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I wonder if this type of equation fits in the formalism of SDE

$$dX_{t}=(\mu_0+\eta_t)dt+\sigma dW_{t}$$

where $\eta_t$ is uncorrelated Gaussian noise.

First I wrote this, but I don't know if it makes sense.

$$dX_{t}=(\mu_0+dW_{1t})dt+\sigma dW_{2t}$$

Wiener Process 1 & 2 are uncorrelated.

Then I though of making a system of two equations,

$$dX_{t}=(\mu_0+\eta_t)dt+\sigma dW_{2t}\\ d\eta_{t}=dW_{1t}$$

But $\eta$ is not uncorrelated, is exactly the Wiener Process.

How should I proceed?

JnxF
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Pablo Riera
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  • Do you know $dW_1(t) dt=0,?$ . We have $$dX_{t}=(\mu_0+dW_{1t})dt+\sigma dW_{2t} \implies dX_{t}=\mu_0dt+\sigma dW_{2t}$$ – Behrouz Maleki Aug 26 '16 at 18:01
  • Well that makes sense. I was hoping that an stochastic drift will have some effect. Suppose we have $\sigma=0$ then the system will be deterministic, isn't it? Then how could it be deterministic if has stochastic drift. – Pablo Riera Aug 26 '16 at 18:29
  • The drift can be stochastic process . For example $dX_t=(\mu_0+X_t)dt+\sigma dW(t)$ or $$dX_{t}=(\mu_0+\eta_t)dt+\sigma dW_{1,t}\ d\eta_{t}=dW_{2,t}$$ – Behrouz Maleki Aug 26 '16 at 19:18
  • True, but in that case the drift is not uncorrelated gaussian noise. Do you know where can I find a demostration for $dW_tdt=0$? – Pablo Riera Aug 26 '16 at 19:21
  • Indeed$$dX_{t}=(\mu_0+W_{1,t})dt+\sigma dW_{2,t}\ dW_{2,t}dW_{1,t}=0$$ – Behrouz Maleki Aug 26 '16 at 19:24
  • Let ${t_i}{i=1}^{n}$ be a partition of $[0,t]$. We want to consider $[W(t),t]=0$ .Set $$A_n=\sum{i=0}^{n-1}(,W_1(t_{i+1})-W_1(t_{i}),)(t_{i+1}-t_{i})$$ $\operatorname{Var}(A_n)=\mathbb{E}[A_n^2]\to 0$ as $\delta_n=\max{t_{i+1}-t_i}_{i=0}^{n-1}\to 0$. This implies that $A_n\to 0$, in probability. – Behrouz Maleki Aug 26 '16 at 19:39

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