Simulating a random process with correlation structure $e^{-|t-s|}$

Question

Consider a random process $X_t$ such that $$dX_t\sim N(0,dt)$$ and $$\operatorname{corr}(dX_t,dX_s)=e^{-k|t-s|}.$$ This question is probably trivial but how can I simulate the path path of $X_t$? Any help is much appreciated. Thank you!

Answer 1

Let us construct a time grid $0=t_0<t_1 < t_2 <\cdots<t_n =T$ with $t_k=k\,\Delta t$ for $k=1,\ldots,n$ and $\Delta t = \frac{T}{n}$.

Case 1: $corr(dX_t,dX_s) = e^{-k|t-s|}$ . The $n$ gaussian variables $\{(X_{t_k}-X_{t_{k-1}}) \}_{k=1,\ldots,n}$ follow the multivariate normal distribution $\mathcal{N}(\mathbf{0}_n,\mathbf{\Sigma})$ with the covariance matrix $\mathbf{\Sigma}$ defined by $$\Sigma_{ij} = e^{-k|i-j|\Delta t} \, \Delta t$$

First, you generate these random vectors $\mathbf{W}$ following $\mathcal{N}(\mathbf{0}_n,\mathbb{I}_{n\times n})$ and calculate the square root matrix $\mathbf{\Sigma}^{\frac{1}{2}}$

Then the vector $\mathbf{Y}$ defined by $Y_k = X_{t_k}-X_{t_{k-1}}$ for $k = 1,...,n$ can be calculated as $$\mathbf{Y}= \mathbf{\Sigma}^{\frac{1}{2}}\mathbf{W}$$

After obtaining the vector $\mathbf{Y}$, the path $X_t$ can be computed as $$X_{t_k} = X_{t_{k-1}}+Y_{k}$$

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Case 2: $corr(X_t,X_s) = e^{-k|t-s|}\implies Cov(X_t,X_s)=e^{-k|t-s|}\sqrt{ts} $$\implies Cov(X_{t_i},X_{t_j})=e^{-k|i-j|\Delta t}\sqrt{ij}\Delta t$

This case is easier, The $n$ gaussian variables $\{X_{t_k}\}_{k=1,\ldots,n}$ follow the multivariate normal distribution $\mathcal{N}(\mathbf{0}_n,\mathbf{\Sigma'})$ with the covariance matrix $\mathbf{\Sigma'}$ defined by $$\Sigma'_{ij} = e^{-k|i-j|\Delta t}\sqrt{ij}\Delta t$$

First, you generate these random vectors $\mathbf{W}$ following $\mathcal{N}(\mathbf{0}_n,\mathbb{I}_{n\times n})$ and calculate the square root matrix $\mathbf{\Sigma'}^{\frac{1}{2}}$

Then the vector $\mathbf{X}$ can be calculated as $$\mathbf{X}= \mathbf{\Sigma'}^{\frac{1}{2}}\mathbf{W}$$