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I was just curious about something...

A general 1-D stochastic differential equation for some process $X$ can be written as:

$dX(t) = a(X,t) dt + b(X,t) dW(t)$,

where $W(t)$ is a Wiener process.

Is there a way to generalize this stochastic differential equation to account for a Wiener process that depends on space AND time, something like: $dW(X,t)$?

Thanks.

Dr. Ikjyot Singh Kohli
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    Not exactly like this, but yes, it is possible. Keyword: stochastic partial differential equation. – zhoraster Dec 26 '16 at 15:35
  • Hi @zhoraster . Yes, I knew about stochastic PDEs that involve a spacetime white-noise term. I know for SDEs one defines a white noise term as $\eta(t) \equiv dW/dt$, where $W$ is a Wiener process. Let's say the space-time white noise is denoted $\xi(x,t)$. Is there a similar definition for this? – Dr. Ikjyot Singh Kohli Dec 26 '16 at 15:55
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    $\xi(x,t) = \partial^2 W/\partial x\partial t$, where $W$ is a Wiener sheet. – zhoraster Dec 27 '16 at 06:09
  • @zhoraster Hi. Okay, that is very helpful thanks. – Dr. Ikjyot Singh Kohli Dec 27 '16 at 15:12

1 Answers1

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Well, to be clear usually the white noise doesn't depend on the solution $X(t)$ because that would make the problem highly nonlinear if not just ill-posed. Even in 1d it is unclear how to study $$dX_{t}=adt+bdW(X_{t},t).$$

Instead, the generalization is that of including more coordinates but for the solution and possibly the noise too i.e. stochastic-pdes

$$du(t,x)=\mathcal{L}u(t,x)dt+b(u(t,x))\xi(t,x),$$

where $\xi(t,x)=\dot{W}(t,x)$ is called the space-time white noise (see below).

\begin{equation} \mathbb{E}\left[\xi(z,t) \right] = 0;\quad \mathbb{E}\left[\xi(z,t)\xi(z',t') \right] = \delta(z-z')\delta(t-t'). \end{equation}

The Gaussian process $\{\dot{W}(A)\}_{A \in \mathscr B(\mathbb R^n)}$ with $E(\dot{W}(A))=0$ and $E(\dot{W}(A)\dot{W}(B))=\lambda^n(A\cap B)$ where $\lambda ^n$ is the n-dimensional Lesbegue measure is called White Noise.

We can consider for $A \in \mathscr B(\mathbb R^n)$ that $\dot{W}(A) = \int 1_A dW$ and for $h\in L^2(\mathbb R^n)$ we define $\dot{W}(h) = \int h(t)dW(t)$ which is the standard Wiener Integral.

For spacetime we consider the white noise $\dot{W}(A)$ over $A\in [0,T]\times \mathbb{R}$.

References:

Thomas Kojar
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