Smoothness of a value function with discontinuous parameters

In summary, the conversation discusses a function v(x) defined by a bounded measurable function and a standard Brownian motion. The function is well-defined and continuous, and the question is whether it is continuously differentiable and satisfies a differential equation when the original functions are discontinuous.
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economicsnerd
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Let ##\mu: \mathbb{R}\to \mathbb{R}##, ##f: \mathbb{R}\to \mathbb{R}##, and ##r: \mathbb{R}\to [1, \infty)## be bounded measurable functions (which may be discontinuous).

I'm interested in the function ##v:\mathbb{R}\to\mathbb{R}## given by ##v(x) = \mathbb E \left[ \int_0^\infty e^{-\int_0^t r(X_\tau) d\tau} f(X_t) \bigg| X_0 = x, \ dX_t = \mu(X_t) dt + dB_t \right]##, where ##\{B_t\}_t## is a standard Brownian motion.

I know that ##v## is well-defined, and I'm confident that it's continuous. I'm wondering if I can get any stronger smoothness results than this. For instance, is ##v## continuously differentiable?
 
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Related question:

If everything above were ##C^2##, then ##v## would satisfy the DE, ##r(x)v(x) = f(x) + \mu(x)v'(x) + \tfrac12 v''(x)##. I'd love to know if, when ##\mu, f, r## are discontinuous, there's still any meaningful sense in which the DE is satisfied.
 
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1. What is a value function with discontinuous parameters?

A value function is a mathematical function used to represent the desirability or utility of a particular outcome. It is typically used in decision-making processes where multiple options are available. A value function with discontinuous parameters is one where the parameters (or variables) used to calculate the value function change abruptly, rather than smoothly.

2. Why is the smoothness of a value function with discontinuous parameters important?

The smoothness of a value function affects the accuracy and reliability of the function's output. A value function with discontinuous parameters can lead to unexpected and potentially incorrect results, which can have significant consequences in decision-making processes.

3. How is the smoothness of a value function with discontinuous parameters measured?

The smoothness of a value function can be measured using mathematical techniques such as differentiability and continuity. In the case of discontinuous parameters, the function may not be differentiable at certain points, and this can be used to assess its smoothness.

4. Can a value function with discontinuous parameters still be useful?

Yes, a value function with discontinuous parameters can still be useful in certain situations. It may accurately represent a decision-making process where abrupt changes in parameters are expected, or it may be the best available representation of a complex system. However, it is important to understand the limitations and potential inaccuracies of such a function.

5. How can the smoothness of a value function with discontinuous parameters be improved?

The smoothness of a value function with discontinuous parameters can be improved by using alternative mathematical techniques, such as piecewise functions, to represent the function. Additionally, collecting more data and refining the parameters used in the function can also help improve its smoothness and accuracy.

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