matlab - How to add white noise process term for a couple of ODEs, assuming the Gaussian distribution? -

matlab - How to add white noise process term for a couple of ODEs, assuming the Gaussian distribution? -

this question has confused me several days. while referred senior students, cannot give reply.

we have 10 odes, each noise term should added. noise defined follows. since find cannot upload picture, formula below maybe not clear. in order understand, can either read explanation or go address: plos one. find description of equations straight above support information in address

the white noise term epislon_i(t) assumed gaussian distribution. epislon_i(t) means equation i, , @ t timepoint, value of noise. the auto-correlation of noise given:

(eq.1)

where delta(t) dirac delta function , diffusion matrix d defined

(eq.2)

our problem focuses on how explain dirac delta function in diffusion matrix. since property of dirac delta function delta(0) = inf , delta(t) = 0 if t neq 0, don't know how calculate epislonif seek sqrt of 2d(x, t)delta(t-t'). assume delta(0) = 1 , delta(t) = 0 if t neq 0; but don't know whether or not right. please tell me how utilize delta function of diffusion equation in matlab?

this question associates stochastic process in matlab. review different stochastic process inspire our ideas. in matlab, wienner process defined a = sqrt(dt) * rand(1, n). n number of steps, dt length of steps. correspondingly, brownian motion can defined as: b = cumsum(a); of these associate stochastic process. however, doesn't related white noise process has constraints on matrix of auto-correlation, noted d.

then consider that, may utilize randn(1, 10) generate vector representing noise. however, since definition of noise must satisfy equation (2), cannot enable noise term in different equation have predefined partial correlation (d_ij). seek utilize mvnrnd generate multiple variable normal distribution @ each time step. unfortunately, function mvnrnd in matlab homecoming matrix. need homecoming vector of length 10.

we rather confused, please give me light? much!

note: see 2 hazy questions in here: 1) how deal stochastic term in de , 2) how deal delta function in de. both of these math related questions , http://www.math.stackexchange.com improve place this. if had question pertaining matlab, haven't been able pin down, , should perhaps add together code examples improve illustrate point. said, i'll reply 2 questions briefly, set on right track.

what have here not odes, stochastic differential equations (sde). i'm not sure how you're using matlab work this, routines ode45 or ode23 not of help. sdes, usual mathematical tools of separation of variables/method of characteristics etc don't work , you'll need utilize itô calculus , itô integrals work them. solutions, might have guessed, stochastic. larn more sdes , working them, can consider stochastic differential equations: introduction applications bernt Øksendal , numerical solutions, numerical solution of stochastic differential equations peter e. kloeden , eckhard platen.

coming delta function part, can deal taking fourier transform of ode. recall fourier transform of delta function 1. simplifies de , can take inverse transform in end homecoming original domain.

matlab noise ode stochastic-process