Hi
I am trying to find steady state solution of a stochastic differential equation
dy/dt=Aydt+B1ydV1+B2ydV2
where A, B1 and B2 are operators , dV1 and dV2 are color noises.
is there any way or any literature , where steady state solution (dy/dt=0) of a stochastic differential equation has been found out. your help will be appreciated.
Dear Arif,
In your case you need to have additional equations for the color noises. About literature.There is a lot of literature on steady state solution of stochastic differential equations in the public domain. For example, follow the link below.
https://users.aalto.fi/~ssarkka/course_s2012/pdf/sde_course_booklet_2012.pdf
Here see point 4.5 Steady State Solutions of Linear SDEs (p.47)
A note: the two above links result into the same chapter of the same pdf file :)
Thanks. But the reference that you have mentioned only talk about mean and covariance, can you guide me how to use mean and covariance to find the steady state solution. i can generate these color noises in my equation with the defined correlation matrix.
@ Arif Ullah
Probably you need a methot of generating a stationary process. This is a special art due to many particular subproblems like the choice of instants of the process. Next, you need to discretise the operators (if they are not multiplication). In any case you need to have the initial distribution which is supposed to be constant, i.e. the same at each next instant chosen for calculations (via genertor of the increments). In the simplest examples, as in the Ornstein-Uhlenbeck stationary processes, where the solution is given by a Brownian motion W(u), u >=0, as follows (just an example)
X(t) = e-t/2 W( et), t \in R; solves dX(t) = - 1/2*X(t) dt + dV(t)
(V is the driving Brownian motion, not the same proces as W(u) ), knowing the stationary distribution = N(0,1) you can generate the future values by simple increments procedure. E.g. for the above O-U process take X(t0=0) with N(0,1) (mean 0, variance 1) and proceed as follows, for n=0,1,2, .. .
X(tn+1) := X(tn) [ 1 - (tn+1- tn)/2] + Vn where Vn is N(0,tn+1- tn) independent of the preceding V-s.
Possibly this hepls. This should be applicable to the case of multidimensional processes with A, B ... multiplication by numerical matrices and if a generation of the driving processes say V1 and V2 is possible. Then Vn are increments of the given processes within the time period [tn,tn+1] However, if the operators are not multiplications (in the example above these were multiplications by numbers 1/2 and 1, respectively), then the problem is much more complex. In particular due to nonexistence of derivatives of some processes. Regards, Joachim
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