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In the mathematics of finance, a stochastic process can be given by a stochastic differential equation: $$dX_t = a(X_t,t)dt + b(X_t,t)dB_t$$ where $dB_t$ is a Wiener process. What is the basic reason that the term $a$ is considered as the mean of process $dX_t$ and $b^2$ the variance?

Thanks.

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  • $\begingroup$ The basic reason is that $\int_0^t \mathbb{E}(a(X_s,s)) \, ds$ is the mean and $\int_0^t \mathbb{E}(b^2(s,X_s)) \, ds$ the variance of $X_t$. $\endgroup$ – saz Jul 17 '17 at 15:52
  • $\begingroup$ @saz Okay thanks, but what is $\mathbb{E}$? Could you elaborate a bit on your response. $\endgroup$ – user116403 Jul 17 '17 at 20:36
  • $\begingroup$ $\mathbb{E}$ is the expected value; i.e. $$\mathbb{E}(Y) := \int Y \, d\mathbb{P}.$$ $\endgroup$ – saz Jul 17 '17 at 20:43
  • $\begingroup$ @saz Oh okay so is the expected value in the integrand with respect to $X_t$ with fixed $t$ in my example, and you then take the integral with respect to $t$? $\endgroup$ – user116403 Jul 17 '17 at 21:08
  • $\begingroup$ @saz Your statement is just plain wrong, unless $a$ and $b$ are both deterministic. $\endgroup$ – encore Aug 5 '17 at 12:28
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For a diffusion process ${X_t,t\geq0}$ we have

$$ \lim_{h\downarrow 0} \frac{1}{h}\mathbb{E}[X(t+h)-X(t)|X(t)=x] = \mu(x,t) $$

so that $\mu(x,t)$ is an infinitesimal mean of the diffusion process (typically referred to as the drift parameter).

Similarly,

$$ \lim_{h\downarrow 0} \frac{1}{h} \mathbb{E}[(X(t+h)-X(t))^2|X(t)=x] = \sigma^2(x,t) $$

so that $\sigma^2(x,t)$ is the infinitesimal variance of the diffusion process (often referred to as the diffusion parameter).

For the SDE $dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dB_t$ one can verify with some effort that the above conditions are true (but for the case where $\mu(X_t,t)=\mu$ (a constant) and $\sigma(X_t,t)=\sigma$ this is easy to see).

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  • $\begingroup$ Just to confirm, by analogy are you stating that $\mu(X_t, t)dt$ and $\sigma^2(X_{t},t)$ are the infinitesimal mean and variance of $X_t$ respectively? How would you define infinitesimal mean and infinitesimal variance? $\endgroup$ – user116403 Jul 23 '17 at 16:40
  • $\begingroup$ The two equations in my posts are the definitions of infinitesimal mean and variance for a diffusion process (see for example Karlin and Taylor "The second course in stochastic processes"). You can verify that if $X_t$ satisfies an SDE then these equations hold. Just try the SDE for a Brownian motion with a drift and convince yourself. $\endgroup$ – Mdoc Jul 23 '17 at 21:29
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    $\begingroup$ It should be pointed out that the limit statements above are true as long as functions $\mu$ and $\sigma$ satisfy certain technical assumptions to guarantee sufficient smoothness and limited growth. $\endgroup$ – encore Aug 5 '17 at 12:32

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