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The Girsanov Theorem Without (So Much) Stochastic Analysis Antoine Lejay

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The Girsanov Theorem Without (So Much) Stochastic Analysis Antoine Lejay The Girsanov theorem without (so much) stochastic analysis Antoine Lejay To cite this version: Antoine Lejay. The Girsanov theorem without (so much) stochastic analysis. 2017. hal-01498129v1 HAL Id: hal-01498129 https://hal.inria.fr/hal-01498129v1 Preprint submitted on 29 Mar 2017 (v1), last revised 22 Sep 2018 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Girsanov theorem without (so much) stochastic analysis Antoine Lejay March 29, 2017 Abstract In this pedagogical note, we construct the semi-group associated to a stochastic differential equation with a constant diffusion and a Lipschitz drift by composing over small times the semi-groups generated respectively by the Brownian motion and the drift part. Similarly to the interpreta- tion of the Feynman-Kac formula through the Trotter-Kato-Lie formula in which the exponential term appears naturally, we construct by doing so an approximation of the exponential weight of the Girsanov theorem. As this approach only relies on the basic properties of the Gaussian distribution, it provides an alternative explanation of the form of the Girsanov weights without referring to a change of measure nor on stochastic calculus. Keywords: Girsanov theorem, Lie-Trotter-Kato formula, Feynman-Kac formula, Stochastic differential equation, Euler scheme, splitting scheme, flow, heat equa- tion, Cameron-Martin theorem. 1 Introduction This pedagogical paper aims at presenting the Girsanov theorem — a change of measure for the Brownian motion — using the point of view of operator analy- sis. We start from the sole knowledge of the Brownian distribution and its main properties. By doing so, stochastic calculus is avoided excepted for identifying the limit. Therefore, in a simplified context, we give an alternative proof of a result which is usually stated and proved using stochastic analysis and measure theory. In 1944, R.H. Cameron and W.T. Martin proved the celebrated theorem on the change of the Wiener measure. They later extend it [7, 8]. Theorem 1 (R.H. Cameron & W.T. Martin [9]). Let 퐹 be a continuous func- tional on the space of continuous functions C([0, 1]; R) with respect to the uniform 1 norm. Let 푏 be a continuous function in C([0, 1]; R) whose derivative 푏′ has bounded variation. Then for a Wiener process (or a Brownian motion)1 W, [︂ (︂ ∫︁ 1 ∫︁ 1 )︂]︂ ′ 1 ′ 2 E[퐹 (W)] = E 퐹 (W + 푏) exp − 푏 (푠) dW푠 − |푏 (푠)| d푠 . 0 2 0 Later in 1960, I.V. Girsanov states in [18] a variant of this theorem for solutions of stochastic differential equations. Even when 휎 is constant, the drift is itself a non-linear functional of the Brownian motion. Theorem 2 (I.V. Girsanov, [18]). Let W be a 푛-dimensional Wiener process on a probability space (Ω, ℱ, P) with respect to a filtration (ℱ푡)푡≥0. Let X be the solution on (Ω, P) to ∫︁ 푡 ∫︁ 푡 X푡(휔) = X0(휔) + 푎(푠, 휔) dW(푠, 휔) + 푏(푠, 휔) d푠 0 0 where 푎 is matrix valued, 푏 is vector valued, and (A1) The applications 푎 and 푏 are measurable with respect to (푠, 휔) ∈ [0, 1] × Ω. (A2) For each 푡 ≥ 0, 푎 is ℱ푡-measurable. 2 ∫︀ 1 2 (A3) Almost everywhere , 0 ‖푎(푡, 휔)‖ d푡 < +∞. ∫︀ 1 (A4) Almost everywhere, 0 |푏(푡, 휔)| d푡 < +∞. Let 휑 = (휑1, . , 휑푛) be a vector-valued function on [0, 1] × Ω such that (A1)- (A3) are satisfied. 1 Let us set P̃︀[ d휔] = exp(Z0(휑, 휔))P[ d휔] where ∫︁ 푡 ∫︁ 푡 (︃ 푛 )︃ 푡 푖 푗 1 ∑︁ 푖 2 Z푠(휑, 휔) = 휑 (푢, 휔)훿푖푗 dW (푢, 휔) − 휑 (푢, 휔) d푢. 푠 2 0 푖=1 ∫︀ 푡 Let us also set W̃︁(푡, 휔) = W(푡, 휔) − 0 휑(푠, 휔) d푠. If P̃︀[Ω] = 1, then W̃︁ is a Wiener process with respect to (ℱ푡)푡≥0 on (Ω, ℱ, P̃︀) and (X, P̃︀) is solution to ∫︁ 푡 ∫︁ 푡 X푡(휔) = X0(휔) + 푎(푠, 휔) dW푠(휔) + (푏(푠, 휔) + 푎(푠, 휔)휑(푠, 휔)) d푠. 0 0 Soon after, these results were extended in many directions, for example to deal with semi-martingales (see e.g. [47]). The study of the weights for the change of measures and their exponential nature gives rise to the theory of Doléans-Dade martingales [12, 33]. 1In the original paper [9], the result is stated for 2−1/2W and 21/2푏. 1/2 2 (︁∑︀푛 2 )︁ The norm of a matrix is ‖푎‖ = 푖,푗=1 |푎푖,푗| while the norm of a vector is ‖푎‖ = (︀∑︀푛 2)︀1/2 푖=1 |푎푖| . 2 These theorems provide us with measures which are equivalent to the Wiener’s one. The converse is also true [39]: Absolute continuity of Wiener or diffusions measures can only be reached by adding terms of bounded variation [39, 42]. The Cameron-Martin and Girsanov theorems have a profound meaning as well as a deep impact on modern stochastic calculus. For example, for example, they are one of the cornerstones of Malliavin calculus [4, 34], likewise a major tool in filtering, statistics of diffusion processes [31, 32], mathematical finance, [24],... The Girsanov theorem has also been extended to some Gaussian processes, including the fractional Brownian motion [16, 22]. The Feynman-Kac formula related the Brownian motion with some PDEs. It involves a probabilistic representation with an exponential weight (see Section 9.1). The Feynman-Kac formula could be proved by many ways, including stochastic calculus from one side and the Trotter-Kato-Lie formula on the other side (see [15, Chap. 3, Sect. 5] or [10] for a nice introduction to this subject, and [19] for a proof of the Feynman-Kac formula with this procedure). To illustrate the latter approach, let us consider three linear matrix-valued equations ˙ ˙ ˙ 푋 = A푋, 푌 = B푌, 푅 = (A + B)푅 with 푋0 = 푌0 = 푅0 = Id, where A and B are 푑 × 푑-matrices. These equations are easily solved by 푋푡 = exp(푡A), 푌푡 = exp(푡B) and 푅푡 = exp(푡(A + B)). The solutions 푋, 푌 and 푅 satisfy the semi-group property: 푋푡+푠 = 푋푡푋푠. If AB = BA, then 푅푡 = 푌푡푋푡. This is no longer true in general. However, as shown first by S. Lie [30], the solution 푅 could be constructed from 푋 and 푌 by the following limit procedure: A푡/푛 B푡/푛 푛 푅푡 = exp(푡(A + B)) = lim 푋푡/푛푌푡/푛푋푡/푛푌푡/푛 ··· 푋푡/푛푌푡/푛 = lim (푒 푒 ) . 푛→∞ 푛→∞ H.F. Trotter [46] and T. Kato [26] have shown that this could be generalized for large families of linear unbounded operators A and B. In this case, (푋푡)푡≥0 and (푌푡)푡≥0 are families on linear operators with the semi-groups property. 푑 On the space of continuous, bounded functions Cb(R , R), consider the (scaled) 1 Laplace operator A = 2 △ and let B be defined by B푓(푥) = 푈(푥)푓(푥) for any 푓 ∈ C(R푑, R), where the continuous function 푈 is called a potential. For a Brownian motion B, 푋푡 = E[푓(푥 + B푡)] whereas 푌푡푓(푥) = exp(푡푈(푥))푓(푥) for any 푓 ∈ 푑 푑 Cb(R , R), 푥 ∈ R and 푡 ≥ 0. With the Trotter-Kato-Lie formula, we compose over short times the semi- group 푋 of the Laplace operator with the one 푌 of the potential term. Using the 3 Markov property of the Brownian motion, for any bounded, measurable function 푓, [︃ (︃ 푛−1 )︃ ]︃ 푡 ∑︁ 푅푡푓(푥) = lim exp 푈(푥 + B푖푡/푛) 푓(푥 + B푡) . (1) 푛→∞ E 푛 푖=0 With the right integrability conditions on 푈, we obtain in the limit an exponential representation of the Feynman-Kac formula [25], that is [︂ (︂∫︁ 푡 )︂ ]︂ 푅푡푓(푥) = E exp 푈(푥 + B푠) d푠 푓(푥 + B푡) , 0 which gives a probabilistic representation to the PDE 1 휕 푅 푓(푥) = △푅 푓(푥) + 푈(푥)푅 푓(푥) with 푅 푓(푥) = 푓(푥). 푡 푡 2 푡 푡 0 The seminal derivation of the Feynman-Kac formula by M. Kac in [25] used an approximation of the Brownian motion by a random walk, which leads to an expression close to (1). What happens now if we use for B the first-order differential operator 푏∇· for a function 푏? This means that we consider giving a probabilistic representation of the semi-group (푅푡)푡≥0 related to the PDE 1 휕 푅 푓(푥) = △푅 푓(푥) + 푏(푥)∇푅 푓(푥) with 푅 푓(푥) = 푓(푥). 푡 푡 2 푡 푡 0 Of course, a probabilistic representation is derived by letting X푡 be the solution ∫︀ 푡 of the SDE X푡 = 푥 + B푡 + 0 푏(X푠) d푠 and 푅푡푓(푥) = E[푓(X푡)]. With the Girsanov theorem (Theorem 2), 푅푡푓(푥) = E[Z푡푓(푥 + B푡)] for the Girsanov weight Z given by (4) below. The Feynman-Kac formula is commonly understood as a byproduct of the Trotter-Kato-Lie formula [20], at least in the community of mathematical physics in relation with the Schrödinger equation [37, 41].
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