Modern Stochastics: Theory and Applications

The main results are formulated on the product Brownian–Poisson filtration. Under the additional nondegeneracy condition $d=k$, with ${\sigma _{t}}$ invertible for Lebesgue-a.e. t and measurable $t\mapsto {\sigma _{t}^{-1}}$, the usual augmentation of the natural filtration of X coincides with it: the jumps of X recover the Poisson coordinate, and the Brownian coordinate is recovered from ${\int _{0}^{t}}{\sigma _{s}^{-1}}\hspace{1.67pt}\hspace{-1.67pt}\mathrm{d}{W_{s}}$ after subtracting drift and jumps.

For each fixed $r\in [0,T]$, Consequently, for every bounded Borel function f and each fixed $r\in [0,T]$,

Fix $r\in [0,T]$ and set ${E_{n}}:=\{z\in {\mathbb{R}_{0}^{d}}:{n^{-1}}\lt |z|\le n\}$. Let ${X_{r}^{(n)}}$ be the truncated Lévy–Itô representative obtained by retaining only jumps with marks in ${E_{n}}$ in the jump part. For every n, the set $[0,T]\times {E_{n}}$ has finite ν-measure. The finite-window add-one identity holds for the chosen representatives: for $\nu \otimes \mathbb{P}$-a.e. $(t,z,\omega )$, after intersecting the corresponding countable family of conull sets. Indeed, on a finite window the identity is pathwise for the Poisson configuration, and Lemma 2.2 transfers it to the fixed canonical representatives.

The truncated variables converge to the fixed Lévy–Itô representative ${X_{r}}$ in probability. The omitted small compensated part has squared ${L^{2}}(\mathbb{P})$ norm equal to and the omitted large-jump part is absent with probability tending to one because $\nu ([0,r]\times \{|z|\gt n\})\downarrow 0$. Choose a subsequence ${n_{j}}$ such that ${X_{r}^{({n_{j}})}}\to {X_{r}}$ $\mathbb{P}$-a.s., and let A be the exceptional set. By Lemma 2.2, Thus, on a common $\nu \otimes \mathbb{P}$-conull set, Passing to the limit in (2.1) and using ${\mathbf{1}_{{E_{{n_{j}}}}}}(z)\to 1$ for every $z\ne 0$ gives The deterministic drift and Brownian coordinate are unchanged by ${\varepsilon _{t,z}^{+}}$, so this is the add-one identity for the full additive path. The assertion for bounded Borel functions follows by composition. □

Let ${\mathcal{C}_{\mathrm{prod}}}$ be the linear span of bounded random variables of the form where $m,\ell \ge 0$, ${h_{i}}\in {L^{2}}([0,T];{\mathbb{R}^{k}})$ are deterministic, each ${\rho _{j}}$ is a bounded Borel function supported on a set of finite ν-measure, and $\varphi \in {C_{b}^{1}}({\mathbb{R}^{m+\ell }})$. Throughout, ${C_{b}^{1}}$ means bounded with bounded first derivative. If vector-valued mark integrals are used, each component is listed as a separate coordinate, so the dimension of φ is the total number of displayed scalar coordinates. Here ${X_{T}^{(n)}}$ denotes the Lévy–Itô variable obtained by retaining marks ${n^{-1}}\lt |z|\le n$ in both jump integrals and leaving the drift and Brownian terms unchanged, as in Lemma 2.5. Thus $f({X_{T}^{(n)}})$, $f\in {C_{b}^{1}}({\mathbb{R}^{d}})$, belongs to the core by expressing the retained jump part through bounded mark integrals.

Let We denote by ${\mathbb{D}_{B,N}^{1,2}}$ the completion, modulo equality in ${L^{2}}(\mathbb{P})$, of the core ${\mathcal{C}_{\mathrm{prod}}}$ from Definition 2.6 under this graph norm. Thus every element is represented by an ${L^{2}}$ limit F, a Brownian derivative ${D^{B}}F\in {L^{2}}([0,T]\times \Omega ;{\mathbb{R}^{k}},\hspace{-1.67pt}\mathrm{d}t\otimes \mathbb{P})$, and a jump difference $\Psi F\in {L^{2}}(\nu \otimes \mathbb{P})$ obtained as graph-norm limits of core approximants. The closedness statement for the raw Poisson add-one operator, recalled in Lemma 2.8, identifies this limit with $F({\varepsilon _{t,z}^{+}}\omega )-F(\omega )$ for a Borel representative of F. The pair $({D^{B}},\Psi )$ is jointly closable on the core: if ${F_{n}}\to 0$ in ${L^{2}}(\mathbb{P})$ and ${D^{B}}{F_{n}}\to U$, $\Psi {F_{n}}\to V$ in the two derivative spaces, then the closability of each component gives $U=0$ and $V=0$.

The operator is closable. Its closed domain is understood with the graph norm induced by this operator. If ${F_{m}}$ belongs to this closed domain, ${F_{m}}\to F$ in ${L^{2}}(\mathbb{P})$, and $\Psi {F_{m}}\to U$ in ${L^{2}}(\nu \otimes \mathbb{P})$, then F belongs to the closed domain and admits a Borel representative, still denoted by F, such that Consequently the Ψ component of the graph-norm completion in Definition 2.7 is the raw add-one difference of the chosen representative.

First consider a core sequence. Let ${G_{m}}\in {\mathcal{C}_{\mathrm{prod}}}$ satisfy ${G_{m}}\to G$ in ${L^{2}}(\mathbb{P})$ and $\Psi {G_{m}}\to V$ in ${L^{2}}(\nu \otimes \mathbb{P})$. Choose a subsequence ${m_{j}}$ such that ${G_{{m_{j}}}}\to G$ $\mathbb{P}$-a.s. and, after passing to a further subsequence if necessary, $\Psi {G_{{m_{j}}}}\to V$ $\nu \otimes \mathbb{P}$-a.e. Let A be the exceptional $\mathbb{P}$-null set for the first convergence. By the Mecke null-set transfer in Lemma 2.2, ${\varepsilon _{t,z}^{+}}\omega \notin A$ for $\nu \otimes \mathbb{P}$-a.e. $(t,z,\omega )$. Hence, on a common $\nu \otimes \mathbb{P}$-conull set, For core variables, ${\Psi _{t,z}}{G_{{m_{j}}}}={G_{{m_{j}}}}({\varepsilon _{t,z}^{+}}\omega )-{G_{{m_{j}}}}(\omega )$ pointwise outside the fixed representative null set. Taking limits along the subsequence gives If $G=0$ and ${G_{m}}\to 0$, this identity gives $V=0$, so Ψ is closable. It also identifies the graph limit of every core graph-convergent sequence with the raw add-one representative.

Now let ${F_{m}}$ be a sequence in the closed domain such that ${F_{m}}\to F$ in ${L^{2}}(\mathbb{P})$ and $\Psi {F_{m}}\to U$ in ${L^{2}}(\nu \otimes \mathbb{P})$. For each m, choose ${G_{m,j}}\in {\mathcal{C}_{\mathrm{prod}}}$ with ${G_{m,j}}\to {F_{m}}$ and $\Psi {G_{m,j}}\to \Psi {F_{m}}$ in graph norm. Choose $j(m)$ so large that Then the core diagonal sequence ${G_{m}}:={G_{m,j(m)}}$ satisfies ${G_{m}}\to F$ in ${L^{2}}(\mathbb{P})$ and $\Psi {G_{m}}\to U$ in ${L^{2}}(\nu \otimes \mathbb{P})$. Applying the core case just proved identifies U with the raw add-one difference of a Borel representative of F, and shows that F belongs to the closed domain. □

Let $\mathcal{C}$ denote the class of functions $\Phi :{\mathbb{R}_{+}}\to \mathbb{R}$ such that Φ is continuous on ${\mathbb{R}_{+}}$, convex, of class ${C^{2}}$ on $(0,\mathrm{\infty })$, and either affine or else ${\Phi ^{\prime \prime }}(u)\gt 0$ for all $u\gt 0$ and $1/{\Phi ^{\prime \prime }}$ is concave on $(0,\mathrm{\infty })$.

For convex Φ, Jensen’s inequality gives ${\operatorname{Ent}_{\Phi }}(F)\ge 0$ whenever the terms are integrable; if Φ is strictly convex on the convex hull of the essential range of F, equality holds only for a.s. constant F.

Fix $\Phi \in \mathcal{C}$ and $\varepsilon \gt 0$. We denote by ${\mathcal{A}_{T,\Phi }^{\varepsilon }}$ the class of all ${\mathcal{F}_{T}}$-measurable functionals $F:\Omega \to (0,\mathrm{\infty })$ such that:

Fix $\Phi \in \mathcal{C}$ and $\varepsilon \gt 0$.

Definition 2.12(i) records the positivity and integrability hypotheses used in the martingale entropy balance. For bounded functionals the product ${L^{2}}$ predictable representation property (PRP) gives a square-integrable representation, and Lemma 2.15 identifies its integrands with those of any finite-horizon integrable representation. Thus all representation energies depend only on the corresponding $\hspace{-1.67pt}\mathrm{d}t\otimes \mathbb{P}$- and $\nu \otimes \mathbb{P}$-classes. The condition ${M_{t-}}+{\xi _{t,z}}\gt 0$ is included because ${A^{\Phi }}(u,v)$ requires $u\gt 0$ and $u+v\gt 0$. When $\xi ={^{p}}(\Psi F)$, Lemma 2.24 gives ${M_{t-}}+{\xi _{t,z}}={^{p}}(F({\varepsilon _{t,z}^{+}}\cdot ))$, so positivity of the canonical representative of F transfers to the compensator class.

Let $M={({M_{t}})_{0\le t\le T}}$ be a square-integrable martingale on the usual augmented Brownian–Poisson product filtration. Then there exist predictable integrands θ and ξ such that with The two stochastic-integral martingales are orthogonal, and the Itô isometry gives The Brownian integrand θ and the jump integrand ξ are unique in these ${L^{2}}$ equivalence classes.

The Brownian and compensated-Poisson stochastic integrals are continuous and purely discontinuous martingales, respectively, and hence are orthogonal. Their isometries show that the map sending a pair of square-integrable predictable integrands to the sum of its terminal integrals is an isometry. Its range $\mathcal{H}$ is therefore a closed subspace of centered ${L^{2}}({\mathcal{F}_{T}})$.

Fix a finite-ν window A. Let U be a bounded Brownian-coordinate variable and V a bounded variable measurable with respect to $N{|_{A}}$. In their coordinate filtrations, the classical Brownian and Poisson representation theorems give where ${U_{t}}$ and ${V_{t}}$ are the coordinate conditional-expectation martingales; see [12] and [10, Theorem 3.1]. Independence of the coordinates and of the Poisson restrictions to A and ${A^{c}}$ gives $\mathbb{E}[\mathit{UV}|{\mathcal{F}_{t}}]={U_{t}}{V_{t}}$. The coordinate martingales retain their martingale property in the product filtration. Since $[U,V]=0$, integration by parts yields The integrands are square-integrable because ${U_{t}}$ and ${V_{t}}$ are bounded. Thus $\mathit{UV}-\mathbb{E}[\mathit{UV}]\in \mathcal{H}$. Finite linear combinations of these products are dense in the finite-window product ${L^{2}}$-space: bounded coordinate indicators generate its sigma-field, and the monotone-class theorem followed by truncation gives density. Closedness of $\mathcal{H}$ therefore gives the representation on each finite window.

Choose increasing finite-ν windows ${A_{n}}\uparrow {E_{T}}$, and let ${\mathcal{G}_{n}}$ be the sigma-field generated by the full Brownian coordinate and $N{|_{{A_{n}}}}$. For $Y\in {L^{2}}({\mathcal{F}_{T}})$, set ${Y_{n}}=\mathbb{E}[Y|{\mathcal{G}_{n}}]$. Then ${Y_{n}}\to Y$ in ${L^{2}}$, since ${\bigvee _{n}}{\mathcal{G}_{n}}$ generates ${\mathcal{F}_{T}}$ up to completion. Let $({\theta ^{n}},{\xi ^{n}})$ represent ${Y_{n}}-\mathbb{E}[Y]$, extending ${\xi ^{n}}$ by zero outside ${A_{n}}$. Orthogonality and the two isometries give The integrands therefore converge in their completed predictable ${L^{2}}$-spaces, and their integrals represent $Y-\mathbb{E}[Y]$. Taking $Y={M_{T}}$ and conditioning at time t proves the assertion; ${\mathcal{F}_{0}}$ is trivial up to null sets. The same isometry gives the stated energy identity and uniqueness. □

Let $(\theta ,\xi )$ and $({\theta ^{\prime }},{\xi ^{\prime }})$ be predictable integrands such that the stochastic integrals appearing below are integrable martingales on $[0,T]$. If for every $0\le t\le T$, up to indistinguishability, then Consequently, if a bounded martingale has a finite-horizon representation in the sense of Definition 2.12(i), its integrands agree with the integrands of the square-integrable representation of Proposition 2.14 on the corresponding null-set classes.

Subtract the two representations and write the difference as $Z={Z^{c}}+{Z^{d}}$, where The local martingale ${Z^{c}}$ is continuous and ${Z^{d}}$ is purely discontinuous. Since $Z\equiv 0$, uniqueness of the decomposition of a local martingale into its continuous and purely discontinuous parts gives ${Z^{c}}\equiv 0$ and ${Z^{d}}\equiv 0$; see, for instance, Jacod–Shiryaev [8, Theorem I.4.18]. Localizing the continuous stochastic integral and applying the Brownian isometry yield for a reducing sequence $({\tau _{n}})$, and monotone convergence gives $\theta ={\theta ^{\prime }}$ $\hspace{-1.67pt}\mathrm{d}t\otimes \mathbb{P}$-a.e.

We first verify that no two Poisson atoms share a time coordinate. On a finite-ν window A with $\nu (A)\gt 0$, conditional on $N(A)=m$, the points have the law of m independent samples from $\nu {|_{A}}/\nu (A)$. Its time marginal is atomless by Assumption 2.1, so any two sampled times agree with probability zero. The assertion is immediate if $\nu (A)=0$. Applying this argument to increasing finite windows covering ${E_{T}}$ gives a single event of probability one on which every pair of distinct atoms has distinct times.

For the jump part, ${Z^{d}}\equiv 0$ implies that every jump is zero. The jump formula defining compensated random-measure integrals [8, Definition II.1.27], together with time-diffuseness and the preceding time-simplicity, gives $\Delta {Z_{t}^{d}}={\xi _{t,z}}-{\xi ^{\prime }_{t,z}}$ at each atom $(t,z)$ of N. With we therefore have ${\int _{{E_{T}}}}{H_{t,z}}\hspace{1.67pt}N(\hspace{-1.67pt}\mathrm{d}t,\hspace{-1.67pt}\mathrm{d}z)=0$ a.s. The compensation formula gives and hence $\xi ={\xi ^{\prime }}$ $\nu \otimes \mathbb{P}$-a.e. The final assertion follows because a bounded terminal martingale is square-integrable, so Proposition 2.14 supplies an ${L^{2}}$ representation and the preceding uniqueness argument compares it with the given integrable representation. □

Let $K:[0,T]\times {\mathbb{R}_{0}^{d}}\times \Omega \to \mathbb{R}$ be bounded and $\mathcal{P}\otimes \mathcal{B}({\mathbb{R}_{0}^{d}})$-measurable, where $\mathcal{P}$ is the predictable σ-field of the usual augmented product filtration. Then

First work with the unaugmented product filtration, denoted by $({\mathcal{F}_{t}^{0}})$. Its predictable sigma-field is generated by sets of the form $(s,u]\times A$, with $A\in {\mathcal{F}_{s}^{0}}$, together with the time-zero sets. For an elementary field the equality is pointwise apart from the harmless endpoint $t=0$: if $t\in (s,u]$, then the map ${\varepsilon _{t,z}^{+}}$ adds a Poisson atom strictly after time s and leaves the Brownian coordinate unchanged, so it does not change the ${\mathcal{F}_{s}^{0}}$-measurable event A. The time-zero exceptional set is ν-null because the deterministic compensator has no fixed time atoms under the standing product realization. A monotone-class argument extends the identity to all bounded unaugmented predictable fields.

For the usual augmentation, the raw-version argument in the proof of Lemma 2.19 gives a bounded unaugmented predictable field ${K^{0}}$ and a $\mathbb{P}$-null set D such that ${K_{t,z}}={K_{t,z}^{0}}$ for all $t\gt 0,z$ whenever $\omega \notin D$. Enlarge D to a raw measurable null set. Lemma 2.2, applied to ${\mathbf{1}_{D}}$, gives ${\varepsilon _{t,z}^{+}}\omega \notin D$ for $\nu \otimes \mathbb{P}$-a.e. $(t,z,\omega )$. Thus K and ${K^{0}}$ agree both before and after insertion, and the raw identity proves the claim. □

Let F be bounded and ${\mathcal{F}_{T}}$-measurable, and let K be bounded predictable with support contained in a deterministic set $A\subset {E_{T}}$ satisfying $\nu (A)\lt \mathrm{\infty }$. Then

All terms are integrable because F and K are bounded and K is supported on a finite-ν set. Mecke’s identity applied to the N-integral gives By Lemma 2.16, the last factor ${K_{t,z}}({\varepsilon _{t,z}^{+}}\omega )$ may be replaced by ${K_{t,z}}(\omega )$. Subtracting the compensator term gives the displayed identity. □

Let $F\in {\mathcal{C}_{\mathrm{prod}}}$, and let H be a bounded product-filtration predictable simple ${\mathbb{R}^{k}}$-valued process. Then

Write ${H_{t}}={\sum _{i}}{h_{i}}{1_{({t_{i}},{t_{i+1}}]}}(t)$, with bounded ${\mathcal{F}_{{t_{i}}}}$-measurable ${\mathbb{R}^{k}}$-valued coefficients. Denote the coordinate filtrations by ${\mathcal{F}_{t}^{B}},{\mathcal{F}_{t}^{N}}$. Up to completion, ${\mathcal{F}_{{t_{i}}}}$ is their product sigma-field. Conditional expectations onto finite product partitions approximate each ${h_{i}}$ in ${L^{2}}$ by finite sums of bounded products $a({\omega ^{B}})b(\eta )$, with a ${\mathcal{F}_{{t_{i}}}^{B}}$-measurable and b ${\mathcal{F}_{{t_{i}}}^{N}}$-measurable, while preserving its uniform bound. For these coefficients, fixing η gives a bounded Brownian predictable simple process and a ${C_{b}^{1}}$ Brownian cylindrical functional $F(\cdot ,\eta )$. Apply Wiener-space integration by parts on each fiber and integrate over the Poisson law. The Brownian isometry and Cauchy–Schwarz inequality permit passage to the ${L^{2}}$ limit. □

Let $({\mathcal{F}_{t}^{0}})$ denote the unaugmented product filtration. Bounded unmarked predictable simple processes of the form are dense in the completed predictable space ${L^{2}}([0,T]\times \Omega ,\hspace{-1.67pt}\mathrm{d}t\otimes \mathbb{P})$. Bounded marked predictable simple fields of the form where ${c_{j}}\in \mathbb{R}$, ${S_{j}}\in \mathcal{B}({\mathbb{R}_{0}^{d}})$, and each ${S_{j}}$ is contained in an annulus $\{{n^{-1}}\lt |z|\le n\}$, are dense in the completed predictable space ${L^{2}}({E_{T}}\times \Omega ,\nu \otimes \mathbb{P})$.

For the unaugmented filtration, the predictable sigma-field is generated by rectangles $(s,u]\times A$, with $A\in {\mathcal{F}_{s}^{0}}$, together with the time-zero sets. On a finite time–mark window, the marked predictable sigma-field is generated by rectangles $(s,u]\times A\times S$. The class of bounded functions that can be approximated in ${L^{2}}$ by the displayed time-simple tests contains these generators and is closed under bounded pointwise monotone limits by dominated convergence. The functional monotone-class theorem therefore gives the approximation of bounded unaugmented predictable processes and fields.

For an augmented predictable rectangle $(s,u]\times D$, with $D\in {\mathcal{F}_{s}}$, choose ${s_{\ell }}\downarrow s$ in $(s,u)$ and ${D_{\ell }}\in {\mathcal{F}_{{s_{\ell }}}^{0}}$ such that $\mathbb{P}(D\triangle {D_{\ell }})=0$. Such sets exist by the definition of usual augmentation. The raw predictable indicators ${\mathbf{1}_{({s_{\ell }},u]}}{\mathbf{1}_{{D_{\ell }}}}$ converge to ${\mathbf{1}_{(s,u]}}{\mathbf{1}_{D}}$ for every time outside one $\mathbb{P}$-null event. Adjoining deterministic mark indicators and applying the monotone-class theorem yields raw versions for all bounded predictable fields, simultaneously in $t\gt 0,z$ outside a $\mathbb{P}$-null event. Time zero is null for both integrand measures. Value truncation and finite-window exhaustion now give the stated ${L^{2}}$ density, componentwise for vector-valued processes. □

Let ${\mathcal{D}_{\mathbb{P}}^{\mathrm{COH}}}$ be the class of all ${\mathcal{F}_{T}}$-measurable functionals F for which the following four hypotheses hold on the product filtration of Assumption 2.1.

In the standard scalar additive setting, (COH2)–(COH3) follow from the mixed Clark–Ocone–Haussmann formula of [6, Section 6] under its domain assumptions. In the present ${\mathbb{R}^{d}}$-valued raw-difference notation, the formulas in this definition specify the normalization used in this paper. Part II gives terminal examples in both finite-first-moment and infinite-activity regimes.

The COH domain contains only representation and derivative-identification hypotheses. Positivity and integrability assumptions for entropy, Poincaré, or exponential estimates are stated in the corresponding analytic cores.

Let $F\in {\mathcal{D}_{\mathbb{P}}^{\mathrm{COH}}}$. Then the martingale admits the predictable representation where the predictable integrands satisfy up to the corresponding null sets. The stochastic integrals in (2.7) are understood in the finite-horizon ${L^{1}}$ representation sense specified by the representation hypotheses. The stochastic integrals in this representation are integrable martingales on $[0,T]$, hence their future increments have zero conditional expectation. When an Itô isometry or square-integrability is used below, it is obtained separately from boundedness of the martingale or imposed in the relevant representation domain. In that case the ${L^{1}}$ Clark–Ocone–Haussmann integrands and the ${L^{2}}$ martingale-representation integrands agree on the corresponding null-set classes by Lemma 2.15.

Assumptions (COH2)–(COH3) give a finite-horizon Clark–Ocone–Haussmann representation on the product filtration, with stochastic integrals that are integrable martingales on $[0,T]$. Taking conditional expectations with respect to ${\mathcal{F}_{t}}$ therefore eliminates the increments over $(t,T]$ and yields (2.7). The Gaussian identification is written with respect to the Brownian driver: ${\theta _{t}}=\mathbb{E}[{D_{t}^{B}}F|{\mathcal{F}_{t-}}]$. □

Let $\Phi \in \mathcal{C}$. For every integer $m\ge 1$ and every symmetric positive semidefinite matrix $A\in {\mathbb{R}^{m\times m}}$, the map is convex. Moreover, with the map is convex on ${D_{\Phi }}$. In particular, taking A to be the identity matrix gives the conditional Jensen argument for the Brownian Gaussian energy term, and the second assertion gives the corresponding argument for the jump remainder term.

If Φ is affine, then both integrands vanish, so there is nothing to prove. Assume that Φ is non-affine, and set By assumption, ψ is positive and concave on $(0,\mathrm{\infty })$.

Fix $m\ge 1$ and a positive semidefinite matrix $A\in {\mathbb{R}^{m\times m}}$, and write ${A^{1/2}}$ for its symmetric square root. For $({u_{i}},{v_{i}})\in (0,\mathrm{\infty })\times {\mathbb{R}^{m}}$ and $\lambda \in [0,1]$, concavity of ψ gives Hence The vector Cauchy–Schwarz inequality in the form with ${a_{i}}={A^{1/2}}{v_{i}}$ and ${b_{i}}=\psi ({u_{i}})$ yields Thus ${\Theta _{1,A}}$ is convex.

For the jump term, Taylor’s formula with integral remainder gives This formula is valid because $\Phi \in {C^{2}}((0,\mathrm{\infty }))$ and the segment $\{u+\mathit{rv}:0\le r\le 1\}$ is contained in $(0,\mathrm{\infty })$ whenever $u\gt 0$ and $u+v\gt 0$. For each fixed $r\in [0,1]$, the map $(u,v)\mapsto {\Phi ^{\prime \prime }}(u+\mathit{rv}){v^{2}}$ is the scalar case of the already proved convexity with $A=1$, composed with the linear transformation $(u,v)\mapsto (u+\mathit{rv},v)$. Hence ${A^{\Phi }}$ is a nonnegative weighted average of convex functions. This is the same convexity mechanism used in the Φ-entropy framework of [4] and, in the canonical Lévy setting, in [16]. □

Let $\Phi \in \mathcal{C}$ and let $\mathcal{G}\subset {\mathcal{F}_{T}}$ be a sub-σ-field. For the Gaussian inequality, let U be positive and let V be ${\mathbb{R}^{m}}$-valued. Assume that U, V, and the displayed conditional expectations are integrable, and that $\mathbb{E}[U|\mathcal{G}]\gt 0$ a.s. Then For the jump inequality, let V be real-valued and assume in addition that $U+V\gt 0$ and $\mathbb{E}[U|\mathcal{G}]+\mathbb{E}[V|\mathcal{G}]\gt 0$ a.s.; then Both inequalities follow from conditional Jensen applied to the convex maps in Proposition 2.30.