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Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le i \le n}$ a family of measurable functions. My goal is to remove jumps from $X$ using a change of variable $$ Y_t = \Phi(t,X_t)$$ where $\Phi$ is yet to be found. When $f_i$'s are affine, it is possible to remove the deterministic term in Itô's lemma by choosing an affine form for $\Phi$ in the space variable and a piecewise constant form in the time variable.

I could not find any result in the litterature regarding the general case, i.e. $f_i$ polynomial, or even discontinuous itself. Is there a general approach for removing deterministic discountinuities from semi-martingales ?

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  • $\begingroup$ I implicitly meant the sum of a local martingale and a pure jump process whose jumps are predictable ($\mathcal{F}_{t_i^-}$-measurable) and jump dates are constant ($\mathcal{F}_0$-measurable). $\endgroup$ – vanna Jun 9 '13 at 9:13
  • $\begingroup$ I clarified my question. I do not assmue $X$ is a local martingale anymore. $\endgroup$ – vanna Jun 9 '13 at 13:24
  • $\begingroup$ Please may you specify precisely what you mean by "remove jumps from $X$"? Which properties do you want $Y$ to have? $\endgroup$ – Ben Derrett Mar 4 '14 at 16:27

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