THE GEOMETRY OF ITERATED STRATONOVICH INTEGRALS 1 ...

THE GEOMETRY OF ITERATED STRATONOVICH INTEGRALS CHRISTIAN BAYER

Abstract. We give a summary on the geometry of iterated Stratonovich integrals. For this exposition, we always have the connection to stochastic Taylor expansion in mind. In particular, we believe that “cubature on Wiener space” is best understood in the setting presented in this text. Besides cubature on Wiener space, we also give a second application regarding the heat kernel on nilpotent free Lie groups.

1. Introduction Although iterated Stratonovich integrals of Brownian motion are interesting objects per se, our motivation for studying them stems from the stochastic Taylor expansion, where they play a similar rˆole as polynomials do in the deterministic Taylor expansion. Let Bt = (Bt1 , . . . , Btd ), t ≥ 0, denote a d-dimensional Brownian motion on the filtered probability space (Ω, F, (Ft ), P ) satisfying the usual conditions. For notational issues, we introduce a 0th component Bt0 = t. Let V0 , . . . , Vd : Rn → Rn be C ∞ -bounded vector fields on Rn , i. e. smooth vector fields such that all their derivatives are bounded. We also require the vector fields themselves to be bounded. Remark 1.1. We identify vector fields with the corresponding first order differential operators,Pi. e. we identify a vector field V = (V 1 , . . . , V n ) as above with the n ∂ 2 operator i=1 V i (x) ∂x denotes the second order differential i . Consequently, V operator n X

V 2 f (x) =

V j (x)

i,j=1 2

n X ∂V i ∂2f ∂f j i V (x)V (x) (x) (x) + (x), ∂xj ∂xi ∂xi ∂xj i,j=1

n

where f ∈ C (R ). Furthermore, let Yty be the (unique strong) solution of the SDE (in Stratonovich formulation)  d  X   dY y = V (Y y )dt + Vi (Yty ) ◦ dBti 0 t t , (1.1) i=1    Yy =y 0

n

for y ∈ R . Given a multi-index I = (i1 , . . . , ik ) ∈ {0, 1, . . . , d}k , k ∈ N, define a degree by deg(I) = k + #{j ∈ {1, . . . , k} | ij = 0} and deg(∅) = 0, ∅ denoting the empty multi-index. This definition takes care of the different orders of the d + 1 √ 0 i components of B in the sense that ∆B = ∆t and ∆B ≈ ∆t, i ∈ {1, . . . , d}. The t t S∞ set k=0 {0, . . . , d}k of all multi-indices is denoted by A. 1

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CHRISTIAN BAYER

Proposition 1.1 (Stochastic Taylor expansion). Let f ∈ Cbm+2 (Rn ) and fix m ∈ N. Then X f (Yty ) = Vi1 · · · Vik f (y)BtI + Rm (t, y, f ), I=(i1 ,...,ik )∈A deg(I)≤m, k∈N

where the iterated Stratonovich integral is defined by Z Zt Ztk Zt2 (i1 ,...,ik ) ik i1 (1.2) Bt = ◦dBt1 ◦ · · · ◦ dBtk = · · · ◦dBti11 ◦ · · · ◦ dBtikk 0