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I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + \mu B_t)$$ I wanted to calculate the expectation of $X_t$. However I think I'm not quite getting it. I thought that I'd just calculate $$E(x_0\exp((\mu - \frac{\sigma^2}{2})t + \mu B_t) = x_0\exp((\mu - \frac{\sigma^2}{2})t)E(\exp(\mu B_t))$$ but the book I'm using gives as answer $E(X_t) = x_0\exp(\mu t)$. I found this quite surprising as I don't quite see how $\sigma$ could just disappear. After reading Wikipedia I see that the result could be either $E(X_t) = x_0\exp((\mu + \frac{\sigma^2}{2})t)$ or $E(X_t) = x_0\exp(\mu t)$, depending on whether you use the Itô interpretation or the Stratanovich interpretation.

Since the book I use only considers the Itô formulation of stochastic integration I am interested in the latter result. But how do I obtain this? Do I just fail in calculating $E(\exp(\mu B_t))$? Thanks in advance.

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Hint: The solution you've given for $X_t$ is incorrect and, based on the subsequent remarks, not simply a typo. – cardinal May 23 '11 at 12:51

3 Answers 3

up vote 7 down vote accepted

The answer is that $E(X_t)=x_0e^{\mu t}$. The easiest way to see it is to start from the SDE and to note that $\mathrm{d}E(X_t)=\mu E(X_t)\mathrm{d}t$ and $E(X_0)=x_0$. Hence $a(t)=E(X_t)$ solves $a'(t)=\mu a(t)$ and $a(0)=x_0$, that is, $a(t)=x_0e^{\mu t}$ as claimed above.

Your solution goes astray when you solve the SDE, the factor of $B_t$ is wrong and, in fact, $$ X_t=x_0e^{(\mu-\sigma^2/2)t+\sigma B_t}. $$ Hence $$ E(X_t)=x_0e^{(\mu-\sigma^2/2)t}E(e^{\sigma B_t}). $$ Since $E(e^{uZ})=e^{u^2/2}$ for every real number $u$ and every standard normal random variable $Z$, the identity $E(e^{\sigma B_t})=e^{\sigma^2 t/2}$ follows from the fact that $\sigma B_t$ is distributed like $\sigma\sqrt{t}Z$. Simplifying, one gets the same expression of $E(X_t)$ than by the direct route.

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I haz a stupid once again. Thanks. – Stijn May 23 '11 at 13:41
@Stijn, note that, when accepting an answer you can also simultaneously upvote it as well. :) – cardinal May 23 '11 at 14:47
Indeed, thanks for reminding me. – Stijn May 23 '11 at 15:27
Can you please say, how this $\mathbb{E}d(X_t) = d\mathbb{E}(X_t)$ can be justified(said) in mathematically correct way? – Ievgenii Nov 28 at 21:51
@Ievgenii Recall that $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ is actually a shorthand for $$X_t = X_0+\mu\int_0^t X_sds + \sigma \int_0^tX_sdB_s$$ which implies $$E(X_t)=E(X_0)+\mu E\left(\int_0^t X_sds\right)+\sigma E\left(\int_0^t X_sdB_s\right)=E(X_0)+\mu \int_0^t E(X_s)ds+0,$$ which implies that $$\frac{d E(X_t)}{dt}=\mu E(X_t).$$ More generally, if $$dX_t=a(t,X_t)dB_t+b(t,X_t)dt,$$ then, under suitable integrability conditions, $$\frac{d E(X_t)}{dt}=E(b(t,X_t)).$$ – Did Nov 28 at 21:54

An easy way to calculate an expected value of SDE's solutions is to find an equation on $m(t) = \mathsf{E}[X_t]$. More precisely, from $$ dX_t = \mu X_tdt+\sigma X_t dB_t $$ taking expectations of both sides (while using that $\mathsf{E}[dB_t] = 0$) we obtain $$ dm(t) = \mu m(t)dt, $$ which gives a desired answer. Indeed, from the latter equation follows that $m(t) = m_0 \mathrm{e}^{\mu t}$. @Did has already described such method, but it may be useful for you to apply such method for other SDEs.

You should be aware of taking the expectation in SDE which is quite informal. On the other hand, you can equivalently write $$ X_t = X_0+ \int\limits_0^t\mu X_sds+\int\limits_0^t\sigma X_s dB_s. $$

Here you can take an expectation since on both sides there are just random variables (at each fixed moment of time). The latter integral has zero expectation since it is Ito integral.

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can you explain more about your second option for calculating expectation straight from SDE? What I conclude from your answer: I can do both ways: 1) taking expectation of Stochastic process; 2) taking expectation of SDE. Are these 2 options equivalent? Are you implicitly using Fubini theorem in second case? (when you are taking $\mathbb{E} \int \sigma X_s dB_s$). And the last question, how do you compute: $\mathbb{E} \int \mu X_s ds$, I am confused. Thank you in advance! – Ievgenii Nov 28 at 21:49

Identify $E(\exp (\mu B_t ))$ with the moment-generating function of the normal distribution with mean zero and variance $t$.

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I'm not downvoting, but it's worth noting that, while you've provided a good hint, applying it to the OP's development still does not yield the stated result. – cardinal May 23 '11 at 12:53
@cardinal: I noticed that; the answer addresses the OP's question "Do I just fail in calculating $E(\exp (\mu B_t ))$?" -- providing a quick way to calculate it. – Shai Covo May 23 '11 at 13:03

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