Function of a stochastic process \(X_t\) - Itô I
Let \(F\) be a function of a stochastic random variable \(X_t\). So \(F = F(X_t)\).
Using a Taylor expansion: \[F(X + dX) \approx F(X) + \frac{dF}{dX} \space dX + \frac{1}{2} \frac{d^2F}{dX^2} dX^2\] Using \(F(X+dX) - F(X) = dF\): \[dF = \frac{dF}{dX} \space dX + \frac{1}{2} \frac{d^2F}{dX^2} dX^2\]
Now, recall \(dX\) is a random variable with mean 0 and standard deviation \(\sqrt{t}\). Also, for small values of \(t\), we have \(\sqrt{dt} \gt dt\), hence it is of higher order than dt. Hence, we re-write the previous equation as
\[dF = \frac{dF}{dX} \space dX + \frac{1}{2} \frac{d^2F}{dX^2} dt\] Usually, we write the deterministic part of the equation first.
\[dF = \frac{1}{2} \frac{d^2F}{dX^2} \space dt + \frac{dF}{dX} \space dX \tag{1}\]
All our Stochastic Differential Equations will have in them these 2 different time steps: \(dt\) and \(\sqrt{dt}\). \[dF = \underbrace {\dots \space dt}_{deterministic-drift} + \underbrace{\dots \space dX}_{random-diffusion}\]
The integral form of this Stochastic Differential Equation is \[\int_0^t \frac{dF}{dX_\tau} \space dX_\tau = \int_0^t dF - \frac{1}{2} \int_0^t \frac{d^2F}{dX_\tau^2} \space d\tau\]
\(F(X_t) = X^2\), where \(X_t\) is stochastic random variable. \[\frac{dF}{dX} = 2X\] \[\frac{d^2F}{dX^2} = 2\]
Hence, using the previous Equation 1 (aka itô I): \[dF = \frac{1}{2} \cdot 2 \space dt + 2X \space dX\] \[dF = dt + 2X \space dX\] Using integral form
We could integrate both side of the previous equation. \[\int_0^t dF = \int_0^t d\tau + \int_0^t 2X(\tau) d{\tau}\] \[F(t) - F(0) = t - 0 + 2 \int_0^t X(\tau) d{\tau}\] And assuming F(0) = 0: \[F(t) = t + 2 \int_0^t X_{\tau} \space d{\tau}\] \[X^2(t) = t + 2 \int_0^t X_{\tau} \space d{\tau}\]
\(F(W_t) = sin \space W_t + cos \space W_t\), where \(X_t\) is stochastic random variable. \[\frac{dF}{dW_t} = cos \space W_t - sin \space W_t\] \[\frac{d^2F}{dW_t^2} = -(sin \space W_t + cos \space W_t)\] Hence, we can now write
\[dF = - \frac{1}{2}(sin \space W_t + cos \space W_t) \space dt + (cos \space W_t - sin \space W_t) \space dW_t\]
Function of a stochastic process \((t, X(t))\) - Itô II
Let \(F\) be a function of both time \(t\) and a stochastic random variable \(X_t\). So \(F = F(t, X_t)\).
Using a Taylor expansion: \[F(t + dt, X + dX) \approx F(t, X) + \frac{\partial F}{\partial t} \space dt + \frac{\partial F}{\partial X} \space dX + \frac{1}{2} \frac{d^2F}{dX^2} dX^2\] Using \(F(t + dt, X+dX) - F(t, X) = dF\) and factoring the \(dt\) and recall \(dX^2=dt\):
\[dF = \left( \frac{\partial F}{\partial t} + \frac{1}{2} \frac{\partial ^2F}{\partial X^2} \right) dt + \frac{\partial F}{\partial X} dX \tag{2}\]
We can transform this Stochastic Differential Equation (SDE) into an itô integral.
\[\int_0^t \frac{\partial F}{\partial X_\tau} dX_\tau = \int_0^t dF - \int_0^t \left( \frac{\partial F}{\partial \tau} + \frac{1}{2} \frac{\partial ^2F}{\partial X_\tau^2} \right) d\tau\] \[\int_0^t \frac{\partial F}{\partial X_\tau} dX_\tau = F(t, X_t) - F(0, X_0) - \int_0^t \left( \frac{\partial F}{\partial \tau} + \frac{1}{2} \frac{\partial ^2F}{\partial X_\tau^2} \right) d\tau \tag{3}\]
Express \(\int_0^t \left( \tau + W_\tau \right) dW_\tau\)
Using Equation 3, we establish that what is being integrated is \(\frac{\partial F}{\partial W_\tau}\), hence \[\frac{\partial F}{\partial W_\tau} = \tau + W_\tau \tag{4}\]
This allows us to find \(F\) by integrating both side in regards to \(W_\tau\). \[F = \tau W_\tau + \frac{1}{2} W_\tau^2 \tag{5}\] and derving Equation 5 for \(\tau\), \[\frac{\partial F}{\partial \tau} = W_\tau \tag{6}\] and deriving the first derivative Equation 4 one more time for \(dW_\tau\), we get \[\frac{\partial ^2F}{\partial W_\tau^2} = 1\] Hence, we can re-write \[\int_0^t \left( \tau + W_\tau \right) dW_\tau = \left( \tau W_\tau + \frac{1}{2} W_\tau^2 \right) - \int_0^t W_\tau + \frac{1}{2} \space d_\tau\]