Define a process by its stochastic differential equation :
In[]:=
proc=ItoProcess[x[t]==-x[t]t+Sqrt[1+x[t]^2]w[t],x[t],{x,1},t,wWienerProcess[]]
Out[]=
ItoProcess{-x[t]},
1+
2
x[t]
,x[t],{{x},{1}},{t,0}
Simulate the process :
In[]:=
RandomFunction[proc,{0.,5.,0.01}]
Out[]=
TemporalData
Time: 0. to 5.
Data points: 501
Paths: 1

In[]:=
ListLinePlot[%,Filling->Axis]
Out[]=