Upper bound of a volume exponent for directed polymers in a random environment

We consider the model of directed polymers in a random environment introduced by Petermann : the random walk is ${\mathbb{R}}^d$-valued and has independent gaussian $N(0,I_d)$-increments, and the random media is a stationary centred Gaussian process $(g(k,x),k\ge 1,x\in {\mathbb{R}}^d)$ with covariance matrix $cov(g(i,x),g(j,y))={\delta }_{ij}\mathrm{\Gamma }(x-y),$ where $\mathrm{\Gamma }$ is a bounded integrable function on ${\mathbb{R}}^d\mathrm{.}$ For this model, we establish an upper bound of the volume exponent in all dimensions $d$.