Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach

Abstract

We explore the functional convex order for various classes of martingales: Brownian or Lévy driven diffusions with respect to their diffusion coefficient, stochastic integrals with respect to their integrand. Each result is bordered by counterexamples. Our approach combines the propagation of convexity results through (simulable) discrete time recursive dynamics relying on a backward dynamic programming principle and powerful functional limit theorems to transfer the results to continuous time models. In a second part, we extend this approach to optimal stopping theory, namely to the réduites of adapted functionals of (jump) martingale diffusions. Applications to various types of bounds for the pricing of pathwise dependent European and American options in local volatility models are detailed. Doing so, earlier results are retrieved in a unified way and new ones are proved. This systematic paradigm provides tractable numerical methods preserving functional convex order which may be crucial for applications, especially in Finance.

Appendices

Appendix 1: Euler Scheme for Brownian Martingale Diffusions

Proposition 10

Let $(\bar{X}_{t}^{n})_{t\in [0,T]}$ be the genuine Euler scheme of step $\frac{T} {n}$ of the SDE ≡ dX_t = σ(t,X_t)dW_t, X₀ = x defined as the solution to

$$\displaystyle{d\bar{X}_{t}^{n} =\sigma (\underline{t}_{ n},\bar{X}_{\underline{t}_{n}}^{n})dW_{ t},\;\bar{X}_{0}^{n} = x,\;t\! \in [0,T].}$$

If $\sigma: [0,T] \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous and satisfies the linear growth assumption

$$\displaystyle{\forall \,t\! \in [0,T],\;\forall \,x\! \in \mathbb{R},\quad \vert \sigma (t,x)\vert \leq C_{\sigma }(1 + \vert x\vert ),}$$

then the sequence $(\bar{X}^{n})_{n\geq 1}$ is C-tight on $\mathcal{C}([0,T], \mathbb{R})$ and any of its limiting distributions is a weak solution to the above SDE. In particular if a weak uniqueness assumption holds, then $\bar{X}^{n}\stackrel{\mathcal{L}(\|\,.\,\|_{\sup })}{\longrightarrow }X$.

Following e.g. [5] (Lemma B.1.2, p. 275, see also [22, 29]), we first show that, owing to the linear growth assumption, the non-decreasing function $\varphi _{p,n}(t) = \mathbb{E}\big(\sup _{s\in [0,t]}\vert \bar{X}_{s}^{n}\vert ^{p}\big)$, p​ ∈ [1, +∞), is finite for every t​ ∈ [0, T]. Using Doob’s Inequality and Gronwall’s Lemma, it follows that

$$\displaystyle{\varphi _{p,n}(t) \leq \varphi _{p}(t):= Ce^{Ct}(1 + \vert x\vert ^{p})}$$

for a real constant C = C′ _p, σ  > 0. Consequently, it follows from the L ^p-B.D.G. and Hölder inequalities, applied successively that, for every for p​ ∈ (2, +∞) and every s, t​ ∈ [0, T], s ≤ t,

$$\displaystyle{\mathbb{E}\vert \bar{X}_{t}^{n} -\bar{ X}_{ s}^{n}\vert ^{p} \leq c_{ p}^{p}\mathbb{E}\left (\int _{ s}^{t}\vert \sigma (\underline{u}_{ n},\bar{X}_{\underline{u}_{n}}^{n})\vert ^{2}du\right )^{\frac{p} {2} } \leq c_{p}^{p}\vert t - s\vert ^{\frac{p} {2} }\big(1 +\varphi _{p}(T)\big).}$$

Kolmogorov’s criterion (see [4, Theorem 12.3, p. 95]) implies that the sequence $M_{n} = (W_{t},\bar{X}_{t}^{n})_{t\in [0,T]}$ is C-tight, i.e. tight on $(\mathcal{C}([0,T], \mathbb{R}^{2}),\|\,.\,\|_{\sup })$. From now on, we mainly rely on the results established in [18]. Let n′ be a subsequence such that $(\bar{X}^{n'},W)$ functionally weakly converges to a probability $\mathbb{Q}$ on $(\mathcal{C}([0,T], \mathbb{R}^{2}),\|\,.\,\|_{\sup })$; hence it satisfies the U. T. (for Uniform Tightness) assumption (see Proposition 3.2 in [18]). The function σ being continuous on $[0,T] \times \mathbb{R}$, the sequence of càdlàg processes $(\sigma (\underline{t}_{n},\bar{X}_{\underline{t}_{n}}^{n}))_{n\geq 1}$ is C-tight on the Skorokhod space since $\big((\underline{t}_{n},\bar{X}_{\underline{t}_{n}}^{n})_{t\in [0,T]}\big)_{n\geq 1}$ clearly is. One derives that, up to a new extraction still denoted (n′), we may assume that $\big((\sigma (\underline{t}_{n'},\bar{X}_{\underline{t}_{n'}}^{n'}))_{t\in [0,T]},\bar{X}^{n'},W\big)_{n\geq 1}$ functionally converges toward a probability $\mathbb{P}$ on $\mathbb{D}([0,T], \mathbb{R}^{3})$. By Theorem 2.6 from [18]—the functional weak convergence of stochastic integrals theorem—we know that

$$\displaystyle{\left (\sigma (\underline{t}_{n'},\bar{X}_{\underline{t}_{n'}}^{n'}),(\bar{X}_{ t}^{n'},W_{ t}),\int _{0}^{t}\sigma (\underline{s}_{ n'},\bar{X}_{\underline{s}_{n'}}^{n'})dW_{ s}\right )_{t\in [0,T]}\stackrel{\mathcal{L}(Sk)}{\longrightarrow }\mathbb{Q}\quad \mbox{ as }\quad n \rightarrow +\infty }$$

where $\mathbb{Q}$ is a probability distribution on $\mathbb{D}([0,T], \mathbb{R}^{4})$ such that the canonical process Y = (Y ⁱ) _{i = 1: 4} satisfies $Y \stackrel{\mathcal{L}}{\sim }\big(Y ^{1},(Y ^{2},B),\int _{0}^{.}Y _{s}^{2}dB_{s})$ where B: = Y ³ is a standard $\mathbb{Q}$-Brownian motion with respect to the $\mathbb{Q}$-completed right continuous canonical filtration $(\mathcal{D}_{t}^{4})_{t\in [0,T]}$ on $\mathbb{D}([0,T], \mathbb{R}^{4})$. Furthermore, we know that Y ¹ = σ(. , Y ²) $\mathbb{Q}$-a. s. since $\sup _{t\in [0,T]}\vert \sigma (\underline{t}_{n'},\bar{X}_{\underline{t}_{n'}}^{n'}) -\sigma (t,\bar{X}_{t}^{n'})\vert$ converges to 0 in probability. The former claim follows from the facts that $\sup _{t\in [0,T]}\vert \bar{X}_{t}^{n}\vert$ is tight and σ(t, ξ) is uniformly continuous on every compact set of $[0,T] \times \mathbb{R}$, with linear growth in ξ uniformly in t​ ∈ [0, T]. On the other hand, we know that $\bar{X}^{n'} = x +\int _{ 0}^{.}\sigma (\underline{s}_{n'},\bar{X}_{\underline{s}_{n'}}^{n'})dW_{s}$, which in turn implies that Y _. ² = x + ∫ ₀ ^. σ(s, Y _s ²)dW _s . This shows the existence of a weak solution to the SDEX _t  = x + ∫ ₀ ^t σ(s, X _s ) dW _s , t​ ∈ [0, T].

Under the weak uniqueness assumption, this distribution is unique, hence is the only functional weak limiting distribution for the tight sequence $(\bar{X}^{n})_{n\geq 1}$. The convergence in distribution on $\mathcal{C}([0,T], \mathbb{R})$ follows.

Remark 10

If the original SDE has a unique strong solution, the same proof leads to establish the convergence in probability of the Euler scheme toward X. One just has to add the process X itself to the sequence $\big((\sigma (\underline{t}_{n},\bar{X}_{\underline{t}_{n}}^{n}))_{t\in [0,T]},\bar{X}^{n},W\big)_{n\geq 1}$.

Appendix 2: Euler Scheme for a Lévy Driven Martingale Diffusion

We consider the following SDE driven by a martingale Lévy process Z with Lévy measure ν:

$$\displaystyle{ X_{t} = x +\int _{(0,t]}\kappa (s,X_{s_{-}})dZ_{s},\;t\! \in [0,T],\;X_{0} = x, }$$

(33)

where κ is a Borel function on $[0,T] \times \mathbb{R}$. Its genuine Euler scheme is defined by

$$\displaystyle{ \bar{X}_{t_{k+1}}^{n} =\bar{ X}_{ t_{k}}^{n} +\ \kappa (t_{ k},\bar{X}_{t_{k}})(Z_{t_{k+1}} - Z_{t_{k}}),\;k = 1,\ldots,n,\;\bar{X}_{0} = X_{0} = x }$$

(34)

at discrete times t _k ⁿ and extended into a continuous time càdlàg process by setting

$$\displaystyle{ \bar{X}_{t}^{n} = x +\int _{ (0,t]}\kappa (\underline{s}_{n-},\bar{X}_{\underline{s}_{n}-}^{n})dZ_{ s},\;t\! \in [0,T]. }$$

(35)

2.1 Convergence of the Euler Scheme Toward a Solution to the Lévy Driven SDE

Proposition 11

• (a) Let p​ ∈ (1,2]. Assume that ν(|z| ^p ) < +∞ and that Z has no Brownian component and κ(t,ξ) has linear growth in ξ, uniformly in t​ ∈ [0,T]. Then

  $$\displaystyle{\sup _{n\geq 1}\big\|\sup _{t\in [0,T]}\vert \bar{X}_{t}^{n}\vert \big\|_{ p} +\big\|\sup _{t\in [0,T]}\vert X_{t}\vert \big\|_{p} < +\infty.}$$

  If moreover κ is continuous, then SDE  (33) has at least one weak solution. Finally, under a weak uniqueness assumption, one has

  $$\displaystyle{\bar{X}^{n}\stackrel{\mathcal{L}(Sk)}{\longrightarrow }X.}$$

• (b) If ν(z ² ) < +∞, the same result remains true mutatis mutandis if Z has a non-zero Brownian component.

Remark 11

In fact, if (33) has a strong solution, one shows using arguments similar to those developed below, the stronger result

$$\displaystyle{\sup _{t\in [0,T]}\vert \bar{X}_{t}^{n} - X_{ t}\vert \stackrel{\mathbb{P}}{\longrightarrow }0\;\mbox{ as }\;n \rightarrow +\infty.}$$

We refer to [15] (devoted to error bounds) for a simpler proof when κ is homogeneous and $\mathcal{C}^{\ni }$ on the real line.

Proof

• (a) We consider the Lévy-Khintchine decomposition of the Lévy process Z = (Z _t ) _{t ∈ [0, T]}, namely

  $$\displaystyle{Z_{t} =\widetilde{ Z}_{t} + Z^{1},\;t\! \in [0,T],}$$

  where $\widetilde{Z}$ is a pure jump, square integrable martingale with jumps of size at most 1 and Lévy measure ν( . ∩{ | z | ≤ 1}) and Z ¹ is a compensated (hence martingale) Poisson process with (finite) Lévy measure ν( . ∩{ | z |  > 1}).

It is clear from (34) that $\bar{X}_{t_{k}^{n}}^{n}\! \in L^{p}$ for every k = 0, …, n. Then, as ν( | z | ^p) < +∞, it follows classically that $\sup _{u\in [t_{k}^{n},t_{k+1}^{n}]}\vert Z_{u}^{1} - Z_{t_{k}^{n}}^{1}\vert \stackrel{d}{\sim }\sup _{[0,\frac{T} {n} ]}\vert Z_{u}^{1}\vert \!\in L^{p}$ (see e.g. [32]). Combining these two results implies that $\varphi _{p,n}(t):=\big\|\sup _{s\in [0,t]}\vert \bar{X}_{s}^{n}\vert \big\|_{p}$ is finite for every t​ ∈ [0, T].

It follows from Eq. (35) satisfied by $\bar{X}$ that

$$\displaystyle{\varphi _{p,n}(t) \leq \vert x\vert +\Big\|\sup _{s\in [0,t]}\Big\vert \int _{(0,s]} \kappa (\underline{u}_{n-},\bar{X}_{\underline{u}_{n-}}^{n})dZ_{ u}\Big\vert \Big\|_{p}}$$

The L ^p-B.D.G. Inequality implies (since p > 1)

$$\displaystyle{\Big\|\sup _{s\in [0,t]}\Big\vert \int _{(0,s]}\kappa (\underline{u}_{n},\bar{X}_{\underline{u}_{n}-})dZ_{u}\Big\vert \Big\|_{p} \leq c_{p}\Big\|\sum _{0<s\leq t}\kappa (\underline{s}_{n},\bar{X}_{\underline{s}_{n-}})^{2}(\varDelta Z_{ s})^{2}\Big\|_{\frac{p} {2} }^{ \frac{1} {2} }.}$$

Using that $\frac{p} {2} \leq 1$, we derive

$$\displaystyle\begin{array}{rcl} \Big\|\sum _{0<s\leq t}\kappa (\underline{s}_{n},\bar{X}_{\underline{s}_{n-}})^{2}(\varDelta Z_{ s})^{2}\Big\|_{\frac{p} {2} }^{ \frac{1} {2} }& \leq & \Big(\mathbb{E}\sum _{0<s\leq t}\vert \kappa (\underline{s}_{n},\bar{X}_{\underline{s}_{ n-}})\vert ^{p}\vert \varDelta Z_{ s}\vert ^{p}\Big)^{\frac{1} {p} } {}\\ & =& \Big(\nu (\vert z\vert ^{p})\,\mathbb{E}\!\int _{ 0}^{t}\vert \kappa (\underline{s}_{ n},\bar{X}_{\underline{s}_{n-}})\vert ^{p}ds\Big)^{\frac{1} {p} } {}\\ & \leq & C_{\kappa,p}^{p}\nu (\vert z\vert ^{p})^{\frac{1} {p} }\Big(\int _{0}^{t}(1 +\varphi _{p,n}(s)^{p})ds\Big)^{\frac{1} {p} } {}\\ \end{array}$$

where C _κ, p is a real constant satisfying $\vert \kappa (s,\xi )\vert \leq C_{\kappa,p}(1 + \vert \xi \vert ^{p})^{\frac{1} {p} }$, $(s,\xi )\! \in [0,T] \times \mathbb{R}$.

Finally, there exists a positive real constant C′ = C′ _{κ, p, ν} such that the function φ _p, n satisfies

$$\displaystyle{\varphi _{p,n}(t)^{p} \leq C'\Big(\vert x\vert ^{p} + t +\int _{ 0}^{t}\varphi _{ p,n}(s)^{p}ds\Big).}$$

One concludes by Gronwall’s Lemma that

$$\displaystyle{\forall \,t\! \in [0,T],\quad \varphi _{p,n}(t)^{p} \leq e^{C't}C'(T + \vert x\vert ^{p})}$$

or, equivalently, there exists a real constant C″ = C″ _{T, κ, p, ν} such that

$$\displaystyle{\forall \,t\! \in [0,T],\quad \varphi _{p,n}(t) \leq \varphi _{p}(t) = e^{C''t}C''(1 + \vert x\vert ).}$$

To establish the Skorokhod tightness of the sequence $(\bar{X}^{n})_{n\geq 1}$, we rely on the Aldous tightness criterion (see Definition 3(b) or [17, Theorem 4.5, p. 356]). Let ρ​ ∈ (0, 1]. Let σ and τ be two [0, T]-valued $\mathcal{F}^{Z}$-stopping times such that σ ≤ τ ≤ (σ +δ) ∧ T.

$$\displaystyle\begin{array}{rcl} \mathbb{E}\vert \bar{X}_{\tau }^{n} -\bar{ X}_{\sigma }^{n}\vert ^{\rho } = \mathbb{E}\,\Big\vert \sum _{\sigma <u\leq \tau }\kappa (\underline{u}_{n},\bar{X}_{\underline{u}_{n-}}^{n})\varDelta Z_{ u}\Big\vert ^{\rho }& \leq & \mathbb{E}\Big(\sum _{\sigma <u\leq \tau }\vert \kappa (\underline{u}_{n},\bar{X}_{\underline{u}_{n-}}^{n})\vert ^{\rho }\vert \varDelta Z_{ u}\vert ^{\rho }\Big) {}\\ & =& \nu (\vert z\vert ^{\rho })\mathbb{E}\int _{\sigma }^{(\sigma +\delta )\wedge T}\vert \kappa (\underline{u}_{ n},\bar{X}_{\underline{u}_{n-}}^{n})\vert ^{\rho }du {}\\ & \leq & \delta \,\nu (\vert z\vert ^{\rho })\,\mathbb{E}\big[\sup _{t\in [0,T]}\vert \kappa (t,\bar{X}_{t}^{n})\vert ^{\rho }\big] {}\\ & \leq & \delta \,\nu (\vert z\vert ^{\rho })\,C_{\kappa }(1 +\varphi _{p}(T))^{ \frac{\rho }{ p} } {}\\ \end{array}$$

where we used that ρ ≤ 1 ≤ p and ν( | z | ^ρ) ≤ ν( | z | ² ∧ 1) +ν( | z | ^p) < +∞. Then

$$\displaystyle\begin{array}{rcl} & & \sup \big\{\mathbb{E}\vert \bar{X}_{\tau }^{n} -\bar{ X}_{\sigma }^{n}\vert ^{\rho } + \mathbb{E}\vert Z_{\tau } - Z_{\sigma }\vert ^{\rho },\;\sigma \leq \tau \leq (\sigma +\delta ) \wedge T,\mathcal{F}^{Z}\mbox{ -stopping times}\big\} {}\\ & & \quad \leq \nu (\vert z\vert ^{\rho })(1 + C_{\kappa }(1 +\varphi _{p}(T))^{ \frac{\rho }{ p} }\big)\delta {}\\ \end{array}$$

which goes to 0 as δ → 0. This implies that the sequence $M_{n} = (\bar{X}^{n},Z)$, n ≥ 1, is Sk-tight. Moreover, following Proposition 3.2 from [18], the sequence (M _n ) _n ≥ 1 satisfies the U. T. condition since it is Sk-tight and

$$\displaystyle\begin{array}{rcl} \mathbb{E}\sup _{t\in [0,T]}\big(\vert \varDelta \bar{X}_{t}^{n}\vert \vee \vert \varDelta Z_{ t}\vert \big)& \leq & \Big[\mathbb{E}\Big(\sum _{0<t\leq T}\vert \varDelta \bar{X}_{t}^{n}\vert ^{p} + \vert \varDelta Z_{ t}\vert ^{p}\Big)\Big]^{\frac{1} {p} } {}\\ & \leq & \Big[\nu (\vert z\vert ^{p})\mathbb{E}\int _{ 0}^{T}\big(1 + \vert \kappa (\underline{t}_{ n},\bar{X}_{\underline{t}_{n}}^{n})\vert ^{p}\big)dt\Big]^{\frac{1} {p} } {}\\ & \leq & \big(\nu (\vert z\vert ^{p})\big)^{\frac{1} {p} }\big(T + C_{\kappa,p}^{p}(1 +\varphi _{p}(T))\big)^{\frac{1} {p} } < +\infty. {}\\ \end{array}$$

On the other hand, the sequence $\left ((\kappa \big(\underline{t}_{n},\bar{X}_{\underline{t}_{n}}^{n}))_{t\in [0,T]},M_{n}\right )_{n\geq 1}$ is Sk-tight, owing to the following lemma.

Lemma 6

Let $\mathcal{V}_{[0,T]}^{+}$ be the set of functions μ: [0,T] → [0,T] such that μ(0) = 0 and μ(T) = T endowed with the sup norm. Assume $\kappa: [0,T] \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Then the mapping $\varPsi: \mathcal{V}_{[0,T]}^{+} \times I\!\!D([0,T], \mathbb{R}^{d}) \rightarrow I\!\!D([0,T], \mathbb{R}^{1+d})$ defined by $\varPsi (\mu,\alpha ) =\big (\kappa (\mu (.),\alpha ^{1}(.)),\alpha \big)$ is continuous (α = (α ¹ ,…,α ^d )) for the product topology.

Proof (Proof of the Lemma)

Let (λ _n ) _n ≥ 1 be a sequence of increasing homeomorphisms of [0, T] such that λ _n  → Id _[0, T] and α _n ∘λ _n  → α uniformly and let μ _n  → μ in $\mathcal{V}_{[0,T]}^{+}$ where Id _[0, T] denotes the identity on [0, T]. Then the closure of (α _n ∘λ _n (t)) _{n ≥ 1, t ∈ [0, T]} is a compact set K of $\mathbb{R}^{d}$ so that the function κ is uniformly continuous on [0, T] × K. On the other hand

$$\displaystyle{\|\mu _{n} \circ \lambda _{n} - Id_{[0,T]}\|_{\sup } \leq \|\mu _{n} - Id_{[0,T]}\|_{\sup } +\|\lambda _{n} - Id_{[0,T]}\|_{\sup }\quad \mbox{ as }\quad n \rightarrow +\infty }$$

and $\|\alpha _{n} \circ \lambda _{n} -\alpha \|_{\sup }\rightarrow 0$ as n → +∞. The conclusion follows.

Up to an extraction, we may assume that the triplet $\big(\big(\kappa (\underline{t}_{n'},\bar{X}_{\underline{t}_{n'}}^{n'}\big)_{t\in [0,T]},M_{n'}\big)_{n\geq 1}$ weakly converges for the Skorokhod topology toward a probability $\mathbb{P}$ on the canonical Skorokhod space $(I\!\!D([0,T], \mathbb{R}^{3}),(\mathcal{D}_{t})_{t\in [0,T]})$.

By Theorem 2.6 from [18] for the functional convergence of stochastic integrals, we know that

$$\displaystyle{\Big(\kappa (\underline{t}_{n'},\bar{X}_{\underline{t}_{n'}}^{n'}),(\bar{X}_{ t}^{n'},Z_{ t}),\int _{0}^{t}\kappa (\underline{s}_{ n'-},\bar{X}_{\underline{s}_{n'-}}^{n'})dZ_{ s}\Big)_{t\in [0,T]}\stackrel{\mathcal{L}(Sk)}{\longrightarrow }\mathbb{Q}}$$

where $\mathbb{Q}$ is a probability on $\mathbb{D}([0,T], \mathbb{R}^{4})$ such that the canonical process Y = (Y ⁱ) _{i = 1: 4} satisfies $Y \stackrel{d}{\sim }\big(Y ^{1},(Y ^{2},Y ^{3}),\int _{ 0}^{.}Y _{ s}^{2}dY _{ s}^{3})$ where Y ³ is a Lévy process with respect to the $\mathbb{Q}$ and the $\mathbb{Q}$-completed right continuous canonical filtration $(\mathcal{D}_{t}^{\mathbb{Q}})_{t\in [0,T]}$ on $\mathbb{D}([0,T], \mathbb{R}^{4})$ having the distribution of Z (i.e. $\mathbb{Q}_{Y ^{3}} = \mathcal{L}(Z)$). Furthermore, we know that Y ¹ = κ(. , Y ². ) $\mathbb{Q}$-a. s. since the mapping (μ, (α ⁱ) _{i = 1: 4}) ↦ α ¹ −κ(μ, α ²) is continuous from $\mathcal{V}_{[0,T]}^{+} \times \mathbb{D}([0,T], \mathbb{R}^{4})$ to $\mathbb{D}([0,T], \mathbb{R})$ (and t _n converges uniformly to Id _[0, T]).

On the other hand we know that $\bar{X}_{t}^{n'} = x +\int _{ 0}^{t}\kappa (\underline{s}_{ n'-},\bar{X}_{\underline{s}_{n'-}}^{n'})dZ_{ s},\,t\! \in [0,T]$ which in turn implies that $(Y _{t}^{2} = x +\int _{ 0}^{t}\kappa (s,Y _{ s_{-}}^{2})dZ_{ s},\,t\! \in [0,T])$ $\mathbb{Q}$-a. s. . This shows the existence of a weak solution to the SDE $X_{t} = x +\int _{ 0}^{t}\kappa (s,X_{s_{-}})dZ_{s},\,t\! \in [0,T]$.

Under the weak uniqueness assumption, the distribution $\mathbb{Q}_{Y ^{2}}$ of Y ² is unique equal, say, to $\mathbb{P}_{X}$.

• (b) We assume that the Lévy measure has a finite second moment ν(z ²) < +∞ on the whole real line. Then one can decompose Z as

  $$\displaystyle{Z_{t} = a\,W_{t} +\widetilde{ Z}_{t},\;t\! \in [0,T],\quad (a \geq 0)}$$

  where a ≥ 0 and $\widetilde{Z}$ is a pure jump martingale Lévy process with Lévy measure ν. Then one shows like in the Brownian case that $\varphi (t) = \mathbb{E}\big(\sup _{s\in [0,t]}\vert \bar{X}_{s}^{n}\vert ^{2}\big)$ is finite over [0, T] using that all $\bar{X}_{t_{k}}$ are square integrable and $\mathbb{E}\big(\sup _{s\in [t_{k},t_{k+1})}\vert Z_{s} - Z_{t_{k}}\vert ^{2}\big) = \mathbb{E}\big(\sup _{s\in [0,\frac{T} {n} ]}\vert Z_{s}\vert ^{2}\big) < +\infty$. Then, using Doob’s Inequality, we show that

  $$\displaystyle{\varphi (t) \leq 4C_{\kappa }^{2}(a^{2} +\nu (z^{2})\big)\Big(t +\int _{ 0}^{t}\varphi (s)ds\Big)}$$

  where C _κ is a real constant satisfying $\kappa (t,\xi ) \leq C_{\kappa }(1 + \vert \xi \vert ^{2})^{\frac{1} {2} }$, $\xi \!\in \mathbb{R}$.

To establish the Skorokhod tightness of the sequence, we rely again on Aldous’ tightness criterion (see Definition 3(b) or[17, Theorem 4.5, p. 356]). Let σ, τ be two [0, T]-valued $\mathcal{F}^{Z}$-stopping times such that σ ≤ τ ≤ (σ +δ) ∧ T. Applying Doob’s Inequality, to the martingale $\Big(\int _{\sigma }^{\sigma +s}\kappa (\underline{u}_{n},\bar{X}_{\underline{u}_{n-}})dZ_{u}\Big)_{s\geq 0}$ yields

$$\displaystyle\begin{array}{rcl} \mathbb{E}\vert \bar{X}_{\tau }^{n} -\bar{ X}_{\sigma }^{n}\vert ^{2}& \leq & 4a^{2}\mathbb{E}\left (\int _{\sigma }^{\tau }\vert \kappa (\underline{u}_{ n},\bar{X}_{\underline{u}_{n-}}^{n})\vert ^{2}du\right ) + 4\,\mathbb{E}\left (\sum _{\sigma <u\leq \tau }\vert \kappa (\underline{u}_{n},\bar{X}_{\underline{u}_{n-}}^{n})\vert ^{2}\vert \varDelta Z_{ u}\vert ^{2}\right ) {}\\ & =& 4\big(a^{2} +\nu (z^{2})\big)\mathbb{E}\left (\int _{\sigma }^{\tau }\vert \kappa (\underline{u}_{ n},\bar{X}_{\underline{u}_{n-}}^{n})\vert ^{2}du\right ) {}\\ & \leq & 4\big(a^{2} +\nu (z^{2})\big)\mathbb{E}\left (\int _{\sigma }^{(\sigma +\delta )\wedge T}\vert \kappa (\underline{u}_{ n},\bar{X}_{\underline{u}_{n-}}^{n})\vert ^{2}du\right ) {}\\ & \leq & 4(a^{2} +\nu (z^{2})\big)\delta \,C_{\kappa }^{2}(1 +\varphi (T)). {}\\ \end{array}$$

It follows that $\mathbb{E}\vert \bar{X}_{\tau } -\bar{ X}_{\sigma }\vert ^{2} + \mathbb{E}\vert Z_{\tau } - Z_{\sigma }\vert ^{2} \leq 4\,C_{\kappa }^{2}(a^{2} +\nu (z^{2})\big)\nu (z^{2})(1 +\varphi (T)\big)\delta$ which clearly implies the Sk-tightness of the sequence $M_{n} = (\bar{X}^{n},Z)$, n ≥ 1.

The sequence satisfies the U. T. condition from [18] since (M _n ) _n ≥ 1 is Sk-tight and (see Proposition 3.2 from [18])

$$\displaystyle\begin{array}{rcl} \mathbb{E}\Big[\sup _{t\in [0,T]}\big(\vert \varDelta \bar{X}_{t}^{n}\vert \vee \vert \varDelta Z_{ t}\vert \big)\Big]& \leq & \Big(\mathbb{E}\Big[\sum _{0<t\leq T}\vert \varDelta \bar{X}_{t}^{n}\vert ^{2} + \vert \varDelta Z_{ t}\vert ^{2}\Big]\Big)^{\frac{1} {2} } {}\\ & \leq & \Big(\nu (z^{2})\mathbb{E}\int _{ 0}^{T}\big(1 + \vert \kappa (\underline{t}_{ n},\bar{X}_{\underline{t}_{n}}^{n})\vert ^{2}\big)dt\Big)^{\frac{1} {2} } {}\\ & \leq & \Big(\nu (z^{2})(T + C_{\kappa }\big(1 +\varphi (T))\big)\Big)^{\frac{1} {2} } < +\infty. {}\\ \end{array}$$

From this point, the proof is similar to that of claim (a).

2.2 Higher Moments

Let $Z_{t} = aW_{t} +\widetilde{ Z}_{t}$, t​ ∈ [0, T], be the decomposition of the Lévy process Z where W is a standard B.M. and $\widetilde{Z}$ is a pure jump Lévy process independent of W.

Proposition 12

Let p​ ∈ [2,+∞). If ν(|z| ^p ) < +∞ then

$$\displaystyle{\sup _{n\geq 1}\Big\|\sup _{t\in [0,T]}\vert \bar{X}_{t}^{n}\vert \Big\|_{ p} < +\infty.}$$

Proof

If p​ ∈ (1, 2], the claim follows from the above Proposition 11. Assume from now on p​ ∈ [2, +∞). Let $\varphi _{p,n}(t) = \mathbb{E}\big(\sup _{t\in [0,T]}\vert \bar{X}_{t}^{n}\vert ^{p}\big)$. Let ℓ _p be the unique integer defined by the inequality $2^{\ell_{p}} < p \leq 2^{\ell_{p}+1}$. It is straightforward, using the same arguments as above, that φ _p, n (T) < +∞ since sup _{t ∈ [0, T]} | Z _t  | ^p​ ∈ L ¹ (see [32, Theorem 25.18, p. 166]) and $X_{t_{k}}\! \in L^{p}$ by induction using (34). For convenience, we set $\kappa _{s_{-}} =\kappa (\underline{s}_{n},\bar{X}_{\underline{s}_{n}-}^{n})$.

Now, combining the integral and the regular Minkowski Inequalities with the B.D.G. Inequality implies

$$\displaystyle\begin{array}{rcl} \varphi _{p,n}(t)^{\frac{1} {p} }& \leq & \vert x\vert + c_{p}\Big\|a^{2}\int _{0}^{t}\kappa _{s_{ -}}^{2}ds +\sum _{ 0<s\leq t}\kappa _{s_{-}}^{2}(\varDelta Z_{ s})^{2}\Big\|_{\frac{p} {2} }^{ \frac{1} {2} } \\ & \leq & \vert x\vert + c_{p}\Big(a\Big\|\int _{0}^{t}\kappa _{ s_{-}}^{2}ds\Big\|_{\frac{p} {2} }^{ \frac{1} {2} } +\Big\|\sum _{0<s\leq t}\kappa _{s_{ -}}^{2}(\varDelta Z_{ s})^{2}\Big\|_{\frac{p} {2} }^{ \frac{1} {2} }\Big){}\end{array}$$

(36)

where we used in the second inequality that $\sqrt{ u + v} \leq \sqrt{u} + \sqrt{v}$, u, v ≥ 0. First note that by two successive applications of Hölder Inequality to dt and $d\mathbb{P}$, we obtain

$$\displaystyle{ \Big\|\int _{0}^{t}\kappa _{ s_{-}}^{2}ds\Big\|_{\frac{p} {2} }^{ \frac{1} {2} } \leq T^{\frac{1} {2} -\frac{1} {p} }\Big(\int _{0}^{t}\mathbb{E}\,\vert \kappa _{s_{ -}}\vert ^{p}ds\Big)^{\frac{1} {p} }. }$$

(37)

Using that for every ℓ​ ∈ { 1, …, ℓ _p }, $\Big(\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{2^{\ell}}\vert \varDelta Z_{ s}\vert ^{2^{\ell}} -\int _{ 0}^{t}\vert \kappa _{ s_{-}}\vert ^{2^{\ell}}ds\,\nu (\vert z\vert ^{2^{\ell}})\Big)_{ t\!\in [0,T]}$, is a true martingale, we have by combining this time the Minkowski inequality, the B.D.G. Inequality applied with $\frac{p} {2^{\ell}} > 1$ and the elementary inequality (u + v)^r ≤ u ^r + v ^r, u, v ≥ 0, r​ ∈ (0, 1] that:

$$\displaystyle\begin{array}{rcl} \Big\|\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{2^{\ell}}(\varDelta Z_{ s})^{2^{\ell}}\Big\|_{\frac{p} {2^{\ell}} }^{ \frac{1} {2^{\ell}} }& \leq & \Big\|\sum _{0<s\leq t}\vert \kappa _{s_{ -}}\vert ^{2^{\ell}}(\varDelta Z_{ s})^{2^{\ell}} -\int _{ 0}^{t}\vert \kappa _{ s_{-}}\vert ^{2^{\ell}}ds\,\nu (\vert z\vert ^{2^{\ell}})\Big\|_{\frac{p} {2^{\ell}} }^{ \frac{1} {2^{\ell}} } {}\\ & & +\Big\|\int _{0}^{t}\vert \kappa _{ s_{-}}\vert ^{2^{\ell}}ds\Big\|_{\frac{p} {2^{\ell}} }^{ \frac{1} {2^{\ell}} }\nu (\vert z\vert ^{2^{\ell} })^{\frac{1} {2^{\ell}} } {}\\ & \leq & c_{\frac{p} {2^{\ell}} }^{ \frac{1} {2^{\ell}} }\Big\|\sum _{0<s\leq t}\vert \kappa _{s_{ -}}\vert ^{2^{\ell+1} }(\varDelta Z_{s})^{2^{\ell+1} }\Big\|_{ \frac{p} {2^{\ell+1}} }^{ \frac{1} {2^{\ell+1}} } {}\\ & & +\Big\|\int _{0}^{t}\vert \kappa _{ s_{-}}\vert ^{2^{\ell}}ds\Big\|_{\frac{p} {2^{\ell}} }^{ \frac{1} {2^{\ell}} }\nu (\vert z\vert ^{2^{\ell} })^{\frac{1} {2^{\ell}} }. {}\\ \end{array}$$

Then two applications of Hölder Inequality applied to dt and $d\mathbb{P}$ successively imply

$$\displaystyle{\Big\|\int _{0}^{t}\vert \kappa _{ s_{-}}\vert ^{2^{\ell}}ds\Big\|_{\frac{p} {2^{\ell}} }^{ \frac{1} {2^{\ell}} } \leq T^{\frac{1} {2^{\ell}} -\frac{1} {p} }\Big(\int _{0}^{t}\mathbb{E}\,\vert \kappa _{s_{ -}}\vert ^{p}ds\Big)^{\frac{1} {p} }.}$$

Summing up these inequalities in cascade finally yields a positive real constant K _{p, ν, a, T} ⁽⁰⁾ such that

$$\displaystyle\begin{array}{rcl} \Big\|\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{2}(\varDelta Z_{ s})^{2}\Big\|_{\frac{p} {2} }^{ \frac{1} {2} }& \leq & K_{p,\nu,a,T}^{(0)}\Big(\Big(\int _{0}^{t}\mathbb{E}\,\vert \kappa _{s_{ -}}\vert ^{p}ds\Big)^{\frac{1} {p} } {}\\ & & \quad +\Big\|\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{2^{\ell_{p}+1} }(\varDelta Z_{s})^{2^{\ell_{p}+1} }\Big\|_{ \frac{p} {2^{\ell_{p}+1}} }^{ \frac{1} {2^{\ell_{p}+1}} }\Big). {}\\ \end{array}$$

Now, as $\frac{p} {2^{\ell_{p}+1}} \leq 1$, one gets by the compensation formula

$$\displaystyle\begin{array}{rcl} \Big\|\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{2^{\ell_{p}+1} }\vert \varDelta Z_{s}\vert ^{2^{\ell_{p}+1} }\Big\|_{ \frac{p} {2^{\ell_{p}+1}} }^{ \frac{1} {2^{\ell_{p}+1}} }& \leq & \Big(\mathbb{E}\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{p}(\varDelta Z_{s})^{p}\Big)^{\frac{1} {p} } {}\\ & =& \Big(\int _{0}^{t}\mathbb{E}\vert \kappa _{ s_{-}}\vert ^{p}ds\Big)^{\frac{1} {p} }\nu (\vert z\vert ^{p})^{\frac{1} {p} }. {}\\ \end{array}$$

Hence, there exists a real constant K _{p, ν, a, T} ⁽¹⁾ > 0

$$\displaystyle{ \Big\|\sum _{0<s\leq t}\vert \kappa _{s_{-}}\vert ^{2}(\varDelta Z_{ s})^{2}\Big\|_{\frac{p} {2} }^{ \frac{1} {2} } \leq K_{p,\nu,a,T}^{(1)}\Big(\int _{0}^{t}\mathbb{E}\vert \kappa _{s_{ -}}\vert ^{p}ds\Big)^{\frac{1} {p} }. }$$

(38)

Finally, plugging (37) and (38) in (36), there exist positive real constants K _{p, ν, a, T} ^(ℓ), ℓ = 2, 3, such that

$$\displaystyle{\varphi _{p,n}(t)^{\frac{1} {p} } \leq K_{p,\nu,a,T}^{(2)}\Big(\vert x\vert +\Big (\int _{0}^{t}\mathbb{E}\vert \kappa _{s_{ -}}\vert ^{p}ds\Big)^{\frac{1} {p} }\Big) \leq K_{p,\nu,a,T}^{(3)}\Big(\vert x\vert + 1 +\Big (\int _{0}^{t}\varphi _{p,n}(s)ds\Big)^{\frac{1} {p} }\Big)}$$

where we used in the second inequality that κ has linear growth. Hence

$$\displaystyle{\varphi _{p,n}(t) \leq 2^{p-1}(K_{ p,\nu,a,T}^{'(3)})^{p}\Big(\big(\vert x\vert + 1\big)^{p} +\int _{ 0}^{t}\varphi _{ p,n}(s)ds\Big).}$$

Gronwall’s lemma completes the proof since it implies that

$$\displaystyle{\varphi _{p,n}(t) \leq e^{2^{p-1}(K_{ p,\nu,a,T}^{'(3)})^{p}\,t }2^{p-1}(K_{ p,\nu,a,T}^{'(3)})^{p}(\vert x\vert + 1)^{p}.}$$