## Itô Process An Itô process $X_t$ is a continuous-time [[Stochastic Process]] that combines a deterministic trend with a random component driven by [[Brownian Motion]]. $ X_t = \underbrace{X_0+ \int_0^T a(t, X_t)\, dt}_{\text{deterministic component}} + \underbrace{\int_0^T b(t, X_t)\, dB_t}_{\text{random component}}$ More commonly it is expressed in [[Derivative in Differential Form|Differential Form]], emphasizing the change in $X_t$. $dX_t = a(t, X_t)\,dt + b(t,X_t)\, dB_t$ where: - $a(t, X_t)$ is the *drift*, i.e. a deterministic function that governs the average change in $X_t$. - $b(t, X_t)$ is the *diffusion*, i.e. a deterministic function that governs the amplitude of the randomness. - $dB_t$ is an increment of the standard [[Brownian Motion]], a random process characterized by the property $(dB_t)^2 \approx dt$. >[!note:] >Since the Brownian motion $B_t$ (which is a component of the Itô process), is nowhere differentiable we cannot directly apply classical [[Differentiation Rules#Chain Rule|Chain Rule]] to find the derivative of $X_t$. This motivates the development of Itô's Lemma. >[!note:] >In simpler cases $a$ and $b$ can be just constants. For simpler notation we will write $a(t, X_t)$ as $a$, which can be a constant or a function. The argumentation does not change. ## Itô's Lemma for $F(X_t)$ Often we are interested in a function $F$ of the Itô process $X_t$. For instance, if $X_t$ represents the stock price, then $F(X_t)$ could represent the payoff of an option written on that stock. Although $X_t$ is not classically differentiable, we assume that the function $F$ itself is smooth (i.e. twice continuously differentiable). To determine how $F(X_t)$ changes, we start with a [[Taylor Series#Taylor Series Expansion|Taylor Series Expansion]]. *Step 1: Taylor Series Expansion:* If $dX_t$ is a small change in $X_t$, then by Taylor's theorem we have: $ F(X_t +dX_t) = F(X_t)+F^\prime(X_t)*dX_t+\frac{1}{2}F^{\prime \prime}(X_t)(dX_t)^2+ \cdots $ In classical (deterministic) calculus, we neglect $(dX_t)^2$ and all higher order terms because $dX_t$ is already infinitesimal. However, for an Itô process, the term $dX_t$ contains the Brownian motion term $dB_t$, whose squared is not negligible. *Step 2: Magnitude of Brownian Motion Term:* To see why $(dX_t)^2$ is not negligible, we take the square of the differential $dX_t$ of the Itô process: $ \begin{align} dX_t&=a \, dt+ b\, dB_t \\[2pt] (dX_t)^2 &= \Big( a\, dt+b\, dB_t\Big)^2\\ &=a^2 (dt)^2+2ab\, dt \,dB_t+b^2 (dB_t)^2 \end{align} $ where: - $(dt)^2$ is of order $dt^2$, which is *negligible*. - $dt \, dB_t$ is of order $dt^{3/2}$ , which is *negligible*. - $(dB_t)^2$ is of order $dt$, since $dB_t$ is of order $\sqrt{dt}$. Thus we drop negligible terms and approximate with the remaining terms. $ (dX_t)^2 \approx b^2 (dB_t)^2 \approx b^2 dt$ *Step 3: Substituting Back into Taylor Expansion:* Since $(dX_t)^2$ is of order $dt$ and therefore not negligible, we retain it in the expansion: $ \begin{align} F(X_t +dX_t) &\approx F(X_t)+F^\prime(X_t)*dX_t+\frac{1}{2}F^{\prime \prime}(X_t)*(dX_t)^2\\ &\approx F(X_t)+F^\prime(X_t)*dX_t+\frac{1}{2}F^{\prime \prime}(X_t)*b(t, X_t)^2 dt\\ \end{align} $ Rewriting in the differential form, we obtain Itô's Lemma when $F$ is a function of $X_t$ only. $ \boxed{\quad dF(X_t) \approx \underbrace{F^\prime(X_t)*dX_t}_{\text{chain rule}}+ \underbrace{\frac{1}{2}F^{\prime \prime}(X_t)*b^2 dt}_{\text{stochastic term}} \quad} $ ## Itô's Lemma for $F(t, X_t)$ When the function $F$ depends explicitly on both time $t$ and the process $X_t$, we must apply the [[Total Derivative|Multivariate Chain Rule]]. Then, the total differential of $F$ is given by: $ dF(t,X_t)= \frac{\partial F}{\partial t} (t, X_t)\, dt + \frac{\partial F}{\partial x} (t, X_t)\, dX_t + \frac{1}{2} \frac{\partial^2F}{\partial x^2} (t, X_t)\, (dX_t)^2 $ Using the approximation $(dX_t)^2 \approx b(t, X_t)^2 dt$, and simplifying notation we obtain the full form of Itô's Lemma. $ \boxed{\quad\begin{align} dF(t,X_t)= \underbrace{ \frac{\partial F}{\partial t}\, dt+ \frac{\partial F}{\partial x}\, dX_t }_{\text{multivariate chain rule}}+ \underbrace{ \frac{1}{2} \frac{\partial^2F}{\partial x^2} \, b^2\, dt }_{\text{stochastic term}}\end{align}\quad} $ >[!note:] >Since we are dealing with a multivariate function $F(t, X_t)$ now, it is more common to use the notation of partial derivatives instead of the differential form. >[!note:] >By the definition of Itô's Lemma we can see that $dF(t, X_t)$ is also an Itô process itself. Hence the differential of a function $F$ of an Itô process, is itself an Itô process!