Feynman-Kac formula under a finite entropy condition

Abstract

Motivated by entropic optimal transport, we are interested in the Feynman-Kac formula associated to the parabolic equation $( {\mathsf {L}}+V)g =0$ with a final nonnegative boundary condition and a Markov generator ${\mathsf {L}}:= \partial _t + {\mathsf {b}}\!\cdot \!\nabla + \Delta _{ {\mathsf {a}}}/2$. It is well-known that when the drift ${\mathsf {b}}$, the diffusion matrix ${\mathsf {a}}$ and the scalar potential V are regular enough and not growing too fast, the classical solution g of this PDE, is represented by the Feynman-Kac formula $g_t(x)=E_R[\exp \left( \int _{[t,T]} V(s,X_s)\,ds\right) g(X_T)\mid X_t=x]$ where R is the Markov measure with generator ${\mathsf {L}}$. We do not assume that g, ${\mathsf {b}}$ and V are regular, and only require that their growth is controlled by a finite entropy condition. These hypotheses are less restrictive than the standard assumptions of the theory of viscosity solutions, and allow for instance V to belong to some Kato class. We prove that g defined by the Feynman-Kac formula belongs to the domain of the extended generator ${\mathcal {L}}$ of the Markov measure R and satisfies the trajectorial identity: $[({\mathcal {L}} +V)g] (t,X_t)=0,\ dtdP\text {-}{a.e.}$ where the path measure P is defined by $P:= f(X_0)\exp \left( \int _{[0,T]}V(t,X_t)\,dt\right) g(X_T)\ R,$ with $f:{\mathbb {R}}^n\rightarrow [0, \infty )$ another nonnegative function. We also show that the forward drift ${\mathsf {b}}^P$ of P satisfies ${\mathsf {b}}^P(t,X_t)=[ {\mathsf {b}}+ {\mathsf {a}}{\widetilde{\nabla }}\log g](t,X_t),$ $dtdP\text {-}{a.e.},$ where ${\widetilde{\nabla }}$ is some extension of the standard derivative. Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov’s theorem and the Hamilton-Jacobi-Bellman equation satisfied by $\log g$.

Appendices

Appendix A. Carré du champ

Lemma A.2 below is a simplified version of [4, Lemma 3.9], which was used during the proof of Lemma 4.30. For the confort of the reader, we give its detailed proof which is slightly simpler, but essentially the same as [4]’s one.

Let $Q\in \mathrm {M}(\Omega )$ be a conditionable path measure. Its forward carré du champ is defined by

$$\begin{aligned} \Gamma ^Q(u,v):= {\mathcal {L}}^Q(uv)-u {\mathcal {L}}^Q v-v {\mathcal {L}}^Q u, \quad \ 0\le t\le T, \end{aligned}$$

for any functions u, v in ${{\,\mathrm{dom}\,}} {\mathcal {L}}^Q$ such that their product uv also belongs to ${{\,\mathrm{dom}\,}} {\mathcal {L}}^Q$.

The quadratic covariation [u(X), v(X)] is a Q-semimartingale. We denote by $\langle u(X),v(X)\rangle ^Q$ its bounded variation part, i.e.

$$\begin{aligned} d[u(X),v(X)]_t=d\langle u(X),v(X)\rangle ^Q_t+ dM_t ^{Q, [u,v]}, \qquad {\overline{Q}}\text {-}{a.e.}\end{aligned}$$

where, here and below, or stands for any local Q-martingale. As next lemma indicates, we are interested in situations where the bounded variation process $\langle u(X),v(X)\rangle ^Q$ is predictable (as a continuous process). Therefore, in the whole article $\langle u(X),v(X)\rangle ^Q$ is the usual sharp bracket (sometimes called conditional quadratic variation) of stochastic process theory.

Lemma A.1

For any $u,v\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q$ such that $uv\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q,$ the process $\langle u(X),v(X)\rangle ^Q$ is absolutely continuous $Q\text {-}{a.e.}$ and

$$\begin{aligned} d\langle u({\overline{X}}),v({\overline{X}})\rangle ^Q_t=\Gamma ^Q(u,v)(t, X _{ [0,t]})\,dt, \qquad Q\text {-}{a.e.}\end{aligned}$$

Proof

As a definition of the forward generator $d(uv)({\overline{X}}_t)= {\mathcal {L}}^Q_t(uv)(t, X _{ [0,t]})\,dt+dM ^{ uv}_t.$ Comparing this expression with (4.23), the Doob-Meyer decomposition theorem gives the announced result. $\square$

We say that a process Y can be localized as a bounded (resp. integrable) process if there exists a sequence of stopping times $( \sigma _k)$ tending almost surely to infinity and such that for each k, the stopped process $Y^{ \sigma _k}$ is bounded almost surely (resp. integrable).

Lemma A.2

For any conditionable path measure $Q\in \mathrm {M}(\Omega )$, almost every $t\in [0,T],$ and any locally bounded functions $u,v\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q$ such that $uv\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q$, and $M ^{ Q,[u,v]}$ as defined at Lemma A.1 can be localized as an integrable Q-martingale, there exist an increasing sequence $(\tau _k)$ of Q-integration times of u and v,  and a sequence $(h_n)$ of positive numbers such that $\lim _{k\rightarrow \infty } \tau _k=\infty ,$ $Q\text {-}{a.e.}$, $\lim _{n\rightarrow \infty } h_n=0$ and

$$\begin{aligned} \begin{aligned} \Gamma ^Q (u,v)&(t, X _{ [0,t]})\\&= \lim _{k\rightarrow \infty } \lim _{n\rightarrow \infty } \frac{1}{h_n}E_Q \left. \Big [\{u( {\overline{X}}^{ \tau _k}_{t+h_n})-u( {\overline{X}}^{ \tau _k}_t)\}\{v( {\overline{X}}^{ \tau _k}_{t+h_n})-v( {\overline{X}}^{ \tau _k}_t)\}\,\right| \, X_{[0,t]} \Big ],\quad Q\text {-}{a.e.}\end{aligned} \end{aligned}$$

Proof

Since u and v are assumed to be locally bounded, $u({\overline{X}})$ and $v({\overline{X}})$ can be localized as bounded processes. Furthermore, the processes and can also be localized as bounded processes. It follows that the local Q-martingales $M^u, M^v$, (where $M^u_t:=u({\overline{X}}_t)-\int _0^t {\mathcal {L}}^Qu(s,X _{ [0,s]})\,ds$) can also be localized as bounded processes. Localizing as in the proof of Proposition 3.14, it is enough to show that

$$\begin{aligned} \begin{aligned}&\lim _{h\rightarrow 0^+}E_Q\int _0 ^{ T-h} \Big |E_Q \big [h ^{ -1}\{u( X_{t+h})-u( X_t)\}\{v( X_{t+h})-v( X_t)\}\,\big |\, X _{[0,t]}\big ]\\&- \Gamma ^Q(u,v)(t,X _{ [0,t]})\,\Big |\,dt =0, \end{aligned} \end{aligned}$$

(A.3)

and we can assume that all the above mentioned processes are bounded.

For each $0\le t\le T-h$ with $0<h\le T,$

$$\begin{aligned}{}[u( X_{ t+h})-&u( X_t)][v( X _{ t+h})-v( X_t)]\\&=\Big [ \int _t ^{ t+h} dM^u_s + \int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\Big ] \Big [ \int _t ^{ t+h} dM^v_s + \int _t ^{ t+h} {\mathcal {L}}^Qv({\overline{X}}_s)\,ds\Big ]\\&= A_t^h+B_t^h+C_t^h+D_t^h, \qquad Q\text {-}{a.e.}, \end{aligned}$$

where

$$\begin{aligned} A_t^h&= \int _t ^{ t+h} dM^u_s\ \int _t ^{ t+h} dM^v_s,B_t^h= \int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\ \int _t ^{ t+h} dM^v_s,\\ C_t^h&= \int _t ^{ t+h} {\mathcal {L}}^Q v({\overline{X}}_s)\,ds\ \int _t ^{ t+h} dM^u_s,D_t^h= \int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\ \int _t ^{ t+h} {\mathcal {L}}^Qv({\overline{X}}_s)\,ds. \end{aligned}$$

Let us control $A^h_t.$ Denoting $U_{t,s}:=M^u_s-M^u_t$ and $V_{t,s}:=M^v_s-M^v_t,$

$$\begin{aligned} A_t^h&= \int _t ^{ t+h} d(U_{t,s}V_{t,s})\\&= \int _t ^{ t+h} U_{t,s} dM^v_s + \int _t ^{ t+h} V_{t,s} dM^u_s + \int _t ^{ t+h} dM ^{Q, [u,v]}_s + \int _t ^{ t+h} d \langle M^u,M^v\rangle ^Q _s, \end{aligned}$$

and with Lemma A.1

$$\begin{aligned} h ^{ -1}E_Q\left. [A^h_t\,\right| \, X _{ [0,t]}]= h ^{ -1}\int _t ^{ t+h} E_Q[\Gamma ^Q(u,v)({\overline{X}}_s)\mid X _{ [0,t]}]\,ds. \end{aligned}$$

(A.4)

Remark that the boundedness properties obtained above by localization, together with the extra assumption that $M ^{Q, [u,v]}$ is integrable, justify the cancelation of the expectations of the martingale terms.

Let us control $B^h$:

$$\begin{aligned} h ^{ -1}E_Q \int _0 ^{ T-h}|B_t^h|\,dt&\le E_Q \int _0 ^{ T-h} h ^{ -1}\Big |\int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\Big |\ |M^v_{ t+h}-M^v_t|\, dt \\&= E_Q \int _0 ^{ T-h} \Big | k^h*( {\mathcal {L}}^Qu)({\overline{X}}_t)\Big |\ |M^v_{ t+h}-M^v_t|\, dt\\&= o _{ h\rightarrow 0^+}(1) , \end{aligned}$$

where we took $k^h:= h ^{ -1} {\mathbf {1}}_{ [-h,0]}$ as our convolution kernel. The last identity is a consequence of Lemma 2.11 under the assumption ${\mathcal {L}}^Qu({\overline{X}})\in L^1({\overline{Q}})$ (because is bounded), the uniform boundedness and right-continuity of $M^v$ and the dominated convergence theorem.

Similarly, $\lim _{h\rightarrow 0^+}h ^{ -1}E_Q \int _0 ^{ T-h}|C_t^h|\,dt=0.$

The control of $D^h$ is analogous:

$$\begin{aligned} h ^{ -1}E_Q \int _0 ^{ T-h}|D_t^h|\,dt&\le E_Q \int _0 ^{ T-h} \Big | k^h*( {\mathcal {L}}^Qu)({\overline{X}}_t)\Big |\ \Big |\int _t ^{ t+h} {\mathcal {L}}^Qv({\overline{X}}_s)\,ds\Big |\, dt\\&= o _{ h\rightarrow 0^+}(1) , \end{aligned}$$

thanks to the uniform boundedness of $\int _{[0,T]}| {\mathcal {L}}^Qv({\overline{X}}_s)|\,ds.$

Putting everything together, we obtain

$$\begin{aligned}&\lim _{h\rightarrow 0^+}E_Q\int _0 ^{ T-h} \Big |E_Q \big [h ^{ -1}\{u( X_{t+h})-u( X_t)\}\{v( X_{t+h})-v( X_t)\}\,\Big |\, X _{ [0,t]}\big ]\\&\qquad -h ^{ -1}\int _t ^{ t+h} E_Q[\Gamma ^Q(u,v)({\overline{X}}_s)\,\big |\, X _{ [0,t]}]\,ds\Big |\,dt =0. \end{aligned}$$

On the other hand, with Corollary 2.16 we obtain

$$\begin{aligned} \lim _{h\rightarrow 0^+}E_Q\int _0 ^{ T-h} \Big |h ^{ -1}\int _t ^{ t+h} E_Q[\Gamma ^Q(u,v)({\overline{X}}_s)\mid X _{ [0,t]}]\,ds - \Gamma ^Q(u,v)(t,X _{ [0,t]})\Big |\,dt =0. \end{aligned}$$

The limit (A.3) follows from these last two limits. $\square$

Appendix B. About Nelson velocities

This section refers to the diffusion measure Q of Sect. 4. Its content is not used directly in this article. We propose it to the reader to stress the importance for our purpose of considering the relative momentum field $\beta ^{ Q|R}$ rather than the absolute velocity ${\mathsf {v}}^Q.$

Denoting by $\mathrm {Id}$ the identity mapping on ${\mathbb {R}}^n,$ we see that the vector field ${\mathsf {v}}^Q$ appearing in the martingale problems satisfy ${\mathsf {v}}^Q= {\mathcal {L}}^Q[\mathrm {Id}]$. Because of the identification ${\mathcal {L}}^Q= L^Q$ which was obtained at Sect. 2, one suspects that ${\mathsf {v}}^Q$ should satisfy

$$\begin{aligned} {\mathsf {v}}^Q_t= L^Q_t[\mathrm {Id}]= \lim _{h\rightarrow 0^+}E_Q\left. \Big ( \frac{X _{ t+h}-X_t}{h}\,\right| \, X _{ [0,t]}\Big ), \end{aligned}$$

whenever this expression is meaningful. The r.h.s. of this identity is the forward Nelson velocity of Q. But in general it is not well defined, due to a possible lack of integrability. In order to give sense to limits of this type in a general setting, one must introduce integration times and work as in Proposition 3.14. Next result presents a situation where integration times can be avoided.

Proposition B.1

Under the hypothesis (4.9), suppose that ${\mathsf {a}}$ is bounded from above. Then, the limit

$$\begin{aligned} {\mathsf {v}}^{ Q|R}(t, \omega )= \lim _{h\rightarrow 0^+}E_Q\left. \Big ( \frac{X _{ t+h}-X_t}{h} - \frac{1}{h} \int _{ [t,t+h]} {\mathsf {b}}({\overline{X}}_s)\,ds\,\right| \, X _{ [0,t]}=\omega _{ [0,t]}\Big ), \quad (t, \omega )\in {\overline{\Omega }}, \end{aligned}$$

takes place in $L^2({\overline{Q}}).$

Proof

Under (4.9), we know that $E _{ {\overline{Q}}}|\beta ^{ Q|R}|^2_ {\mathsf {a}}< \infty .$ Because of the assumed upper boundedness of ${\mathsf {a}}$, this implies that $E _{ {\overline{Q}}}| {\mathsf {v}}^{ Q|R}|^2< \infty ,$ where ${\mathsf {v}}^{ Q|R}:= {\mathsf {a}}\beta ^{ Q|R}.$ Rewrite the assertion: $Q\in {{\,\mathrm{MP}\,}}( {\mathsf {a}}, {\mathsf {b}}+ {\mathsf {v}}^{ Q|R})$ as:

$$\begin{aligned} X _{ t+h}-X_t-\int _{ [t,t+h]} {\mathsf {b}}({\overline{X}}_s)\,ds =\int _{ [t,t+h]} {\mathsf {v}}^{ Q|R}({\overline{X}}_s)\,ds +M^Q _{ t+h}-M^Q_t, \end{aligned}$$

where $M^Q$ is a local Q-martingale. The assumption $E _{ {\overline{Q}}}| {\mathsf {v}}^{ Q|R}|^2< \infty ,$ expressed with the Euclidean norm rather than the Riemannian norm permits us to apply the convolution Lemma 2.11 to $v=a \beta$ componentwise with $p=2$. The critical step where this is used is Jensen’s inequality right below (2.12). With this at hand, proceeding as in the proof of Proposition 2.18 leads us to the announced result. $\square$

Remarks B.2

 

• (a) In the setting of this proposition, if the Nelson velocity $L^R[\mathrm {Id}]$ is ill defined because $E_R\int _{[0,T]}| {\mathsf {b}}_t|\,dt= \infty$, it might happen that $L^Q[\mathrm {Id}]$ is also ill defined. Nevertheless, we have: $\int _{[0,T]}| {\mathsf {b}}_t|\,dt<\infty , R\text {-}{a.e.},$ and $Q\in {{\,\mathrm{MP}\,}}( {\mathsf {a}}, {\mathsf {v}}^Q)$ where ${\mathsf {v}}^Q= {\mathsf {b}}+ {\mathsf {v}}^{ Q|R}$ satisfies $\int _{[0,T]}| {\mathsf {v}}^Q_t|\,dt< \infty , Q\text {-}{a.e.}$

• (b) Requiring that the diffusion matrix field ${\mathsf {a}}$ is upper bounded is not a strong restriction for the applications, because in general temperature is upper bounded.

• (c) If ${\mathsf {a}}$ is only locally bounded, then there exists a sequence $(h_n)$ of positive numbers such that $\lim _{n\rightarrow \infty } h_n=0$ and the limit

  $$\begin{aligned}&{\mathsf {v}}^{ Q|R}(t, \omega ) = \lim _{k\rightarrow \infty } \lim _{n\rightarrow \infty } E_Q\\&\left. \Big ( \frac{X ^{ \tau _k}_{ t+h_n}-X^{\tau _k} _t}{h_n} -\frac{1}{h_n} \int _{ [t,t+h_n]}{\mathbf {1}}_{ \left\{ s\le \tau _k\right\} } {\mathsf {b}}({\overline{X}}_s)\,ds\,\right| \, X _{ [0,t]}=\omega _{ [0,t]}\Big ) \end{aligned}$$

  holds ${\overline{Q}}\text {-}{a.e.},$ where for each integer $k\ge 1,$ $\tau _k:= \inf \{t\in [0,T]: |X_t|\ge k\}.$ The proof of this statement is similar to Proposition 3.14’s proof.