The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the second argument. We then perform a Taylor series expansion as follows: $$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} dt^2 + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2 + \frac{\partial^2 f}{\partial t \partial x} dt dx + \ldots $$

We then substitute $x=X_t$ where $X_t$ is a stochastic process such as an Ito process: $$dX_t = \mu dt + \sigma dW_t$$ where $W_t$ is a Wiener process. Realizing that $dX_t^2 = dt$ we obtained Ito's formula.

I have several questions regarding this procedure:

1. How should we interpret differentials of stochastic terms e.g. $dW_t$ or derivatives with respect to stochastic processes like $\frac{\partial}{\partial X_t}$ which appear in the Taylor series expansion when we substitute $x=X_t$. This seems to be undefined since it's not a smooth function
2. I am confused by what we mean when we say $f$ is smooth if it's a function of a stochastic process? I understand it's continuous differentiable in terms of its arguments, but as soon as we replace $x=X_t$ doesn't it become non-differentiable in time?
3. How can we replace $x=X_t$ if $X_t$ is a function of $t$? Wouldn't this require us to define the time derivative of $X_t$, which by definition is non-differentiable? This is the same discussion as: https://math.stackexchange.com/questions/2252734/confusion-about-second-partial-derivative-term-in-itos-lemma-with-a-constraint

I understand that we're taking the Taylor series of $f$ (some ordinary function) and which has nothing to do with $X_t$. But treating the argument as $x$ and then substituting it with a time dependent argument $X_t$ seems a little bit un-intuitive. However, I do understand that substituting $X_t$ is the same as substituting any time-dependent process, regardless of it being non-differentiable or not in terms of time. It just seems that when we substitute $x=X_t$ the Taylor series just makes a bit less sense.

The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the second argument. We then perform a Taylor series expansion as follows: $$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} dt^2 + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2 + \frac{\partial^2 f}{\partial t \partial x} dt dx + \ldots $$

We then substitute $x=X_t$ where $X_t$ is a stochastic process such as an Ito process: $$dX_t = \mu dt + \sigma dW_t$$ where $W_t$ is a Wiener process. Realizing that $dX_t^2 = dt$ we obtained Ito's formula.

I have several questions regarding this procedure:

1. How should we interpret differentials of stochastic terms e.g. $dW_t$ or derivatives with respect to stochastic processes like $\frac{\partial}{\partial X_t}$ which appear in the Taylor series expansion when we substitute $x=X_t$. This seems to be undefined since it's not a smooth function
2. I am confused by what we mean when we say $f$ is smooth if it's a function of a stochastic process? I understand it's continuously differentiable in terms of its arguments, but as soon as we replace $x=X_t$ doesn't it become non-differentiable in time?
3. How can we replace $x=X_t$ if $X_t$ is a function of $t$? Wouldn't this require us to define the time derivative of $X_t$, which by definition is non-differentiable? This is the same discussion as: https://math.stackexchange.com/questions/2252734/confusion-about-second-partial-derivative-term-in-itos-lemma-with-a-constraint

I understand that we're taking the Taylor series of $f$ (some ordinary function) and which has nothing to do with $X_t$. But treating the argument as $x$ and then substituting it with a time dependent argument $X_t$ seems a little bit un-intuitive. However, I do understand that substituting $X_t$ is the same as substituting any time-dependent process, regardless of it being non-differentiable or not in terms of time. It just seems that when we substitute $x=X_t$ the Taylor series just makes a bit less sense.

Edit: $d W_t^2 = dt$ not $d X_t^2 = dt$

The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the second argument. We then perform a Taylor series expansion as follows: $$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} dt^2 + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2 + \frac{\partial^2 f}{\partial t \partial x} dt dx + \ldots $$

We then substitute $x=X_t$ where $X_t$ is a stochastic process such as an Ito process: $$dX_t = \mu dt + \sigma dW_t$$ where $W_t$ is a Wiener process. Realizing that $dX_t^2 = dt$ we obtained Ito's formula.

I have several questions regarding this procedure:

1. How should we interpret differentials of stochastic terms e.g. $dW_t$? This seems to be undefined since it's not a smooth function
2. I am confused by what we mean when we say $f$ is smooth if it's a function of a stochastic process? I understand it's continuous differentiable in terms of its arguments, but as soon as we replace $x=X_t$ doesn't it become non-differentiable in time?
3. How can we replace $x=X_t$ if $X_t$ is a function of $t$? Wouldn't this require us to define the time derivative of $X_t$, which by definition is non-differentiable? This is the same discussion as: https://math.stackexchange.com/questions/2252734/confusion-about-second-partial-derivative-term-in-itos-lemma-with-a-constraint

I understand that we're taking the Taylor series of $f$ (some ordinary function) and which has nothing to do with $X_t$. But treating the argument as $x$ and then substituting it with a time dependent argument $X_t$ seems a little bit un-intuitive. However, I do understand that substituting $X_t$ is the same as substituting any time-dependent process, regardless of it being non-differentiable or not in terms of time. It just seems that when we substitute $x=X_t$ the Taylor series just makes a bit less sense.

The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the second argument. We then perform a Taylor series expansion as follows: $$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} dt^2 + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2 + \frac{\partial^2 f}{\partial t \partial x} dt dx + \ldots $$

We then substitute $x=X_t$ where $X_t$ is a stochastic process such as an Ito process: $$dX_t = \mu dt + \sigma dW_t$$ where $W_t$ is a Wiener process. Realizing that $dX_t^2 = dt$ we obtained Ito's formula.

I have several questions regarding this procedure:

1. How should we interpret differentials of stochastic terms e.g. $dW_t$ or derivatives with respect to stochastic processes like $\frac{\partial}{\partial X_t}$ which appear in the Taylor series expansion when we substitute $x=X_t$. This seems to be undefined since it's not a smooth function
2. I am confused by what we mean when we say $f$ is smooth if it's a function of a stochastic process? I understand it's continuous differentiable in terms of its arguments, but as soon as we replace $x=X_t$ doesn't it become non-differentiable in time?
3. How can we replace $x=X_t$ if $X_t$ is a function of $t$? Wouldn't this require us to define the time derivative of $X_t$, which by definition is non-differentiable? This is the same discussion as: https://math.stackexchange.com/questions/2252734/confusion-about-second-partial-derivative-term-in-itos-lemma-with-a-constraint

I understand that we're taking the Taylor series of $f$ (some ordinary function) and which has nothing to do with $X_t$. But treating the argument as $x$ and then substituting it with a time dependent argument $X_t$ seems a little bit un-intuitive. However, I do understand that substituting $X_t$ is the same as substituting any time-dependent process, regardless of it being non-differentiable or not in terms of time. It just seems that when we substitute $x=X_t$ the Taylor series just makes a bit less sense.