Existence/uniqueness of solution and Ito's formula

[mbp]

Hi everybody,

I have an Ito's stochastic differential equation

$dX_t = a(X_t,t) dt + b(X_t,t) dB_t$

where $a(X_t,t)$ and $b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions.
Given a function $f(X_t,t) \in C^2$ using Ito's formula I can derive the SDE

$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dX_t^2$

where $dX_t^2$ is computed using Ito's lemma.

The question is: are there any requirements that $f(X_t,t)$ must satisfy to guaranty
the existence and uniqueness of solutions (I would say yes).

Any reference is welcome. Thanks in advance.

 

[jvicens]

Hi there,

Thank you for sharing your Ito's stochastic differential equation and the derived SDE using Ito's formula. It is great to see your interest in understanding the requirements for the existence and uniqueness of solutions for this type of equation.

To answer your question, yes, there are certain requirements that f(X_t,t) must satisfy to guarantee the existence and uniqueness of solutions. In fact, the existence and uniqueness of solutions for SDEs depend on the properties of the coefficients a(X_t,t) and b(X_t,t), and the initial condition X_0.

For example, as you have mentioned, the coefficients a(X_t,t) and b(X_t,t) must satisfy the Lipschitz condition. This means that there exists a constant K such that for any two points (x,t) and (y,t) in the domain of the coefficients, the following inequality holds:

|a(x,t) - a(y,t)| + |b(x,t) - b(y,t)| < K|x-y|

Moreover, the initial condition X_0 must be a random variable that is adapted to the underlying Brownian motion process, which means that its value at time t depends only on the information up to time t. This ensures that the solution X_t is also adapted to the same filtration.

Apart from these requirements, there are also other conditions that must be satisfied, such as the integrability conditions and the growth conditions for the coefficients. These are essential for the existence and uniqueness of solutions, and you can find more information about them in any standard textbook on stochastic calculus and SDEs.

I hope this helps answer your question. If you need any further clarification or references, please do not hesitate to ask. Keep up the good work in exploring Ito's stochastic differential equation and its applications!