ODE for mean of SDE

Asked Dec 04 '23 at 04:16

Active Dec 04 '23 at 04:16

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Given a stochastic differential equation

$$ dX_t = f(t,X_t) dt + \sigma(t,X_t)dB_t$$

where $B$ represents standard Brownian motion, if I wanted an ODE for the mean of the SDE, I could write for any $T > 0$

$$ X_T = X_0 + \int_{0}^{T}f(t,X_t)dt + \int_{0}^{T}\sigma(t,X_t)dB_t $$

then by taking expectation on both sides and since the integral with respect to $B$ is a martingale and has mean zero, one has

$$ \mathbb{E}(X_T) = \mathbb{E}(X_0) + \mathbb{E}\left(\int_{0}^{T}f(t,X_t)dt\right). $$

My question is, when can one exchange the expectation and integral with respect to $t$ to obtain $$ \frac{d}{dt}\mathbb{E}(X_t) = \mathbb{E}(f(t,X_t)) $$

asked Dec 04 '23 at 04:16

sixtyTonneAngel

1,186

1

Does this answer your question? When can we interchange the derivative with an expectation? – Thomas Kojar Dec 04 '23 at 04:32
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If you want an ODE for the mean $\mathbb E[X_t]$ you want a lot more than just exchanging expectation and integral. What properties must $f$ have so that you can pull the expectation inside it? – Kurt G. Dec 04 '23 at 09:06
@KurtG. Was there ever an answer to this? – AB_IM Oct 17 '24 at 21:38
@ABIM this was a fairly simple socratic question to OP who disappeared. The answer is that OP's $f(t,X_t)$ must be linear, or at least affine linear, in $X_t$ . – Kurt G. Oct 18 '24 at 03:11
That's what i thought also... – AB_IM Oct 18 '24 at 18:33

ODE for mean of SDE

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