I want to solve the following SDE:

$$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$

Integrating, I get:

$$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$ $$ X_{t} = x_{0} + \mu \int_{0}^{t} X_{s} ds + \sigma [W_{t} - W_{0}] $$ $$ X_{t} = x_{0} + \mu \int_{0}^{t} X_{s} ds + \sigma W_{t} $$

This is where I get stuck -- How do I proceed?

I know that the solution to:

$$ dX_{t} = \mu X_{t} dt + \sigma X{t} dW_{t} \quad X_{0} = x_{0}$$

Is given by:

$$ X_{t} = X_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t} + \sigma W_{t}$$

This is given by the fundamental matrix solution. (Though honestly I am not sure how this is derived, so I can't specialize it to my case).

Sorry for such a basic question. I am trying to learn this from resources on the web. So if anyone could recommend a good book dealing with how to actually solve SDEs, I would really appreciate it.

  • $\begingroup$ Consider $d\big(e^{-\mu t} X_t \big)$. $\endgroup$ – Gordon Oct 11 '19 at 12:57
  • $\begingroup$ Okay so then: $ d(e^{\mu t} X_{t})$ = $X_{t} d(e^{\mu t}) + e^{\mu t} dX_{t} = \mu X_{t} e^{\mu t} dt + e^{\mu t} dX_{t}$. Solving this for $X_{t}$ I get: $X_{t} = \sigma \frac{dW}{dt}$. That doesn't seem right. Or is it? I am not sure. $\endgroup$ – the_src_dude Oct 11 '19 at 13:07
  • $\begingroup$ there is a minus sign $\endgroup$ – Gordon Oct 11 '19 at 13:08
  • $\begingroup$ @Gordon. In this case I get: $d(e^{- \mu t} X_{t})= 0$. What am I doing wrong? Could you please explain how to solve this for an idiot like me? And why do you want to consider that function anyway? I am an engineer trying to learn SDEs from ground up because I realized I could use them to specify prediction models in Kalman filters. However I have no idea what I am doing -- just piecing it together piece by piece. $\endgroup$ – the_src_dude Oct 11 '19 at 13:11
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    $\begingroup$ To eliminate the term $\mu X_t dt$, an integrating factor of the form $e^{-\mu t}$ is the only choice. $\endgroup$ – Gordon Oct 13 '19 at 22:48

It appears that you need to read some books such as Stochastic Differential Equations. For such type of equations, you need to use something called integrating factor such as the function $e^{-\mu t}$ here. Note that \begin{align*} d\big(e^{-\mu t} X_t \big) &= X_t d\big(e^{-\mu t}\big) + e^{-\mu t} dX_t\\ &=-\mu e^{-\mu t} X_t dt + e^{-\mu t} (\mu X_t dt + \sigma dW_t)\\ &=\sigma e^{-\mu t} dW_t. \end{align*} Then, for $t > s \ge 0$, \begin{align*} e^{-\mu t} X_t = e^{-\mu s} X_s+ \int_s^t \sigma e^{-\mu v} dW_v. \end{align*} Consequently, \begin{align*} X_t &= X_s e^{\mu (t-s)} + \sigma\int_s^t e^{\mu (t-v)} dW_v\\ &\sim N\Big(X_s e^{\mu (t-s)}, \, \sigma^2\int_s^t e^{2\mu (t-v)} dv \Big). \end{align*}

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  • $\begingroup$ Thank you for the reference. So, is there anyway I can simplify that last integral? $\endgroup$ – the_src_dude Oct 11 '19 at 13:54
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    $\begingroup$ No, unless $\mu=0$. But from here, you can do many thing such as computing the mean and variance etc. $\endgroup$ – Gordon Oct 11 '19 at 13:56
  • $\begingroup$ Gordon is obviously a whiz at this stuff ( thanks for great answer ) and I don't know of a better one but, to me, Oskendal is not a good recommedation for someone to get a handle on basic SDE. I've tried it. Maybe check out Evans's book. $\endgroup$ – mark leeds Oct 11 '19 at 13:56
  • $\begingroup$ @Gordon Okay, basically what you are saying is, if I want to simulate this process, I am going to need numerical methods. $\endgroup$ – the_src_dude Oct 11 '19 at 13:58
  • $\begingroup$ @markleeds Thank you for the other reference. They both seem to have their strengths and weaknesses. Gordon's suggestion seems to talk about methods of solution, while Evan's seems to just give you the answer. Both are helpful. $\endgroup$ – the_src_dude Oct 11 '19 at 14:04

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