We know that if W(t) is a Brownian Motion, then the Ito Process is given by,
X(t)= X(0) + $\int_0^t \Delta(u)dW(u)$ + $\int_0^t \Phi(u)du$
Could someone explain briefly what the two components are in this expression. If I'm not wrong, the first is an Ito Integral with respect to a Brownian Motion, but what's the second component and why is it included in the Ito Process? Or why is the Ito Process structured in this way?
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$\begingroup$The Ito process written in integral form (as you have it) is $$ X_t = X_0 + \int_0^t \mu(X_u,u) du + \int_0^t \sigma(X_u,u) dB_u $$ Or more succintly in "differential" form via $$ dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t $$ In the latter case, notice that $dB_t$ is the only source of noise. Hence if $\sigma\rightarrow 0$, we get: $$ \frac{dX_t}{dt} = \mu(X_t,t) $$ I.e. an ordinary differential equation. In essence, $\mu$ is a deterministic "push" on the position of the stochastic process. Separately, $\sigma$ is essentially a "scaling" function for the noise component.
For this reason, we can call $\mu$ the "drift" function and $\sigma$ the "diffusion" component. Think of a particle in water undergoing stochastic motion: the drift describes pushing (e.g. of the flowing water), while $\sigma$ controls its movement under natural diffusion.
This analogy is somewhat deep: notice that if $\mu\rightarrow 0$ and $\sigma \rightarrow 1$, then $X_t = B_t$. I.e. we can view the process as a Brownian particle under scaling and force. The relation to diffusion under drift can be seen via the Fokker-Planck equation (in 1D): $$ \frac{\partial}{\partial t} p_t(x) = -\frac{\partial}{\partial x}[\mu(x,t)p_t(x)] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma(x,t)^2 p_t(x)] $$ where $p_t(x)$ is the probability that the particle is at position $x$ at time $t$. Notice that $\mu\rightarrow 0$ and $\sigma \rightarrow 1$ shows that you exactly reproduce the classical diffusion (or heat) PDE in this limit. Recall also that the fundamental solution to the diffusion PDE is the Gaussian function, and so unsurprisingly, we can understand an Ito process in terms of it being normally distributed over a small time step. More specifically, the transition density function of the process over a small time step is given by: $$ p_t(x|y) \approx \mathcal{N}(x|y+\mu(y)t,\sigma^2(y)t) $$ where I'm letting $\mu$ and $\sigma$ just depend on space, not time (e.g. see here). This also explains my notation: $\mu$ (the drift) controls the mean of the process at a a given time, while $\sigma$ controls the variance over a small time step.
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