Definition of Itô process

Question

Let $\lambda_t$ and $r_t$ be predictable processes and suppose that $\int_{0}^t | \lambda_s |^2 \,ds < +\infty$, for all $t>0$. We define \begin{equation} Y_t = Y_0 \,\text{exp} \bigg\{ -\int_{0}^{t} r_s + \frac{|\lambda_s| ^2 }{2} \,ds - \int_{0}^{t} \lambda_s \,dW_s \bigg\} \end{equation} By Itô's lemma, we know that $dY_t = Y_t ( -r_t dt - \lambda_t dW_t)$.

My textbook claims that $\{Y_t\}$ is an Itô process. However, by definition, we have to check that $Y_t$ is predictable. How can we show that?

Answer 1

Hints:

1. Continuous (adapted) processes are predictable.
2. (Stochastic) integrals are continuous, i.e. both mappings $$t \mapsto \int_0^t b(s) \, ds \qquad \quad t \mapsto \int_0^t \sigma(s) \, dW_s$$ are continuous whenever the integrals make sense.