The Girsanov Theorem Without (So Much) Stochastic Analysis

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 interpretation 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.

Appendices

Appendix 1: Almost Sure Convergence of the Euler Scheme

The convergence property (8.20) of the Euler scheme follows from the recursive inequality

$$\displaystyle \begin{aligned} |\xi_{i+1}(x)-\mathbb{X}_{t_{i+1},0}(x)| &\leq |\xi_{i}(x)-\mathbb{X}_{t_{i},0}(x)| +\rho_i+\frac{T}{n}|b(\xi_i(x))-b(\mathbb{X}_{t_i,0}(x))|\\ &\leq \exp\left(T\|\nabla b\|{}_{\infty}\right)\sum_{j=1}^i \rho_j \end{aligned} $$

(8.36)

with

$$\displaystyle \begin{aligned} \rho_i=\int_{t_i}^{t_{i+1}} |b(\mathbb{X}_{r,0}(x))-b(\mathbb{X}_{t_i,0}(x))|\,\mathrm{d} r \leq \frac{T}{n}\|\nabla b\|{}_{\infty}\sup_{r\in[t_i,t_{i+1}]}|\mathbb{X}_{r,0}(x)-\mathbb{X}_{t_i,0}(x)|. \end{aligned}$$

For this, we have used the fact that b is Lipschitz continuous and $1+\frac {T}{n}\|\nabla b\|{ }_{\infty }\leq \exp (T\|\nabla b\|{ }_{\infty }/n)$.

It can be proved that similarly to the Brownian path, each path of $\mathbb {X}_{t,0}(x)$ is α-Hölder continuous for any α < 1∕2. This is a direct consequence of the Kolmogorov lemma on the regularity of stochastic processes. This proves that the right hand side of (8.36) converges to 0 at rate α < 1∕2.

Appendix 2: The Heat and the Transport Semi-group

The underlying Banach space is $\mathrm {B}=\mathrm {C}_{\mathrm {z}}(\mathbb {R}^d,\mathbb {R})$, the space of continuous, bounded functions that vanish at infinity. The norm on B is $\|f\|=\sup _{x\in \mathbb {R}^d} |f(x)|$.

The Heat Semi-group

The heat semi-group is

$$\displaystyle \begin{aligned} P_{t}f(x)=\int \frac{1}{(2\pi t)^{d/2}}\exp\left(\frac{-|x-y|{}^2}{2t}\right)f(y)\,\mathrm{d} y \end{aligned}$$

for a measurable function g which is bounded or in $\mathrm {L}^2(\mathbb {R}^d)$, the space of square integrable functions.

Since the marginal distribution of the Brownian motion B at any time t is normal one with mean 0 and variance t, $P_tf(x)=\mathbb {E}[f(x+B_t)]$.

Using Fourier transform or computing derivatives,

$$\displaystyle \begin{aligned} \partial_t P_t f(x)=\frac{1}{2}\triangle P_t f(x),\ x\in\mathbb{R}^d,\ t>0. \end{aligned}$$

Multiplying the above equation by $g(x)\in \mathrm {C}_{\mathrm {c}}^2(\mathbb {R}^d,\mathbb {R})$, performing an integration by parts then integrating between 0 and t lead to

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^d} (P_tg(x)-g(x))f(x)\,\mathrm{d} x=\frac{1}{2}\int_0^t \int_{\mathbb{R}^d} P_tf(x)\triangle g(x)\,\mathrm{d} x. \end{aligned}$$

Then, passing to the limit and since f is freely chosen,

$$\displaystyle \begin{aligned} \lim_{t\to 0} \frac{P_tg(x)-g(x)}{t}=\frac{1}{2}\triangle g(x), \ \forall x\in\mathbb{R}^d,\ g\in\mathrm{C}_{\mathrm{c}}^2(\mathbb{R}^d,\mathbb{R}). \end{aligned} $$

(8.37)

Thus, if $(\mathfrak {L}, \operatorname {\mathrm {Dom}}(\mathfrak {L}))$ is the infinitesimal generator of (P _t)_t≥0 (this is necessarily a close operator), $\mathfrak {L}=\frac {1}{2}\triangle$ on $\mathbb {C}_{\mathrm {c}}^2(\mathbb {R}^d,\mathbb {R})\subset\operatorname {\mathrm {Dom}}(\mathfrak {L})$. The latter space being dense in the underlying space $\mathrm {C}_{\mathrm {z}}(\mathbb {R}^d,\mathbb {R})$ with respect to ∥⋅∥_∞ and $\operatorname {\mathrm {Dom}}(\mathfrak {L})$ with respect to the graph norm $\|\cdot \|{ }_{\infty }+\|\mathfrak {L}\cdot \|{ }_{\infty }$, (8.37) characterizes $(\mathfrak {L}, \operatorname {\mathrm {Dom}}(\mathfrak {L}))$ when $\mathrm {C}_{\mathrm {z}}(\mathbb {R}^d,\mathbb {R})$ is the ambient Banach space.

In other words, we recover that the infinitesimal generator of the Brownian motion is $\mathfrak {L}=\frac {1}{2}\triangle$ with a suitable domain. This could of course be easily obtained from the Itô’s formula. Also, we see the link between the heat equation (8.10), the Brownian motion and the heat semi-group.

The Transport Semi-group

Let us consider now the flow $(\mathbb {Y}_t)$. Since $\mathbb {Y}(x)$ is solution to $\mathbb {Y}_t(x)=x+\int _0^t b(\mathbb {Y}_s(x))\,\mathrm {d} s$, the Newton formula for $f\in \mathrm {C}_{\mathrm {c}}^1(\mathbb {R}^d,\mathbb {R})$ implies that

$$\displaystyle \begin{aligned} f(\mathbb{Y}_t(x))-f(x)=\int_0^t b(\mathbb{Y}_s(x))\nabla f(\mathbb{Y}_s(s))\,\mathrm{d} s. \end{aligned}$$

Hence, the infinitesimal generator of (Q _t)_t≥0 is $\mathfrak {B}=b\nabla \cdot$ whose domain $\operatorname {\mathrm {Dom}}(\mathfrak {B})$ is the closure of $\mathrm {C}_{\mathrm {c}}^1(\mathbb {R}^d,\mathbb {R})$ to the graph norm (see e.g. [17, § II.3.28, p. 91]).

The semi-group (Q _t)_t>0 is also linked to a PDE, called the transport equation. We have seen in Proposition 8.2 that $x\mapsto \mathbb {Y}_t(x)$ is differentiable. It is actually of class $\mathrm {C}^1(\mathbb {R}^d,\mathbb {R})$. Thus, $Q_t f(x)=f\circ \mathbb {Y}_t(x)$ is also differentiable. Applying the Newton formula to $f\circ \mathbb {Y}_t=Q_t f$ and using the flow property of $\mathbb {Y}_t$ leads to

$$\displaystyle \begin{aligned} f(\mathbb{Y}_{t+\epsilon}(x))-f(\mathbb{Y}_t(x))=Q_t f(\mathbb{Y}_\epsilon(x))-Q_t f(x) =\int_0^{\epsilon} b(\mathbb{Y}_s(x))\nabla Q_t f(\mathbb{Y}_s(x))\,\mathrm{d} s. \end{aligned}$$

Dividing each side by 𝜖 and passing to the limit implies that

$$\displaystyle \begin{aligned} \partial_t Q_t f(x)=b(x)\nabla Q_t f(x). \end{aligned}$$

Conversely, it is also possible to start from the transport equation ∂ _tu(t, x) = b(x)∇u(t, x) to construct the flow $\mathbb {Y}$ through the so-called method of characteristics, that is to find the paths $\mathbb {Z}:\mathbb {R}_+\to \mathbb {R}^d$ such that $u(t,\mathbb {Z}_t)$ is constant over the time t.