Applying Itô’s Formula Locally and Piecewise in Time

Summary

The document asks whether Itô’s formula remains valid when a function is smooth only on a restricted time and state domain, or is piecewise differentiable in time. The answer explains that the formula for diffusions is local: it can be applied while the process remains inside a region where the required smoothness conditions hold.

To handle a bounded region, choose a smaller set lying strictly inside the open ball and stop the process when it exits that set. Apply Itô’s formula from the starting time until the earlier of this exit time and the desired end time. This localizes the calculation away from the boundary, where the function may not be defined or smooth. The response does not address the separate piecewise-time case in detail, nor does it state conditions for crossing a discontinuity or give a modified formula; its guidance is limited to stopping within a smooth region.

Key ideas

• Itô’s formula can be applied locally when its smoothness conditions hold locally.
• Choose a region strictly inside the function’s domain to maintain a margin from the boundary.
• Stop the process at its exit time and apply the formula up to that stopping time or the target time, whichever comes first.
• The response does not explain how to handle piecewise differentiability in time.

Tags

# Piecewise Ito formula


# Piecewise Ito formula












Usually Ito's lemma is stated for $C^{1,2}(\mathbb{R}^{d+1},\mathbb{R})$ functions.

My question is does Ito still hold if the domain is restricted. That is if the semi-martingale $Z_t$ is only considered on $[t_1,t_2]\times B$ where B is an open ball in $\mathbb{R}^d$ and $f$ is $C^{1,2}$ thereon then does the Ito formula still hold?

More generally,what i need precisely is to know if $f$ is a piecewise $C^{1}$ function in the time parameter and twice differentiable in the space parameter, then does Ito still hold? If not can it be modified, if so how?

References are always welcome.

## Answer by M. Jeunesse (score 2)

https://quant.stackexchange.com/a/29700

Ito for diffusion is local so it holds locally if conditions are local.

Let B a open ball of $\mathbb{R}^d $ Let I be a open time interval. Let f be $C^{1,2}(I,B) $. Let $A\subset B $ strictly in $B $ (you take margins wrt to the boundary). Let $t_1,t_2\in I $ with $t_1 <t_2$ and $x\in B $

Then define $\tau$ the exit time of $A $ starting from $t_1,x$

You can now write Ito lemma between $t_1$ and $\min (\tau ,t_2)$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.