If you have a function \( f \) that depends on time and another variable \( x \), and if you differentiate it with respect to both time and \( x \), you get the change in \( f \). For our purposes, \( x \) is going to be our Brownian motion \( W(t) \).
We're interested in the function \( f(t, W(t)) = W(t)^2 - t \).
For a function \( f(t, X(t)) \), the differential \( df \) using Itô's lemma is:
\[ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial X} dX + \frac{1}{2} \frac{\partial^2 f}{\partial X^2} (dX)^2 \]
Where:
- \( \frac{\partial f}{\partial t} \) is the partial derivative of \( f \) with respect to \( t \).
- \( \frac{\partial f}{\partial X} \) is the first partial derivative of \( f \) with respect to \( X(t) \).
- \( \frac{\partial^2 f}{\partial X^2} \) is the second partial derivative of \( f \) with respect to \( X(t) \).
To determine if \( Y(t) \) is a martingale, we'll find its differential and check its properties.
1. First Partial Derivative with respect to t:
The function \( Y(t) \) has a term \( -t \), which directly depends on \( t \). So, its derivative with respect to \( t \) is \( -1 \).
2. First Partial Derivative with respect to W(t):Differentiating \( W(t)^2 \) with respect to \( W(t) \) gives \( 2W(t) \).
3. Second Partial Derivative with respect to W(t):Differentiating the result from the previous step, \( 2W(t) \), again with respect to \( W(t) \) gives \( 2 \).
Now, plug these values into Itô's Lemma:
\[ dY(t) = (-1)dt + 2W(t)dW(t) + \frac{1}{2} \times 2 d(W(t))^2 \]
\[ \Rightarrow dY(t) = -dt + 2W(t)dW(t) + d(W(t))^2 \]
\[ \Rightarrow dY(t) = -dt + 2W(t)dW(t) + dt = 2W(t)dW(t) \]
(given that \( d(W(t))^2 \) equals \( dt \))
For \( Y(t) \) to be a martingale, its expected change should be zero. From the equation above, the expectation of \( 2W(t)d(W(t)) \) is zero, as \( d(W(t)) \) has a mean of 0.
Thus, we have a process \( Y(t) \) whose change consists solely of a diffusion term with zero expectation, which means that \( Y(t) \) does not drift up or down over time. This is precisely the property of a martingale.
This means \( Y(t) = W(t)^2 - t \) is a martingale.
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