  Using a statistically rigorous analysis method, we place limits on the
existence of an isotropic stochastic gravitational wave background using pulsar
timing observations. We consider backgrounds whose characteristic strain
spectra may be described as a power-law dependence with frequency. Such
backgrounds include an astrophysical background produced by coalescing
supermassive black-hole binary systems and cosmological backgrounds due to
relic gravitational waves and cosmic strings. Using the best available data, we
obtain an upper limit on the energy density per unit logarithmic frequency
interval of \Omega^{\rm SMBH}_g(1/8yr) h^2 <= 1.9 x 10^{-8} for an
astrophysical background which is five times more stringent than the earlier
Kaspi et al. (1994) limit of 1.1 x 10^{-7}. We also provide limits on a
background due to relic gravitational waves and cosmic strings of \Omega^{\rm
relic}_g(1/8yr) h^2 <= 2.0 x 10^{-8} and \Omega^{\rm cs}_g(1/8yr) h^2 <= 1.9 x
10^{-8} respectively. All of the quoted upper limits correspond to a 0.1% false
alarm rate together with a 95% detection rate. We discuss the physical
implications of these results and highlight the future possibilities of the
Parkes Pulsar Timing Array project. We find that our current results can 1)
constrain the merger rate of supermassive binary black hole systems at high red
shift, 2) rule out some relationships between the black hole mass and the
galactic halo mass, 3) constrain the rate of expansion in the inflationary era
and 4) provide an upper bound on the dimensionless tension of a cosmic string
background.
