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This might be a very simple question, but I couldn't find an answer yet.

Consider a multidimensional stochastic SDE of the form

$S_t = x + b(S_t) dt + \sigma(S_t) dW_t $

where $b: \mathbb{R}^n \to \mathbb{R}^n $, $\sigma: \mathbb{R}^{n \times n} \to \mathbb{R}^{n \times n}$ are measurable functions, $x \in \mathbb{R}^n$, $W_t $ is a vector of independent Wiener processes and it is assumed that a strong solution for $S_t$ exists. Is it then possible to calculate the covariance matrix of $S_{\tilde{t}}$ for some time $\tilde{t}$ by

$Cov(S_{\tilde{t}}) = \int_{0}^{\tilde{t}} \frac{d Cov(S_t)}{dt}dt = \mathbb{E} [\int_{0}^{\tilde{t}} \sigma(S_t) \sigma(S_t)^T dt] \quad .$

Heuristically the change of the covariance matrix should depend on

$\sigma(S_t) dW_t \sim N(0, \sigma(S_t) \sigma(S_t)^T dt) \quad .$

So my first question would be wether the formula for $Cov(S_{\tilde{t}})$ is correct and if that's the case, wether someone knows a more formal argument for that.

B23
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1 Answers1

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First, for the covariance we can indeed use Itô isometry Quadratic Variations and the Itô Isometry to get

$$E[S_{t}S_{r}]=E[\int^t b(S_s) ds\int^r b(S_s) ds]+\int^{\min(t,r)} E\sigma(S_s)^{2} ds$$

and so (using that Itô integrals have zero mean when the integrable is adapted The expected value of the Ito integral of functions in $\mathcal{V}$ is zero, $\mathbb{E}[\int_S^T f dB_t] = 0$ for $f\in\mathcal{V}$ and that the cross-variation of a semimartingale and an increasing process is zero Quadratic Variation of Increasing Process?)

$$Cov[S_{t},S_{r}]=\int^{\min(t,r)} E\sigma(S_s)^{2} ds.$$

If we take $t=r$, we see that $g(t)=Cov[S_{t},S_{t}]=\int^{t}E\sigma(S_s)^{2} ds$ is differentiable and so we also get

$$g(t)=\int_{0}^{t}g_{s}(s)ds.$$

Thomas Kojar
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