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Mathematical Finance: What are Multidimensional Diffusions, Quadratic Covariation, and Ito's Formula?
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Multidimensional Diffusions, Quadratic Covariation, and Ito's Formula

If $X := (X_1, X_2, \ldots, X_n)'$ is a $n$-dimensional diffusion process with form \begin{equation*} X(t) = X(0) + \int_0^t \mu(s)\,ds + \int_0^t \Sigma(s)\,dW(s), \end{equation*} where $\Sigma(t) \in R^{n \times m}$ and $W$ is a $m$-dimensional Brownian motion. The \emph{quadration covariation} of the components $X_i$ and $X_j$ is \begin{equation*} QV{X_i,X_j}(t) = \int_0^t \Sigma_i(s)' \Sigma_j(s)\,ds, \end{equation*} or in differential form $dQV{X_i,X_j}(t) = \Sigma_i(t)'\Sigma_j(t)\,dt$, where $\Sigma_i(t)$ is the $i^\text{th}$ column of $\Sigma(t)$. The \emph{quadratic variation} of $X_i(t)$ is $QV{X_i}(t) = \int_0^t \Sigma_i(s)' \Sigma_i(s)\,ds$. The multi-dimensional It\^{o} formula for $Y(t) = f(t, X_1(t), \ldots, X_n(t))$ is \begin{align*} d Y(t) = &\frac{\partial f}{\partial t} (t,X_1(t), \ldots, X_n(t)) dt + \sum_{i=1}^n \frac{\partial f}{\partial x_i} (t,X_1(t), \ldots, X_n(t))dX_i(t) \\ &+ \frac{1}{2}\sum_{i,j\,=1}^n \frac{\partial^2 f}{\partial x_i \partial x_j} (t,X_1(t), \ldots, X_n(t)) d QV{X_i, X_j}(t). \end{align*} The (vector-valued) multi-dimensional It\^{o} formula for \begin{equation*} Y(t) = f(t,X(t)) = (f_1(t,X(t)), \ldots, f_n(t,X(t)))' \end{equation*} where $f_k(t, X) = f_k(t, X_1, \ldots, X_n)$ and $Y(t) = (Y_1(t), Y_2(t), \ldots, Y_n(t))'$ is given component-wise (for $k=1, \ldots, n$) as \begin{align*} d Y_k(t) = \frac{\partial f_k(t,X(t))}{\partial t} dt &+ \sum_{i=1}^n \frac{\partial f_k(t,X(t))}{\partial x_i} dX_i(t) \\ &+ \frac{1}{2}\sum_{i,j\,=1}^n \frac{\partial^2 f_k(t,X(t))}{\partial x_i \partial x_j} d QV{X_i, X_j}(t). \end{align*}
by Wooden (1,878 points)

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