Deterministic interpretation of stochastic differential equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:01:08Z http://mathoverflow.net/feeds/question/20091 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20091/deterministic-interpretation-of-stochastic-differential-equation Deterministic interpretation of stochastic differential equation vonjd 2010-04-01T18:03:41Z 2010-04-01T18:03:41Z <p>In <em>Paul Wilmott on Quantitative Finance Sec. Ed.</em> in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$\delta S=\mu\ S\ \delta t\ +\sigma \ S\ \phi \ (\delta t)^{1/2}$$ where $\phi$ is drawn from a standardized normal distribution.</p> <p>This reasoning which seems to follow naturally from the definition of a <a href="http://en.wikipedia.org/wiki/Wiener_process" rel="nofollow">Wiener process</a> triggered some thoughts and questions I cannot solve. Think of the following general diffusion process: $$dS=a(t,S(t))\ dt\ +b(t,S(t))\ dX$$ Now transform the second term in a similar fashion as above and drop the stochastic component:$$dS=a(t,S(t))\ dt\ +b(t,S(t))\ (dt)^{1/2}$$</p> <p><strong>NB:</strong> The last term is <em>no</em> <a href="http://en.wikipedia.org/wiki/Riemann-Stieltjes" rel="nofollow">Riemann-Stieltjes integral</a> (that would e.g. be $d(t)^{1/2}$)</p> <p><strong>My questions</strong><br>(1) How would you interpret the last formula and how can the (now deterministic) differential equation be solved analytically? Will you get an additional term with a second derivative like in <a href="http://en.wikipedia.org/wiki/Ito%27s_lemma" rel="nofollow">Ito's lemma</a>?<br>(2) Is the last term a <a href="http://mathworld.wolfram.com/FractionalIntegral.html" rel="nofollow">fractional integral</a> of order 1/2 (which is a <a href="http://mathworld.wolfram.com/Semi-Integral.html" rel="nofollow">Semi-Integral</a>)? Or is this a completely different concept?<br>(3) Will there be a different result in the construction of the limit like with Ito integrals (from left endpoint) and <a href="http://en.wikipedia.org/wiki/Stratonovich_integral" rel="nofollow">Stratonovich integrals</a> (average of left and right endpoint)?</p>