An example to be a local martingale but not a martingale

Question

What is a good example for $\varphi_{s}$, to $$\int_{0}^{t}\varphi_{s}dW_{s}$$ be a local martingale, but not a martingale?

A simplier question: what should I choose for $\varphi_{s}$, if I don't want $\int_{0}^{t}\varphi_{s}dW_{s}$ to be a martingale? Here $W$ denotes a Wiener process, and $\varphi$ a continuous process.

Answer 1

Take some random variable $Y$ which is not integrable (e.g. Cauchy distributed) and which is independent of $(W_t)_{t \geq 0}$. Define $\varphi_s(\omega):=Y(\omega)$, then

$$\int_0^t \varphi_s \, dW_s = Y W_t$$

is a local martingale. However, it is not a true martingale since

$$\mathbb{E}(|Y \cdot W_t|) = \mathbb{E}(|Y|) \mathbb{E}(|W_t|)=\infty.$$