Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$.

I think that these are uncorrelated. But Why?

So thanks

Answer

if you talk about correlation then:

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  compute expectation: $$\mathbb{E}(W_t)=0\text{ and }\mathbb{E}(\int_0^tW_d ds)=0$$

  
  


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  variance: $$\text{Var}(W_t)=t\text{ and }\text{Var}(\int_0^tW_s ds)=\frac{t^3}{3}$$

  
  


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  covariance: $$\mathbb{E}(W_t\int_{0}^tW_sds)=\int_{0}^t\mathbb{E}(W_tW_s)ds=\int_0^tsds=\frac{t^2}{2}$$

  
  

then you get: $$\text{Corr}(W_t,\int_0^tW_s ds)= \frac{\sqrt{3}}{2}$$

hint

$$\mathbb{E}(W_uW_s)=\min(u,s)$$

$$\text{Var}(\int_0^tW_sds)=\mathbb{E}(\int_0^t\int_0^tW_sW_u duds)$$