By Andreas Ro?ler
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Additional resources for Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations (Berichte Aus Der Mathematik)
Example text
VN∗ ; Z, U0 ) and Yt = f (Ut ) = πUt . Differentiating, we have dYt = Vα∗ f (Ut ) ◦ d Ztα = Vα (Yt ) ◦ d Ztα . This is the equation for X. , U is the horizontal lift of X. 4 dWt = Ut−1 Pβ (Xt ) ◦ dXtβ . In our case dXt = Vα (Xt ) ◦ d Ztα , or in components dXtβ = Vαβ (Xt ) ◦ d Ztα , where Vαβ are the component of the vector Vα in RN . Since Vα is tangent to M , it is easy to verify that N Vαβ Pβ . Vα = β=1 It follows that N dWt = Ut−1 Vαβ (Xt )Pβ (Xt ) ◦ d Ztα = Ut−1 Vα (Xt ) ◦ d Ztα , β=1 as desired.
Since x˙ t ∈ Txt M , we have u−1 ˙ t ∈ Rd . 7) wt = u−1 ˙ s ds. s x 0 Note that w depends on the choice of the initial frame u0 at x0 but in a simple way: if {vt } is another horizontal lift of {xt } and u0 = v0 g for a g ∈ GL(d, R), then the anti-development of {vt } is {gwt }. From ut w˙ t = x˙ t we have Hw˙ t (ut ) = ut w˙ t = x˙ t = u˙ t . 8) u˙ t = Hi (ut ) w˙ ti . We can also start from a curve {wt } in Rd and a frame u0 at a point x0 . The unique solution of the above equation is a horizontal curve {ut } in F(M ) and is called the development of {wt } in F(M ), and its projection t → xt = πut is called the development of {wt } in M .
8) interpreted properly—this usually means replacing the usual integral by the corresponding Stratonovich stochastic integral—we can develop a semimartingale W on Rd into a horizontal semimartingale U on F(F ), and then project it down to a semimartingale on M (“rolling without slipping”). Conversely, we can lift a semimartingale X on M to a horizontal semimartingale U on F(M ) and then to a semimartingale W on Rd . , πU0 = X0 ), the correspondence W ←→ X is one-to-one. Because euclidean semimartingales are easier to handle than manifold-valued semimartingales, we can use this geometrically defined correspondence to our advantage.
