# What is the difference between stochastic discount factor and stochastic discount factor process?

What is the difference between stochastic discount factor and stochastic discount factor process and how are they both related?

-Pricing kernel: It is a stochastic process $\{M_t\}_{t=1}^\infty$ such that the pricing process is a martingale, i.e., $M_t S_t =E_t[M_{t+1}S_{t+1}] \quad \forall t$
-Stochastic discount factor: you can divide both sides of the previous equation by $M_t$ to obtain that $P_t=E_t\left[\frac{M_{t+1}}{M_t}S_{t+1}\right]=E[SDF_{t+1}\times Payoff_{t+1}]$ . The SDF tells you about the ratio between marginal utility of wealth at time t+1 and at time t, i.e., $SDF_{t+1}=\left(\frac{\partial U( W)}{\partial W_{t+1}}\right)/\left(\frac{\partial U( W)}{\partial W_{t}}\right)$
-Stochastic discount factor process: it is the stochastic process of $\left\{SDF_t\right\}_{t=1}^\infty$