Talk:Stochastic differential equation

Created a new page here from what was previously all in Langevin equation - which is actually more specific than just an SDE. More work is needed. --SgtThroat 15:04, 4 Jan 2005 (UTC)

I don't get it: "A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space."

I read this to say that a weak solution satisfies A&B, while a strong solution satisfies B&A. Please clarify.

Terminology?
The terminology subsection currently says that SDEs can be written in one of three forms, the third of which is "as an SDE". Could someone clarify that?

Also, under "use in physics", it says that the noise is multiplicative if $$g_i(x)$$ is not constant. Isn't it only multiplicative if $$g_i(x)\propto x$$? LachlanA (talk) 05:29, 15 January 2008 (UTC)

weak vs strong solution, a typo
I think the formulation in the text is correct and slightly different from what you quoted, namely ``... strong solution ... in the given probability measure". Where ``the" refers to the given probability measure of the SDE. This requires especially that the solution process $$X_t$$ is defined for all elementary events ω. While for a weak solution you can first restrict the probability measure (and especially its underlying sigma-algebra) to a convenient subspace (possibly rescaling the prob-measure).

Towards the end of the current article I guess there is a typo in the formulation of the Ito-SDE. Namely you should not set equal the process $$X_t$$ (for all $$t$$) to the random variable $$Z$$. It would make sense if you wrote $$X_0=Z$$. Before I change this I'd prefer someone had a look in one of the literature sources and check if that is correct.

[melli]64.178.100.37 (talk) 04:10, 16 March 2008 (UTC)

History of SDE
This is the first time I am making a posting. I may not be doing this the best way possible.

The article correctly cites Bachelier (1900) as one who wrote the stochatic DE before Enstein.

It turns out that prior to Bachelier, Francis Ysidro Edgeworth (1883) and Lord Rayleigh (1880, 1894)

Edgeworth, F.Y., The law of error, Phil. Mag., Fifth ser., 16, 300-309, 1883.

Rayleigh, Lord, On the resultant of a large number of vibrations of the same pitch and of arbitrary phase, Phil. Mag., 10, 73-78, 1880. Rayleigh, Lord, The Theory of Sound, MacMillan and Co., London, Vol. 1, Second Edition, 1894.

Historical details can be found in,

Narasimhan, T. N., Fourier's Heat Conduction Equation: History, Influence, And Connections, Reviews of Geophysics, 37(1), 151-172, 1999

Narasimhan —Preceding unsigned comment added by Tnnarasimhan (talk • contribs) 23:30, 3 May 2008 (UTC)

Use in probability and financial mathematics
Here:


 * This equation should be interpreted as an informal way of expressing the corresponding integral equation


 * $$ X_{t+s} - X_{t} = \int_t^{t+s} \mu(X_u,u) \mathrm{d} u + \int_t^{t+s} \sigma(X_u,u)\, \mathrm{d} B_u . $$


 * The equation above characterizes the behavior of the continuous time stochastic process Xt as the sum of an ordinary Lebesgue integral and an Itō integral.

I think the enhanced adjective is misleading: there's nothing ordinary in the integration of an expression that includes a stochastic process! For example, what is the ordinary integral of:


 * $$\int_{t_1}^{t_2} B_t \ \mathrm{d} t\,$$ ?

Albmont (talk) 18:30, 29 July 2009 (UTC)

You can regard $$\int_{t_1}^{t_2} B_t(\omega) \ \mathrm{d} t\,$$ pathwise as an ordinary Lebesgue integral (or, since Brownian paths are almost surely continuous, as a Riemann integral for almost every fixed $$\omega$$ 129.177.32.74 (talk) 09:54, 2 April 2014 (UTC) Georgy

Hi Georgy, 1) I concur with the comment above yours, "ordinary Lebesgue integral" reads very strangely in this context; 2) reg your explanation about the integral of B_t, can you not say the same of the solution to any SDE? that is to say, you pick up a path omega and the corresponding B_t(omega) and then there is nothing random left in the SDE and X_t(omega) is just a normal integral; or am I missing something? 129.199.97.192 (talk) 10:13, 27 January 2017 (UTC) Vincent


 * No. If $$f$$ and $$g$$ are arbitrary continuous functions, there is no natural definition for the integral $$\int f\, dg$$. In particular, this is not how the Itô integral is defined. (You might be interested in reading up on the differences between Itô and Stratonovich integrals...) Hairer (talk) 14:47, 27 January 2017 (UTC)

Regarding "The stochastic process Xt is called a diffusion process", why is this the case if it includes a diffusion component? — Preceding unsigned comment added by 162.212.205.234 (talk) 01:23, 7 December 2014 (UTC)

What is the meaning of the $$\eta_m$$ function? If this is related to the noise function $$\xi^\alpha$$ of the preceding section 'Use in physics', could someone maybe clarify this? Jeroenvanriel (talk) 13:44, 5 March 2021 (UTC)

Use in physics
The overall article is already very technical. Why not have a juicy example here instead of the general form of the equation? I pasted a suggestion below.

More remarks: - does the chain rule for nonlinear SDEs with additive noise? I doubt it. - There are two statements about misleading terminology (THE Langevin ..., MULTIPLICATIVE noise): lets put less emphasis

Application in physical sciences: The OUP is a prototype of a noisy relaxation process. Consider for example a Hookean spring with spring constant $$k$$ whose dynamics is highly overdamped with friction coefficient $$\gamma$$. In the presence of thermal fluctuations with temperature $$T$$, the length $$x(t)$$ of the spring will fluctuate stochastically around the spring rest length $$x_0$$; its stochastic dynamic is described by an OUP with $$\theta=k/\gamma$$, $$\mu=x_0$$, $$\sigma=\sqrt{2k_B T/\gamma}$$. (The equation for the effective diffusion constant $$D=\sigma^2$$ is the famous Einstein relation.) In physical sciences, the stochastic differential equation of an OUP is rewritten as a Langevin equation
 * $$ \gamma\dot{x} = - k( x - x_0 ) + \xi $$

where $$\xi(t)$$ is Gaussian white noise with $$<\xi(t_1)\xi(t_2)>= 2 k_B T \delta(t_1-t_2). $$ If temperature is a function of position $$T=T(x)$$, the noise term is said to be multiplicative and care has to be taken in manipulating the equation to avoid the Ito-Stratonovich dilemma.


 * "does the chain rule for nonlinear SDEs with additive noise?" I was thinking of the same thing. If you have a nonlinear change of variable, you get an additional term in Ito calculus that you don't get in other types of calculus, regardless of whether it is an additive or multiplicative noise. Am I right? Can someone confirm this, and if correct, remove the the incorrect statement please? Sprlzrd (talk) 22:16, 10 May 2016 (UTC)


 * I've fixed that paragraph. Actually, the part about being able to use the chain rule was completely misleading: if the equation is linear then one can write down the solution explicitly using the variation of constants formula, if it is not then there is typically no closed form expression anyway. Hairer (talk) 08:30, 11 May 2016 (UTC)

Parameter Estimation
Can someone add some discussion on methods for estimating parameters of an SDE? Jay (User talk:Shantham11) 06:15, 11 October 2015 (UTC)

SDEs and supersymmetry
The connection between SDEs and supersymmetry, seems to be a very marginal topic; not very central to the theory of SDEs. Perhaps this section should be removed from the article? — Preceding unsigned comment added by 80.203.112.253 (talk) 05:10, 23 January 2019 (UTC)


 * Either way, I would strongly be in favour of at least removing the silly "resolving the Itô-Stratonovich dilemma" comment. Hairer (talk) 00:08, 25 January 2019 (UTC)

Inline citations and further references

 * I added a number of inline citations and removed the banner, I hope it's ok but I defer to more experienced editors for this.
 * I saw that some of the above comments in this talk page mentioned Louis Bachelier but he was not present in the article, so I mentioned him as one of the early authors on modeling Brownian motion and referred to the related wiki article.
 * I added references to Ito SDEs on manifolds (Schwartz morphism, jet bundle) because they led to ways to optimally project on submanifolds according to clear criteria as opposed to projecting Stratonovich SDEs whose projection, while more straightforward, satisfies no sensible minimization criteria.
 * I added SDEs distributed as mixtures of lognormals as a generalization of geometric Brownian motion in mathematical finance.
 * I added SDEs seen as pure rough paths as I think this is an interesting aspect but I defer to the authorities like Martin Hairer here, I hope it's ok.