Abstract
We prove existence and uniqueness of solutions to Fokker–Planck equations associated with Markov operators multiplicatively perturbed by degenerate timeinhomogeneous coefficients. Precise conditions on the timeinhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be either globally bounded or bounded away from zero. The approach is based on constructing random timechanges and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.
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Notes
That means \(\mathcal {D}\subset C_0(E)\) is an algebra with respect to the addition and multiplication induced by \(C_0(E)\).
For instance the functions defined for \(n \in \mathbb {N}\) by \(\gamma _n(s) = 1\) for \(s \in [0,n]\), \(\gamma _n(s) = 0\) for \(s \in [n+1,\infty )\) and \(\gamma _n(s) = 2(sn)^33(sn)^2+1\) for \(s \in [n,n+1]\) satisfy these properties.
To check the assumptions of [13, Chap. 4, Thm. 4.1] in more detail (see [13] for unexplained definitions), note that \([0,\infty )\) is locally compact, separable, \(D(\partial _t)\) is dense in \(C_0[0,\infty )\) and \(C_0[0,\infty )\) is convergence determining (see [13, Chap. 3, Prop. 4.4]), hence separating. Furthermore, whenever \(\gamma \in D(\partial _t)\), \(t^* \ge 0\) satisfy \(\gamma (t^*) = \sup _{t\ge 0} \gamma (t)\), then \(\gamma '(t^*) = 0\). Thus \(\partial _t\) satisfies the positive maximum principle and is hence dissipative by [13, Chap. 4, Lem. 2.1]. Finally, fix \(\lambda > 0\), then for any \(g \in C_c^1[0,\infty )\), the function \(\gamma (t):=\exp (\lambda t)\int _t^\infty g(s)\exp (\lambda s) \,\mathrm {d}s \) satisfies \(\gamma \in D(\partial (t))\) and \(\lambda \gamma  \partial _t \gamma = g\) so that the range of the operator \(\lambda  \partial _t\) is \(C_c^1[0,\infty )\) and in particular dense in \(C_0[0,\infty )\).
See footnote 3 for a discussion why (ii) implies that \(\mathcal {L}_y^0\) is a pregenerator (in the terminology of [20]).
References
Bass, R.F.: Skorokhod Imbedding via Stochastic Integrals, Seminar on Probability, XVII, Lecture Notes in Mathematics, vol. 986, pp. 221–224. Springer, Berlin (1983)
Bass, Richard F.: Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79(2), 271–287 (1988)
Bentata, A., Cont, R: Mimicking the marginal distributions of a semimartingale, Preprint arXiv:0910.3992, pp. 1–36 (2009)
Bhatt, Abhay G., Karandikar, Rajeeva L.: Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21(4), 2246–2268 (1993)
Bhatt, Abhay G., Karandikar, Rajeeva L.: Martingale problems and path properties of solutions. Sankhyā Indian J. Stat (2003–2007) 65(4), 733–743 (2003)
Barbu, Viorel, Röckner, Michael, Russo, Francesco: Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case. Probab. Theory Relat. Fields 151(1–2), 1–43 (2011)
Böttcher, Björn, Schilling, René, Wang, Jian: Lévy Matters III. Springer, Berlin (2013)
Carr, Peter, Geman, Hélyette, Madan, Dilip B., Yor, Marc: From local volatility to local Lévy models. Quant. Finance 4(5), 581–588 (2004)
Cox, Alexander M.G., Hobson, David, Obłój, Jan: Timehomogeneous diffusions with a given marginal at a random time. ESAIM Probab. Stat. 15, 11–24 (2011)
Döring, Leif, Gonon, Lukas, Prömel, David J., Reichmann, Oleg: On Skorokhod embeddings and poisson equations. Ann. Appl. Probab. 29(4), 2302–2337 (2019)
Dupire, B.: Pricing with a Smile. Risk Mag. 7, 18–20 (1994)
Ekström, Erik, Hobson, David, Janson, Svante, Tysk, Johan: Can timehomogeneous diffusions produce any distribution? Probab. Theory Relat. Fields 155(3–4), 493–520 (2013)
Ethier, Stewart N., Kurtz, Thomas G.: Markov Processes. Characterization and Convergence. Wiley, New York (1986)
Engelbert, HansJürgen, Schmidt, Wolfgang: On solutions of onedimensional stochastic differential equations without drift. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 68(3), 287–314 (1985)
Figalli, Alessio: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254(1), 109–153 (2008)
Filippov, A.F.: : Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications, vol. 18. Springer, Berlin (1988)
Hirsch, F., Profeta, C., Roynette, B., Yor, M.: Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series, vol. 3. Springer, Milan (2011)
Karatzas, Ioannis, Shreve, Steven E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)
Kurtz, Thomas, Stockbridge, Richard: Stationary solutions and forward equations for controlled and singular martingale problems. Electron. J. Probab. 6(17), 1–52 (2001)
Kurtz, Thomas: Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3(9), 1–29 (1998)
Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence (2009)
Lumer, Gunter: Perturbation de générateurs infinitésimaux du type changement de temps. Annales de l’institut Fourier 23(4), 271–279 (1973)
Rogers, L.C.G., Williams, David: Diffusions, Markov Processes, and Martingales, vol. 1, 2nd edn. Cambridge University Press, Cambridge (2000)
Revuz, Daniel, Yor, Marc: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)
Sato, KenIti: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Stroock, Daniel W.: Diffusion processes associated with Lévy generators. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 32(3), 209–244 (1975)
Veraguas, J.B., Beiglböck, M., Huesmann, M., Källblad, S.: Martingale Benamou–Brenier: a probabilistic perspective, Preprint arXiv:1708.04869 (2017)
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L.G. and D.J.P acknowledge generous support from ETH Zürich, where a major part of this work was completed.
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Appendix A: Auxiliary Results for the Construction of TimeChanges
Appendix A: Auxiliary Results for the Construction of TimeChanges
In order to ensure the existence of the timechange \(\tau \) as defined by the random differential equation (3.1), we used the following lemma concerning socalled Carathéodory differential equations.
Lemma A.1
Let \(t_0> 0\) and consider the Carathéodory differential equation
where \(\gamma :[0,t_0] \times [0,\infty ) \rightarrow [0,\infty )\) and the integral is understood in the Lebesgue sense. For some \(S \in (0,t_0]\) and \(T>0\), suppose that \(\gamma (r,\cdot )\) is measurable for each \(r \in [0,S]\), \(\gamma (0,\cdot )\) is integrable on [0, T] and there exists an integrable function \(f:[0,T]\rightarrow [0,\infty )\) such that
Then, there exists a unique absolutely continuous function \(\mathcal {T}:I \rightarrow [0,S]\) satisfying (A.1) for some interval \(I \subset [0,\infty )\), where either there exists \(T_0 \in (0,T]\) such that we may take \(I =[0,T_0]\) and we have \(\mathcal {T}(T_0) = S\) or we may take \(I =[0,T]\) and have \(\mathcal {T}(t) < S\) for all \(t \le T\).
Proof
Since
and the righthand side is integrable on [0, T], \(\gamma \) satisfies the Carathéodory conditions in [16, Chap. 1] and thus [16, Chap. 1, Thm. 1] guarantees the existence of a solution \(\mathcal {T}\) on an interval \([0,T_0]\) for some \(T_0 >0\). The solution \(\mathcal {T}\) can be extended either to the whole interval [0, T] provided \(\mathcal {T}(t)\le S\) for all \(t\in [0,T]\) or to the interval \([0,T_0]\) for some \(T_0 \in (0,T]\) with \(\mathcal {T}(T_0) = S\) (see e.g. [16, Chap. 1, Thm. 4]). Uniqueness of the solution follows by [16, Chap. 1, Thm. 2]. \(\square \)
Next, we provide a condition that is useful in verifying regularity of H (see Definition 2.4) needed for the existence of the timechange in Lemma 3.1 and the uniqueness in Lemma A.3 below.
Proposition A.2
Let \(\mathcal {D} \subset C_0(E)\) dense in \(C_0(E)\) and \(\mathcal {A}:\mathcal {D} \rightarrow C_0(E)\) be linear. Suppose M is a solution on \((\varOmega ,\mathcal {F},\mathbb {P})\) to the RCLLmartingale problem for \((\mathcal {A},\mu _0)\), for some \(\mu _0 \in \mathcal {P}(E)\). Denote by P the law on \(D_E[0,\infty )\) of M. Then, any \(H \in \mathcal {D}\) with \(H \ge 0\) is regular for P.
Proof
Define \(\rho \) as in (3.2) and recall that, by Definition 2.4, (3.7) and (3.8) have to be verified. Set
Since H is continuous and M is RCLL, \(H(M_{\rho })=0\) on \(\{\rho < \infty \}\) and \(\rho _0 \le \rho \), \(\mathbb {P}\)a.s. In particular, \(\rho _0 = \rho \) on \(\{\rho _0 = \infty \}\), and if \(\{\rho _0 < \infty \}\) is a \(\mathbb {P}\)null set, this already establishes the claim. Otherwise, the probability measure \(\tilde{\mathbb {P}}(\,\cdot \,):=\mathbb {P}(\,\cdot \, \{\rho _0 <\infty \})\) is welldefined, \(\rho _0 < \infty \), \(\tilde{\mathbb {P}}\)a.s. and to prove the proposition we only need to show \(\tilde{\mathbb {P}}(\rho _0 \ge \rho )=1\). To do so, on \(\{\rho _0 < \infty \}\) define for any \(t \ge 0\) the random time
Since \(H(M_{\rho _0})=0\) and \(\rho _0 \le \rho \), \(\tilde{\mathbb {P}}\)a.s., it suffices to establish that \(\tilde{\mathbb {P}}\)a.s. for any \(t \ge 0\), \(\tau (t)=0\).
For the proof of the last statement one proceeds as follows: Since H is bounded, \(H(M_\rho )=0\) on \(\{ \rho < \infty \}\), \(\tilde{\mathbb {P}}\)a.s., and by footnote 2, Lemma 3.1 can be applied to the RCLL process \((M_{u+\rho _0})_{u \ge 0}\) on \((\varOmega ,\mathcal {F},\tilde{\mathbb {P}})\) with \(\tilde{\sigma }=1\) and \(\sigma =H\). This yields \(\tilde{\mathbb {P}}\)a.s.,
and \(\tau (t) < \infty \) for any \(t \ge 0\). Denote by \((\mathcal {F}_t)_{t \ge 0}\) the \(\mathbb {P}\)usual augmentation of the filtration generated by M. Then, \(\rho \) and \(\rho _0\) (possibly modified on a \(\mathbb {P}\)null set, see [13, Chap. 4, Cor. 3.13]) are \((\mathcal {F}_t)_{t \ge 0}\)stopping times and thus
shows that also \(\tau (t)+\rho _0\) is a stopping time. By assumption on H, M and \(\mathcal {A}\) the process
is an \((\mathcal {F}^M_t)_{t\ge 0}\)martingale and thus, by [23, Lem. II.67.10], also an \((\mathcal {F}_t)_{t \ge 0}\)martingale. By the optional sampling theorem, for any \(r \ge 0\), \(\mathbb {P}\)a.s.,
or equivalently
Multiplying by \(\mathbb {1}_{\{ \rho _0 \le r \}}\), using \(\{\rho _0 \le r\} \in \mathcal {F}_{\rho _0 \wedge r}\) and taking expectations gives
By assumption \(\mathcal {D} \subset C_0(E)\) is dense and thus separating (see [13, Chap. 3, Ex. 11]) in the terminology of [13, Chap. 3, Sec. 4]. Therefore, by quasileft continuity, [13, Chap. 4, Thm. 3.12],
and so using dominated convergence, boundedness and nonnegativity of H, \(H(M_{\rho _0})=0\) on \(\{\rho < \infty \}\) and setting \(C:=\Vert \mathcal {A} H\Vert \), one estimates
Using (A.2) and Tonelli’s theorem for the first and (A.3) for the second equality yields
and so Gronwall’s lemma implies that the lefthand side of (A.4) is 0 for any \(t \ge 0\). But this implies that \(\tilde{\mathbb {P}}\)a.s., \(\tau (t) = 0\) for all \(t \ge 0\) as desired. \(\square \)
1.1 A.1. Pathwise Uniqueness
To verify that the random times \((\tau (t))_{t\in [0,t_0]}\) solving the differential equation (3.1) are indeed stopping times with respect to the filtration generated by the process M, we show pathwise uniqueness of the timechanged Markov process \(X_t:=M_{\tau (t)}\) for \(t\in [0,t_0]\).
Lemma A.3
Let \(\sigma \) and M be given as in Lemma 3.1. \((\tau (t))_{t \in [0,t_0]}\) is the family of random times from Lemma 3.1 with \(\tau (t): = \tau (t_0)\) for \(t >t_0\) and the timechanged process X is given by \(X _{t} :=M_{\tau (t)}\) for \(t \ge 0\). Suppose M is \((\mathcal {F}_t)\)adapted. Then, the following holds:

(i)
The timechanged process X has RCLL sample paths, \(\mathbb {P}\)a.s.

(ii)
Any RCLL process \(\tilde{X}\) satisfying
$$\begin{aligned} \tilde{X}_t = M_{\int _0^t \sigma (u,\tilde{X}_u) \,\mathrm {d}u},\quad t\in [0,\infty ),\, \mathbb {P}\text {}a.s., \end{aligned}$$(A.5)is indistinguishable from X.

(iii)
The random times \((\tau (t))_{t \in [0,t_0]}\) are \((\mathcal {F}_t)\)stopping times.
Proof

(i)
Since M has RCLL sample paths and \(\tau \) is nondecreasing and absolutely continuous by Lemma 3.1, the timechanged process \(X_{}\) has RCLL sample paths.

(ii)
Let \(\tilde{X}\) be an RCLL process satisfying equation (A.5). Define the random time
$$\begin{aligned} \tilde{\rho } := t_0\wedge \inf \big \{ t \ge 0 : H(\tilde{X}_t) = 0 \big \} \end{aligned}$$and set
$$\begin{aligned} \tilde{\tau }(s) := \int _0^s \sigma (u,\tilde{X}_u ) \,\mathrm {d}u, \quad s\in [0,\infty ). \end{aligned}$$Notice that the integral is welldefined since \(\sigma \) is bounded on compacts and \(\tilde{X}\) is RCLL. Since \(X_t=M_{\tau (t)}\) and \(\tilde{X}_t= M_{\tilde{\tau }(t)}\), to verify that X and \(\tilde{X}\) are indistinguishable, it is sufficient to show that \(\tau (t)= \tilde{\tau }(t)\) for every \(t\in [0,\infty )\), \(\mathbb {P}\)a.s.
By [21, Lem. 3.31] \(\tilde{\tau }\) is absolutely continuous with weak derivative \(\tilde{\tau }'(u) = \sigma (u, \tilde{X}_u )\) for \(u \in [0,\infty )\) and invertible on \([0,\tilde{\rho }\wedge t_0]\) by the definition of \(\tilde{\rho }\). The inverse of \(\tilde{\tau }\) is denoted by \(\tilde{\mathcal {T}}\) with domain \([0,\tilde{\tau }(\tilde{\rho }\wedge t_0)]\). Because \(\tilde{\mathcal {T}}\) is also strictly increasing and absolutely continuous, the chain rule (see [21, Thm. 3.44]) gives
$$\begin{aligned} 1 = \frac{\,\mathrm {d}}{\,\mathrm {d}t} \tilde{\tau } (\tilde{\mathcal {T}}(t)) =\sigma (\tilde{\mathcal {T}}(t), \tilde{X}_{\tilde{\mathcal {T}}(t)})\frac{\,\mathrm {d}}{\,\mathrm {d}t}\tilde{\mathcal {T}}(t)\quad \text {for almost all } t \in {[}0,\tilde{\tau }(\tilde{\rho }\wedge t_0)]. \end{aligned}$$Combining this with fundamental theorem of calculus (see [21, Thm. 3.30]), one has that \(\tilde{\mathcal {T}}\) satisfies the integral equation
$$\begin{aligned} \tilde{\mathcal {T}}(t) = \int _0^t \sigma (\tilde{\mathcal {T}}(s), \tilde{X}_{\tilde{\mathcal {T}}(s)})^{1} \,\mathrm {d}s,\quad t \in {[}0,\tilde{\tau }(\tilde{\rho }\wedge t_0)]. \end{aligned}$$Moreover, notice that \(M_t= M_{\tilde{\tau }(\tilde{\mathcal {T}}(t))}=\tilde{X}_{\tilde{\mathcal {T}}(t)}\) for \(t \in [0,\tilde{\tau }(\tilde{\rho }\wedge t_0)]\). Therefore, \(\mathcal {T}(t) = \tilde{\mathcal {T}}(t)\) for \(t \in [0,\tilde{\tau }(\tilde{\rho }\wedge t_0)\wedge \tau (t_0)]\) since the solution to this equation is unique on \([0,\tau (t_0)]\), see (3.4). Furthermore, we have \(\tilde{\tau }(\tilde{\rho }\wedge t_0) \le \rho \wedge \tilde{\tau } (t_0)\) since \(t < \tilde{\tau }(\tilde{\rho }\wedge t_0)\) implies \(\tilde{\mathcal {T}}(t) < \tilde{\rho }\wedge t_0\) and thus \(t <\rho \wedge \tilde{\tau } (t_0)\), where we recall \(\rho \) from (3.2) and that (3.7), (3.8) holds. In conclusion, \(\mathcal {T}(t) =\tilde{\mathcal {T}}(t)\) for \(t \in [0,\tilde{\tau }(\tilde{\rho }\wedge t_0)]\), which leads to \(\tilde{\tau }(s) = \tau (s)\) for \( s \in [0,\tilde{\rho }\wedge t_0]\).
To see \(\tilde{\tau }(s) = \tau (s)\) for \(s>\tilde{\rho }\wedge t_0\), we first observe that \(1/(H(M_s)\vee \varepsilon )\) is bounded for every \(\varepsilon >0\) and \(\tilde{\sigma }\) is bounded on compacts by Assumption 2.6. Applying a change of variables ([21, Cor. 3.57]) and using monotone convergence gives
$$\begin{aligned} C t \ge \lim _{\varepsilon \rightarrow 0} \int _0^t \frac{\sigma (s,\tilde{X}_s)}{H(\tilde{X}_s) \vee \varepsilon } \,\mathrm {d}s = \lim _{\varepsilon \rightarrow 0} \int _0^{\tilde{\tau }(t)} \frac{1}{H(M_s) \vee \varepsilon } \,\mathrm {d}s = \int _0^{\tilde{\tau }(t)} \frac{1}{H(M_s)} \,\mathrm {d}s, \end{aligned}$$which ensures \(\tilde{\tau }(t) \le \rho \) for all \(t \ge 0\). Assuming \(\tilde{\rho }< t_0\), there exist \(\{t_n\}_{n \in \mathbb {N}} \subset [\tilde{\rho },t_0]\) with \(t_n \downarrow \tilde{\rho }\) and \(H(\tilde{X}_{t_n}) = 0\) and so \(\rho \le \tilde{\tau }(\tilde{\rho })\) by (A.5) and (3.7), (3.8). In this case, \(\tilde{\tau }(t) = \tilde{\tau }(\tilde{\rho }) = \rho =\tau (t) \) for all \(t \ge \tilde{\rho }\). Assuming \(\tilde{\rho } \ge t_0\), we have \(\tilde{\tau }(t) = \tilde{\tau }(t_0)\) for all \(t \ge t_0\) due to \(\sigma (t,\cdot ) = 0\) for \(t > t_0\) and in particular \(\tilde{\tau }(t) = \tilde{\tau }(t_0)=\tau (t)\) for \(t \ge t_0\).

(iii)
In order to apply a result from [13], we consider the twodimensional process \(Y_t:=(t,M_t)\) and the timechanged process \((t, X_t)\) for \(t\in [0,T]\). Hence, [13, Chap. 6, Thm. 2.2 (b)] implies that \(\tau (t)\) is a stopping time with respect to the usual augmentation of the filtration generated by M, and thus also an \((\mathcal {F}_t)\)stopping time, where we keep in mind that the first component of Y generates a trivial filtration.
\(\square \)
Corollary A.4
Let \(\sigma \), M and \(\tilde{X}\) be given as in Lemma A.3 and denote by P the law (on \(D_E[0,\infty )\)) of M under \(\mathbb {P}\). Then, the law of \(\tilde{X}\) under \(\mathbb {P}\) is uniquely determined by P and \(\sigma \).
Proof
By Lemma A.3(ii), the law of \(\tilde{X}\) is identical to the law of X under \(\mathbb {P}\). To show explicitly that the latter is uniquely determined by P and \(\sigma \), one proceeds as follows: Let \(n \in \mathbb {N}\), \(t_1,\ldots ,t_n \in [0,\infty )\), \(B_1,\ldots ,B_n \in \mathcal {B}(E)\) and let \(\pi _1:D_E[0,\infty )\times D_E[0,\infty ) \rightarrow D_E[0,\infty )\) be the projection map on the first component. We have seen in the Proof of Lemma A.3 that there exist a unique solution to the timechange equation for \(\mathbb {P}\)a.e. sample path \(M(\omega )\). Hence, as in the proof of [13, Chap. 6, Lem. 2.1], the map
is Borel measurable and the set
is in \(\mathcal {B}(D_E[0,\infty )^2)\). Then, [13, Appendix 11, Thm. 11.3] implies that \(\pi _1 C\) is in the Pcompletion of \(\mathcal {B}(D_E[0,\infty ))\) and thus
is indeed uniquely determined by P and \(\sigma \). \(\square \)
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Döring, L., Gonon, L., Prömel, D.J. et al. Existence and Uniqueness Results for TimeInhomogeneous TimeChange Equations and Fokker–Planck Equations. J Theor Probab 34, 173–205 (2021). https://doi.org/10.1007/s1095901900969y
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DOI: https://doi.org/10.1007/s1095901900969y
Keywords
 Fokker–Planck equation
 Forward Kolmogorov equation
 Markov process
 Martingale problem
 Random timechange
 Timedependent operator