Interpreting Left Limits in Jump-Diffusion Price Dynamics

# Simple question on jump-diffusion


# Simple question on jump-diffusion












In the textbook by Shreve in sec. 11.7.2 a jump-diffusion process is introduced. More precisely

$$ dS_t = \alpha\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,d\left(Q_t-\beta\,\lambda\,t\right)\quad (1) $$

where $Q_t = \sum_{i=1}^{N_t}Y_i$ and $N_t$ is Poisson with intensity $\lambda$. The process is re-written as

$$ dS_t = (\alpha-\beta\,\lambda)\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,dQ_t\quad(2). $$

The problem is that, a part from time instants in which there is no jump and hence $S_t=S_{t-}$, I cannot go from (1) to (2), because if there is a jump of size $Y_i$ at time $t$ it holds that

$$ \frac{S_t-S_{t-}}{S_{t-}} = Y_i\rightarrow S_t = S_{t-}\,(Y_i+1). $$

and so I get

$$ dS_t = \alpha\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,dQ_t-S_{t-}\,\beta\,\lambda\,dt\neq (\alpha-\beta\,\lambda)\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,dQ_t. $$

Here there is a snapshot of the textbook.

## Answer by Quantuple (score 2, accepted)

https://quant.stackexchange.com/a/25288

Could it be that your problem is only due to the $t^-$ notation convention?

Think of it that way, it is only worth distinguishing $S_{t^-}$ from $S_t$ at a jump time. Elsewhere, knowing that Brownian motion paths are continuous, you'll always have $S_t = S_{t^-}$.

Thus you could also write the SDE:

$$\frac {dS_t}{S_{t^-}} = \alpha dt+\sigma dW_t+ d\left(Q_t-\beta\,\lambda\,t\right)$$

Or simply drop the $t^-$ notation altogether if it bothers you. Just remember that we are dealing with processes with càdlàg paths (right continuous $S_t $, left limit $S_{t^-} $)