Uniqueness of the nonlinear Schrödinger equation driven by jump processes

Abstract

In a recent paper by the first two authors, existence of martingale solutions to a stochastic nonlinear Schrödinger equation driven by a Lévy noise was proved. In this paper, we prove pathwise uniqueness, uniqueness in law and existence of strong solutions to this problem using an abstract uniqueness result of Kurtz.

Appendices

Appendix A: Uniqueness of the stochastic integral

Let X and E be two separable Banach spaces. Later on we will take X to be one of the spaces E or $L^p(S,\nu ,E)$. Let ${\mathfrak {A}}=(\Omega ,{{ {\mathcal {F}} }},({{ {\mathcal {F}} }}_t)_{t\in [0,T]},{{\mathbb {P}}})$ be an arbitrary filtered probability space and $\eta$ be a Poisson random measure defined over ${\mathfrak {A}}$. Let ${\mathcal {N}}(\Omega \times [0,T];X)$ be the space of (equivalence classes of) progressively measurable functions $\xi :\Omega \times [0,T]\rightarrow X$.

For $q\in (1,\infty )$ we set

$$\begin{aligned} \;\; {\mathcal {N}}^q(\Omega \times [0,T],{\mathcal {F}};X)= & {} \left\{ \xi \in {\mathcal {N}}(\Omega \times [0,T],{\mathcal {F}};X): \; \int _0^\infty \vert \xi (t)\vert ^q\,dt<\infty \text{ a.s. } \right\} , \nonumber \\ \end{aligned}$$

(A.1)

$$\begin{aligned} \;\; \;\; {\mathcal {M}}^q(\Omega \times [0,T],{\mathcal {F}};X)= & {} \left\{ \xi \in {\mathcal {N}}(\Omega \times [0,T],{\mathcal {F}};X):{\mathbb {E}}\int _ 0^\infty \vert \xi (t)\vert ^q\,dt<\infty \right\} . \nonumber \\ \end{aligned}$$

(A.2)

Let E be a space of martingale type p, put $X=L^p(S, \nu ;E)$, and let $\xi \in {{ {\mathcal {N}} }}^ p(\Omega \times [0,T],{{ {\mathcal {F}} }};X)$. In particular, $\xi : \Omega \times [0,T]\times S\rightarrow E$ is a progressively measurable process such that ${{\mathbb {P}}}$-a.s.

$$\begin{aligned} \int _0^T \int _S\vert \xi (r,z)\vert _E^p\, \nu (dz)\,dr< & {} \infty . \end{aligned}$$

(A.3)

Let us consider the law of the triplet $(\eta ,\xi ,I)$, where I is the Itô integral of $\xi$ with respect to $\eta$ as defined on page 5. It was proved in Theorem 2.4 of [6] that the law is unique in case

$$\begin{aligned} {\mathbb {E}}\int _0^T \int _S\vert \xi (r,z)\vert _E^p\, \nu (dz)\,dr< & {} \infty . \end{aligned}$$

In this “Appendix” we want to extend this result to all progressively processes $\xi$ satisfying only (A.3). However, before stating the Theorem we want to define uniqueness in law. Here, it is important that evaluating the integral, the underlying probability space can be changed with having any consequence for the laws.

Theorem A.1

Let $(\Omega _i,{{ {\mathcal {F}} }}_i,{{\mathbb {P}}}_i)$, $i=1,2$, be two probability spaces and $({{ {\mathcal {F}} }}^ i_t)_{t\in [0,T]}$ a filtration of $(\Omega _i,{{ {\mathcal {F}} }}_i)$. Assume that $\{(\xi _i,\eta _i),n\in {\mathbb {N}}\}$, $i=1,2$, are two $L ^ p([0,T];L^p(S,\nu ,E)) \times {{M}_{{\bar{{\mathbb {N}}}}}( \{S_n\times (0,T]\})}$ valued random variables defined on $(\Omega _i,{{ {\mathcal {F}} }}_i,{{ {\mathcal {F}} }}^ i_t,{{\mathbb {P}}}_i)$, $i=1,2$, respectively. Assume that $\eta _1$ is a time homogeneous Poisson random measure over $(\Omega _1,{{ {\mathcal {F}} }}_1,({{ {\mathcal {F}} }}^ 1_t)_{t\in [0,T]},{{\mathbb {P}}}_1)$ with intensity $\nu$. Furthermore, assume that $\xi _1\in {{ {\mathcal {N}} }}(\Omega _1\times [0,T];L^p(S,\nu ,E))$ with respect to $({{ {\mathcal {F}} }}^ 1_t)_{t\in [0,T]}$.

Let

$$\begin{aligned} I _i (t): =I(\xi _i,\eta _i)(t)= \int _0 ^ t \int _S \xi _i(s,z)\, {\tilde{\eta }}_i(dz,ds). \end{aligned}$$

If ${{ {\mathcal {L}} }}aw( (\xi _1,\eta _1))={{ {\mathcal {L}} }}aw((\xi _2,\eta _2))$ on

$$\begin{aligned} L ^ p([0,T];L ^ p(S,\nu ;E)) \times {{M}_{{\bar{{\mathbb {N}}}}}( \{S_n\times (0,T]\})}, \end{aligned}$$

then ${{ {\mathcal {L}} }}aw( (I _1,\xi _1,\eta _1))={{ {\mathcal {L}} }}aw((I _2,\xi _2,\eta _2))$ on

$$\begin{aligned} L^p([0,T];E)\cap {\mathbb {D}}([0,T];E)\times L ^ p({{\mathbb {R}}}_+;L ^ p(S,\nu ;E))\times {{M}_{{\bar{{\mathbb {N}}}}}( \{S_n\times (0,T]\})}. \end{aligned}$$

Proof

In fact, Theorem A.1 follows from Theorem 2.4 in [6] by localization. First, for a fixed $R>0$ let us introduce the stopping times

$$\begin{aligned} \tau ^R_i=\inf _{t>0} \left\{ \int _0^t|\xi (s)|_X^p \, ds \ge R\right\} . \end{aligned}$$

Put $\xi ^R_i := 1_{[0,\tau ^R_i]} \, \xi$. Observe, using the shifted Haar projection defined in (B.2) one obtains a sequence of simple functions $\{{\mathfrak {h}} ^ s_n \xi ^R_i:n\in {\mathbb {N}}\}$ such that ${{\mathbb {P}}}^i$-a.s. ${\mathfrak {h}} ^ s_n \xi ^R_i\rightarrow \xi ^R_i$ in $L^p([0,T];E)$. In addition

$$\begin{aligned} {\mathbb {E}}\int _0^T \int _S\vert \xi _i^R(r,z)\vert _E^p\, \nu (dz)\,dr\le R. \end{aligned}$$

Thus, Theorem 2.4 in [6] is applicable and we have ${{\mathcal {L}}aw}( (I ^R_1,\xi ^R _1,\eta _1))={{\mathcal {L}}aw}((I ^R_2,\xi ^R _2,\eta _2))$ on

$$\begin{aligned} ({\mathbb {D}}([0,T];X)\cap L^p([0,T];X))\times L^p([0,T];L^p(Z,\nu ;E))\times {{M}_{{\bar{{\mathbb {N}}}}}( \{S_n\times (0,T]\})}, \end{aligned}$$

where

$$\begin{aligned} I^R_i(t) = \int _0 ^ t \int _S \xi ^R_i(s,z)\, {\tilde{\eta }}_i(ds,dz), \quad t\in [0,T]. \end{aligned}$$

(A.4)

Let

$$\begin{aligned} A_i^R:= \left\{ \omega \in \Omega _i: \int _0^T \left| \xi _i(s) \right| _{L^p(Z,\nu ;E)}^p \, ds\le R\right\} . \end{aligned}$$

Then, first, on $A^R_i$, $\xi _i^R=\xi _i$, secondly, $A^ {R_1}\supset A^ {R_2}$ for $R_1>R_2$, and, thirdly, by Lemma 1.14 [15], $\lim _{R\rightarrow \infty } {{\mathbb {P}}}\left( A^R_i\right) ={{\mathbb {P}}}\left( \Omega \right) =1$. Take a set

$$\begin{aligned}&B_1\times B_2\times B_3 \in {{ {\mathcal {B}} }}({\mathbb {D}}([0,T];X)\cap L^p([0,T];B)) \\&\quad \times {{ {\mathcal {B}} }}( L^p([0,T];L^p(Z,\nu ;E)))\times {{ {\mathcal {B}} }}( {{M}_{{\bar{{\mathbb {N}}}}}( \{S_n\times (0,T]\})}). \end{aligned}$$

Since $\xi ^ R_i\le \xi _i$, we have by the dominated convergence Theorem

$$\begin{aligned}&{{\mathbb {P}}}_1\left( (u_1,\xi _1,\eta )\in B_1\times B_2\times B_3\right) \\&\quad = \lim _{R\rightarrow \infty } {{\mathbb {P}}}_1\left( (u_1,\xi _1,\eta )\in B_1\times B_2\times B_3, \tau _1^R>T \right) \\&\quad = \lim _{R\rightarrow \infty } {{\mathbb {P}}}_1\left( (u ^R_1,\xi ^R_1,\eta )\in B_1\times B_2\times B_3\right) \\&\quad = \lim _{R\rightarrow \infty } {{\mathbb {P}}}_2\left( (u ^R_2,\xi ^R_2,\eta )\in B_1\times B_2\times B_3\right) \\&\quad = \lim _{R\rightarrow \infty } {{\mathbb {P}}}_2\left( (u ^R_2,\xi ^R_2,\eta )\in B_1\times B_2\times B_3, \tau _1^R>T \right) \\&\quad = {{\mathbb {P}}}_2\left( (u _2,\xi _2,\eta )\in B_1\times B_2\times B_3\right) . \end{aligned}$$

Now, the assertion follows. $\square$

Appendix B: the Haar projection

1.1 The Haar projection onto $L^ q$-spaces

For $n\in {\mathbb {N}}$, let $\Pi ^n=\{ s^n_0=0<s^n_1<\cdots <s^n_{2^n}\}$ be a partition of the interval [0, T] defined by $s_j ^ n=j\,2 ^ {-n}T$, $j=1,\ldots , 2^n$. Each interval of the form $(s_{j-1} ^ n,s_j ^ n]$, where $n\in {\mathbb {N}}$ and $j=1,\ldots , 2^n$ is called a dyadic interval. For $n\in {\mathbb {N}}$, the $j^ {th}$ element, for $j=1,\ldots , 2^n$, of the Haar system of order n is the indicator function of the interval $(s_{j-1}^n,s_j^{n}]$, i.e. $1_{(s_{j-1}^{n},s_j^{n}]}$. First, given a function $x:[0,T]\rightarrow Y$, Y a Banach space, let us define the averaging operator $\iota _{j,n}: L ^ p(0,T,Y)\rightarrow Y$ over the interval $(s^n_{j-1},s^n_{j}]$ by

$$\begin{aligned} \iota _{j,n}(x) := \frac{1}{s^n_{j}-s^n_{j-1}}\; \int _{s^ n_{j-1}}^{s ^ n_{j}} x(s)\,ds, \; x\in L ^ p([0,T],Y). \end{aligned}$$

(B.1)

For $n\in {\mathbb {N}}$, let ${\mathfrak {h}} ^ s_n: L ^ p([0,T],Y)\rightarrow L ^ p([0,T],Y)$ be the shifted Haar projection of order n, i.e.

$$\begin{aligned} {\mathfrak {h}} ^ s_n x =\sum _{j=1} ^{2^n -1} 1_{(s^n_j,s^n_{j+1}]}\otimes \iota _{j,n}(x), \;\; x\in L ^ p([0,T];Y), \end{aligned}$$

(B.2)

where we put ${\iota }_{0,n}=0$ for every $n\in {\mathbb {N}}$. In the above, for $f\in L ^ p([0,T],{\mathbb {R}})$ and $y\in Y$, by $f\otimes y$ we mean an element of $L ^ p([0,T],Y)$ defined by $[0,T]\ni t\mapsto f(t) y\in Y$. For completeness, let us cite the following results taken from [6, Appendix B, Theorem B.2].

Proposition B.1

The following holds:

• (i) For any $n\in {\mathbb {N}}$, the shifted Haar projection ${\mathfrak {h}} ^ s_n:L ^ p([0,T] ;Y)\rightarrow L ^ p([0,T];Y)$ is a continuous operator.

• (ii) For all $x\in L ^ p([0,T] ;Y)$, ${\mathfrak {h}} ^ s_n x\rightarrow x$ in $L ^ p([0,T] ;Y)$.

Remark B.1

Observe, for any $\xi \in {{ {\mathcal {N}} }}_p([0,T] ;X)$, the process $[0,T ]\ni t\mapsto {\mathfrak {h}} ^ s_n x(t)$ is simple, left continuous and predictable and the sequence $\{ {\mathfrak {h}} ^ s_n \xi :n\in {\mathbb {N}}\}$ converges to $\xi$ in $L ^p([0,T];Y)$.

Appendix C: Polish measure spaces

Lemma C.1

Let $(S,{{ {\mathcal {S}} }})$ be a Polish space and the family $\{S_n\in {{ {\mathcal {S}} }}\}$ satisfy $S_n\uparrow S$. Then $(M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),{{ {\mathcal {M}} }}_{\bar{\mathbb {N}}}(\{S_n\}))$ is a Polish space.

Let $(S,{{ {\mathcal {S}} }})$ be a Polish space and let $S_n\in {{ {\mathcal {S}} }}$ satisfy $S_n\uparrow S$. Then there exists a metric $\varrho$ on S such that $(S,\varrho )$ is a complete separable metric space, $\mathscr {B}(S,\varrho )={{ {\mathcal {S}} }}$ and $S_n$ is closed for every $n\in \mathbb {N}$, see, e.g. [16, (13.5) p. 83]. Consider the Lévy-Prokhorov metric on $M_+(S,\varrho )$

$$\begin{aligned} \pi (\mu ,\nu )=\inf \,\{\varepsilon >0:\mu (A)\le \nu (A^\varepsilon )+\varepsilon ,\,\nu (A)\le \mu (A^\varepsilon )+\varepsilon ,\,\forall A\in {{ {\mathcal {S}} }}\} \end{aligned}$$

where $A^\varepsilon =\{x\in S:\exists a\in A,\,\varrho (x,a)<\varepsilon \}$. Then $(M_+(S,\varrho ),\pi )$ is a complete separable metric space and $\pi (\mu _n,\mu )\rightarrow 0$ iff

$$\begin{aligned} \int _Sf\,d\mu _n\rightarrow \int _Sf\,d\mu ,\qquad \forall f\in C_b(S,\varrho ), \end{aligned}$$

see, e.g., [3, pp. 72–73], where the proof for probability measures can be quite easily adapted to finite non-negative measures.

Lemma C.2

The $\sigma$-algebra ${\mathcal {M}}_+(S)$ on $M_+(S)$ generated by the mappings $\mu \mapsto \mu (A)$, $A\in {{ {\mathcal {S}} }}$ coincides with the Borel $\sigma$-algebra ${\mathscr {B}}(M_+(S),\pi )$.

Proof

The mapping $\mu \mapsto \mu (A)$ is upper semicontinuous on $(M_+(S),\pi )$ for every $A\subseteq S$ closed and lower semicontinuous for every $A\subseteq S$ open, hence Borel measurable for every $A\in {{ {\mathcal {S}} }}$. In particular, ${\mathcal {M}}_+(S)\subseteq {\mathscr {B}}(M_+(S),\pi )$. On the other hand, let ${\mathcal {G}}$ be a countable basis of open sets in $(S,\varrho )$ closed under finite unions. Then

$$\begin{aligned} \{\mu :\,\pi (\mu ,\theta )<r\}= & {} \bigcup _{\varepsilon \in \mathbb {Q}\cap (0,r)}\bigcap _{A\in \mathcal G}\{\mu :\,\mu (A)\le \theta (A^\varepsilon ) \\&+\varepsilon ,\,\theta (A)\le \mu (A^\varepsilon )+\varepsilon \}\in \mathcal M_+(S). \end{aligned}$$

Hence open balls in $(M_+(S),\pi )$ belong to ${\mathcal {M}}_+(S)$ and since $(M_+(S),\pi )$ is separable, every open set is a countable union of open balls. Consequently, every open set in $(M_+(S),\pi )$ belong to ${\mathcal {M}}_+(S)$, hence $\mathscr {B}(M_+(S),\pi )\subseteq {\mathcal {M}}_+(S)$. $\square$

Lemma C.3

The set of integer-valued measures $M_{\mathbb {N}}(S)$ is closed in $(M_+(S,\varrho ),\pi )$.

Proof

Let $\pi (\mu _n,\mu )\rightarrow 0$, $\mu _n$ be integer-valued and $k<\mu (A)<k+1$ for some integer k and some $A\in {{ {\mathcal {S}} }}$. By regularity, we can find a compact $C\subseteq A$ such that $k<\mu (C)\le \mu (A)<k+1$ and $\delta >0$ such that $k<\mu (C)-\delta$ and $\mu (C^{2\delta })+\delta <k+1$. If $\pi (\mu _n,\mu )<\delta$ then

$$\begin{aligned} k<\mu (C)-\delta \le \mu _n(C^\delta )\le \mu (C^{2\delta })+\delta <k+1, \end{aligned}$$

which cannot happen as $\mu _n(C^\delta )$ is an integer. $\square$

Lemma C.4

$(M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),{{ {\mathcal {M}} }}_{{\bar{{\mathbb {N}}}}}(\{S_n\}))$ is a Polish space.

Proof

Consider the metric

$$\begin{aligned} \rho (\mu ,\nu )=\sum _{n=1}^\infty 2^{-n}\min \,\{1,\pi (\mu (\cdot \cap S_n),\nu (\cdot \cap S_n))\},\qquad \mu ,\nu \in M_{{\bar{{\mathbb {N}}}}}(\{S_n\}). \end{aligned}$$

Then $(M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),\rho )$ is a metric space and the mapping

$$\begin{aligned} I:(M_{{\bar{{\mathbb {N}}}}}(\{S_n\},\rho )\rightarrow (M_{\mathbb {N}}(S),\pi )^{\mathbb {N}}:\mu \mapsto (\mu (\cdot \cap S_n))_{n\in \mathbb {N}} \end{aligned}$$

is a homeomorphism onto a closed set in $(M_{\mathbb {N}}(S),\pi )^{\mathbb {N}}$, hence $(M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),\rho )$ is a complete separable metric space by Lemma C.3. To show closedness, let $\lim _{j\rightarrow \infty }\pi (\mu _j(\cdot \cap S_n),\theta _n)=0$ for every $n\in \mathbb {N}$, and let $m<k$. If $f\in C_b(S_m)$ and we extend f by zero on $S{\setminus } S_m$ then $f\in C_b(S)$ since $S_m$ is clopen. So $\partial S_n=\emptyset$,

$$\begin{aligned} \theta _n(S{\setminus } S_n)=\lim _{j\rightarrow \infty }\mu _j((S{\setminus } S_n)\cap S_n)=0,\qquad \forall n\in \mathbb {N}, \end{aligned}$$

and

$$\begin{aligned} \int _Sf\,d\theta _k=\lim _{j\rightarrow \infty }\int _Sf\,d\mu _j(\cdot \cap S_k)=\lim _{j\rightarrow \infty }\int _Sf\,d\mu _j(\cdot \cap S_m)=\int _Sf\,d\theta _m \end{aligned}$$

so $\theta _m(\cdot )=\theta _k(\cdot \cap S_m)$. In particular, $\theta _n(A)\uparrow$ for every $A\in {{ {\mathcal {S}} }}$ and $\theta (A)=\lim _n\theta _n(A)$ is a $\sigma$-additive, $\bar{\mathbb {N}}$-valued measure on ${{ {\mathcal {S}} }}$ and $\theta _m(\cdot )=\theta (\cdot \cap S_m)$.

Now, by Lemma C.2, the mapping $\mu \mapsto \mu (A\cap S_k)$ is ${\mathscr {B}}(M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),\rho )$ measurable for every $k\in \mathbb {N}$ and every $A\in {{ {\mathcal {S}} }}$ since I is Borel measurable. Hence ${{ {\mathcal {M}} }}_{{\bar{{\mathbb {N}}}}}(\{S_n\})\subseteq {\mathscr {B}}(M_{\bar{\mathbb {N}}}(\{S_n\}),\rho )$. On the other hand, the mapping

$$\begin{aligned} (M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),{{ {\mathcal {M}} }}_{{\bar{{\mathbb {N}}}}}(\{S_n\}))\rightarrow (M_{\mathbb {N}}(S),{\mathscr {B}}(M_{\mathbb {N}}(S))):\mu \mapsto \mu (\cdot \cap S_k) \end{aligned}$$

is measurable for every $k\in \mathbb {N}$ by Lemma C.2. So, if $\theta \in M_{{\bar{{\mathbb {N}}}}}(\{S_n\})$ is fixed, the mapping

$$\begin{aligned} (M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),{{ {\mathcal {M}} }}_{{\bar{{\mathbb {N}}}}}(\{S_n\}))\rightarrow \mathbb {R}:\mu \mapsto \pi (\mu (\cdot \cap S_k),\theta (\cdot \cap S_k)) \end{aligned}$$

is measurable for every $k\in \mathbb {N}$. Consequently, the mapping

$$\begin{aligned} (M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),{{ {\mathcal {M}} }}_{{\bar{{\mathbb {N}}}}}(\{S_n\}))\rightarrow \mathbb {R}:\mu \mapsto \rho (\mu ,\theta ) \end{aligned}$$

is measurable. Since $(M_{{\bar{{\mathbb {N}}}}}(\{S_n\}),\rho )$ is a separable metric space and every open set is a countable union of open balls, we conclude that ${\mathscr {B}}(M_{\bar{\mathbb {N}}}(\{S_n\}),\rho )\subseteq {{ {\mathcal {M}} }}_{{\bar{{\mathbb {N}}}}}(\{S_n\})$. $\square$