Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define $$ Y^r_t := \int_0^t f(r,s) dW_s $$ For each fixed $r$, $(Y^r_t)_{t \geq 0}$ is a martingale and we can apply the following martingale inequality $$ \mathbb{P} \left( \sup_{t \in (0,T]} |Y^r_t| \geq K_1, \langle Y^r_. \rangle _T \leq K_2\ \right) \leq 2 \exp \left( - \frac{K_1^2}{2K_2} \right) $$ The process I am really interested in is $ \int_0^t f(t,s) dW_s$, i.e. $Y^t_t$. I would like to have a similar sort of inequality (and in fact it seems to be used in some papers that such an inequality holds) but $Y^t_t$ is not a martingale.
Can anyone explain why such an inequality would hold (e.g. if it is automatic from the above) or provide a reference or counterexample?
For anyone interested, the process I described above is known as a (Gaussian) Volterra process and there is a reasonable amount of literature on the subject. In particular, it is a Gaussian process and so some deviation inequalities for Gaussian processes apply such as those found here:
http://www.math.udel.edu/~wli/papers/01-survey-LiShao.pdf