1. E(sin(X(t)) and E(sin(X(t))) ?

2. Properties of the unknown solution X(t) are usually NOT assumed, since they follow from the properties of the given function. or: Is X already known?

• 1-first moment and second moment ?
• 2-

Do I understand correctly that you pose the following problem?

Find the first and the second moment of the solution X(t) of the Ito equation dX(t) = sin(X(t)) dt + dW(t), where W is the standard Wiener process.

yes Dear

Dear Waleed Hajo ,

Thus, what is your reason to guess that X(t)t are normally distributed? One should take into account that usually, for W(t) being the Wiener process the solutions to nonlinear SDE are NOT normally distributed. Therefore, perhaps, you have some special reasons to admit normality of X(t). Are they implied by application of the equation to some real processes? If yes, then W(t) probably cannot be assumed to be Wiener, which IS gaussian (normally distributed).

Best, Joachim

What if we assume X(t) follows Poisson distribution?

Hi Waleed,

Starting with assumptions on X(t) is not the correct way of analysis after you already have assummed that W(t) is gaussian (as it is for the Wiener process).

In other formulation, if you assume some conditions on X(t) then this will restrict the class of admitted processes W(t). In particular:

1. In case X is assumed to be a process with discontinuities, then W(t) in general cases will possess discontinuities too. In particular, in such cases W cannot be a process with continuous trajectories (in particular W cannot be Wiener).

2. In case X is assumed gaussian (normal), its increments governed by the sin part of your equation become bounded, which are not normal. Consequently, the second terms governed by W cannot possess gaussian distribution (which again excludes W as a Wiener process).

Conclusion: Don't try to assume a'priori any probability distribution of X, unless you want to resign of the assumption that W is the Wiener process.

Best, Joachim

I can't do without the Wiener process but how do I find it E(sin(X(t)) ?

To Waleed,

Let me suggest that first you study the Ito formula to obtain possibility to trace the examples from the book. Feel free to ask any detailed question related to the intermediate steps omitted in the book. Perhaps there will be a chance to complete them afterwords:-)

Best, JoaD

I understand the Ito formula in detail

Dear Waleed,

OK, so I would try to derive from your Ito equation the Ito differential for sin(X(t)), i.e. to find

d{sin(X(t))} = . . .

This suggestion is a repetition of the step illustrated within the presented by you pages of the book. Will this work?

Good luck, Joachim

Let F be a smooth function and assume

that the conditions c4 and c5

I thought that we want to cooperate toward solving the problem:

Find the first and the second moment of the solution X(t) of the Ito equation dX(t) = sin(X(t)) dt + dW(t), where W is the standard Wiener process.

according to the pattern cases given by you from some book.

I have suggested to start with obtaining/deriving the differential d{sin(X(t))} = . . . from the equation for dX(t) given in the formulation of the problem.

Are you going to try this way? or: Do you have another proposal?

Joachim

If you suggest this suggestion, do not benefit from anything because I need a second moment

We should suggest that

d(x^2)=......

OK, but I have no constructive Idea for getting the second moment.

Best, JoaD