# Removing deterministic discontinuities from semi-martingales

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 ?

• 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). Jun 9, 2013 at 9:13
• I clarified my question. I do not assmue $X$ is a local martingale anymore. Jun 9, 2013 at 13:24
• Please may you specify precisely what you mean by "remove jumps from $X$"? Which properties do you want $Y$ to have? Mar 4, 2014 at 16:27