x̄ - > Stochastic processes \[ dX_t = 2dt + 3dW_t + 2dW_t^2 \] \[ dY_t = 3dt - 7dW_t - 4dW_t^2 \]

Using Ito's Lemma for \(XY\) and correctly derived that \(d[X,Y]_t = 0\). If you want to perform this computation in R, you can use the `itoDifferential()` function from the `sde` package. Here's how you can do it:

```R

# Install and load the sde package if not already installed

if (!requireNamespace("sde", quietly = TRUE)) {

  install.packages("sde")

}

library(sde)

# Define the coefficients of dX_t and dY_t

dX <- expression(2*dt + 3*dWt + 2*(dWt)^2)

dY <- expression(3*dt - 7*dWt - 4*(dWt)^2)

# Compute the differential of the product XY using Ito's Lemma

dXY <- itoDifferential(dX, dY)

# Print the result

print(dXY)

```

This code will compute the differential \(d[X,Y]_t\) and print the result, which should be \(0\) based on your manual calculation.

To calculate \(d[X,Y]_t\), we first need to compute the differential of the product \(XY\) using the Ito's Lemma. 

Let's denote \(X_t\) and \(Y_t\) as stochastic processes given by:

\[ dX_t = 2dt + 3dW_t + 2dW_t^2 \]

\[ dY_t = 3dt - 7dW_t - 4dW_t^2 \]

Using Ito's Lemma for \(XY\), we have:

\[ d(X_t Y_t) = X_tdY_t + Y_tdX_t + dX_tdY_t \]

Now, we'll compute each term:

1. \( X_tdY_t \):

\[ X_tdY_t = (2dt + 3dW_t + 2dW_t^2)(3dt - 7dW_t - 4dW_t^2) \]

2. \( Y_tdX_t \):

\[ Y_tdX_t = (3dt - 7dW_t - 4dW_t^2)(2dt + 3dW_t + 2dW_t^2) \]

3. \( dX_tdY_t \):

\[ dX_tdY_t = (2dt + 3dW_t + 2dW_t^2)(3dt - 7dW_t - 4dW_t^2) \]

After computing these terms, we can find the stochastic differential \(d[X,Y]_t\) by subtracting \(dX_tdY_t\) from \(d(X_t Y_t)\):

\[ d[X,Y]_t = d(X_t Y_t) - dX_tdY_t \]

Let's compute these terms:

1. \( X_tdY_t \):

\[ (2dt + 3dW_t + 2dW_t^2)(3dt - 7dW_t - 4dW_t^2) \]

\[ = 6dt^2 + 9dW_tdW_t - 14dW_t^3 - 12dW_tdW_t^2 + 6dW_tdt + 9dW_t^2dt + 6dW_t^2dW_t - 21dW_tdW_t^2 - 8dW_tdW_t^3 \]

\[ = 6dt^2 + 9dt - 14dW_t^3 - 12dW_t^2dt - 21dW_t^2dt \]

2. \( Y_tdX_t \):

\[ (3dt - 7dW_t - 4dW_t^2)(2dt + 3dW_t + 2dW_t^2) \]

\[ = 6dt^2 + 9dW_tdW_t - 14dW_t^3 - 12dW_tdW_t^2 + 6dW_tdt + 9dW_t^2dt + 6dW_t^2dW_t - 21dW_tdW_t^2 - 8dW_tdW_t^3 \]

\[ = 6dt^2 + 9dt - 14dW_t^3 - 12dW_t^2dt - 21dW_t^2dt \]

3. \( dX_tdY_t \) is the same as \( X_tdY_t \).

Now, subtracting \( dX_tdY_t \) from \( d(X_t Y_t) \):

\[ d[X,Y]_t = (6dt^2 + 9dt - 14dW_t^3 - 12dW_t^2dt - 21dW_t^2dt) - (6dt^2 + 9dt - 14dW_t^3 - 12dW_t^2dt - 21dW_t^2dt) \]

\[ = 0 \]

So, \(d[X,Y]_t = 0\).

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