defining $Y_t \equiv h(X_t)$, with $h(x)$ given by: +This is done by defining $$Y_t \equiv h(X_t)$$, with $$h(x)$$ given by: $$\begin{aligned} \boxed{ @@ -197,8 +197,8 @@ $$\begin{aligned} \end{aligned}$$ The goal is to make the parenthesized first term (see above) -of Itō's lemma disappear, which this $h(x)$ does indeed do. -Note that $x_0$ and $x_1$ do not enter: +of Itō's lemma disappear, which this $$h(x)$$ does indeed do. +Note that $$x_0$$ and $$x_1$$ do not enter: $$\begin{aligned} 0 @@ -212,8 +212,8 @@ $$\begin{aligned} It is worth knowing under what condition a solution to a given SDE exists, in the sense that it is finite on the entire time axis. -Suppose the drift $f$ and intensity $g$ satisfy these inequalities, -for some known constant $K$ and for all $x$: +Suppose the drift $$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) @@ -222,8 +222,8 @@ $$\begin{aligned} \end{aligned}$$ 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 $t$: +on an Itō process $$X_t$$, +which clearly implies that $$X_t$$ is finite for all $$t$$: $$\begin{aligned} \boxed{ @@ -237,7 +237,7 @@ $$\begin{aligned} <label for="proof-existence">Proof</label> <div class="hidden" markdown="1"> <label for="proof-existence">Proof.</label> -If we define $Y_t \equiv X_t^2$, +If we define $$Y_t \equiv X_t^2$$, then Itō's lemma tells us that the following holds: $$\begin{aligned} @@ -253,7 +253,7 @@ $$\begin{aligned} = Y_0 + \mathbf{E}\! \int_0^t 2 X_s f(X_s) + g^2(X_s) \dd{s} \end{aligned}$$ -Given that $K (1 \!+\! x^2)$ is an upper bound of $x f(x)$ and $g^2(x)$, +Given that $$K (1 \!+\! x^2)$$ is an upper bound of $$x f(x)$$ and $$g^2(x)$$, we get an inequality: $$\begin{aligned} @@ -267,7 +267,7 @@ $$\begin{aligned} We then apply the [Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/), -noting that $(Y_0 \!+\! 3 K t)$ does not decrease with time, leading us to: +noting that $$(Y_0 \!+\! 3 K t)$$ does not decrease with time, leading us to: $$\begin{aligned} \mathbf{E}[Y_t] @@ -275,12 +275,13 @@ $$\begin{aligned} \\ &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big) \end{aligned}$$ + </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 $y$: +Suppose that $$f$$ and $$g$$ satisfy the following inequalities, +for some constant $$K$$ and for all $$x$$ and $$y$$: $$\begin{aligned} \big| f(x) - f(y) \big| \le K \big| x - y \big| @@ -288,10 +289,10 @@ $$\begin{aligned} \big| g(x) - g(y) \big| \le K \big| x - y \big| \end{aligned}$$ -Let $X_t$ and $Y_t$ both be solutions to a given SDE, +Let $$X_t$$ and $$Y_t$$ both be solutions to a given SDE, but the initial conditions need not be the same, -such that the difference is initially $X_0 \!-\! Y_0$. -Then the difference $X_t \!-\! Y_t$ is bounded by: +such that the difference is initially $$X_0 \!-\! Y_0$$. +Then the difference $$X_t \!-\! Y_t$$ is bounded by: $$\begin{aligned} \boxed{ @@ -305,8 +306,8 @@ $$\begin{aligned} <label for="proof-uniqueness">Proof</label> <div class="hidden" markdown="1"> <label for="proof-uniqueness">Proof.</label> -We define $D_t \equiv X_t \!-\! Y_t$ and $Z_t \equiv D_t^2 \ge 0$, -together with $F_t \equiv f(X_t) \!-\! f(Y_t)$ and $G_t \equiv g(X_t) \!-\! g(Y_t)$, +We define $$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} @@ -322,9 +323,9 @@ $$\begin{aligned} = Z_0 + \mathbf{E}\! \int_0^t 2 D_s F_s + G_s^2 \dd{s} \end{aligned}$$ -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: +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: $$\begin{aligned} \mathbf{E}[Z_t] @@ -333,12 +334,12 @@ $$\begin{aligned} &\le Z_0 + \int_0^t (2 K \!+\! K^2) \: \mathbf{E}[Z_s] \dd{s} \end{aligned}$$ -Where we have implicitly used that $D_s F_s = |D_s F_s|$ -because $Z_t$ is positive for all $G_s^2$, -and that $|D_s|^2 = D_s^2$ because $D_s$ is real. +Where we have implicitly used that $$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$$ is real. We then apply the [Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/), -recognizing that $Z_0$ does not decrease with time (since it is constant): +recognizing that $$Z_0$$ does not decrease with time (since it is constant): $$\begin{aligned} \mathbf{E}[Z_t] @@ -346,13 +347,14 @@ $$\begin{aligned} \\ &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big) \end{aligned}$$ + </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. +which is $$\mathcal{F}_t$$-adapted, continuous, and exists for all times. |