A Note to Multivalued Martingales

H. Fritsch Martin-Luther-Universität Halle-Wittenberg Halle, Germany

Abstract In the theory of setvalued random processes we focus our attention to setvalued martingales and submartingales. The aim of this short note is to derive a setvalued analogy to the well-known maximal inequality of Doob. Key words: setvalued martingale, setvalued submartingale, maximal inequality. AMS subject classifications: 60G44, 60G48.

1 Introduction

In the last decades several authors like [3], [8], [9], [4] have considered Banach space valued multifunctions over finite measure spaces. Conceptions like integral, conditional expectation, martingale and measure became a new setvalued interpretation. These concepts were used in papers about the existence of solutions of stochastic differential inclusions ([6], [7], [1]) or setvalued stochastic control problems (e.g. [2]). For uniform estimation of the moments of multivalued random processes an instrument like the maximal 𝕃_{p}-inequality of Doob (for finite p) is needed very often. For a setvalued martingale an analogy can be shown by writing down the traditional way of proving in the speech of multifunctions. Let 𝕏 be a separable real Banach space and (Ω, 𝔉, P) be a complete probability space. By P_{fc}(𝕏) we define the set of all nonempty, closed, convex subsets of 𝕏. For a multifunction B : Ω → P_{fc}(𝕏) and p ≥ 1 we define the closed subset of 𝕃_{p}(Ω; 𝕏)

S_{B}^{p} = {b ∈ 𝕃_{p}(Ω; 𝕏) : b(ω) ∈ B(ω) a.e.} .

For a nonempty, closed subset A of 𝕏 we denote |A| = sup_{x∈A} ∥x∥. Following [4]

we introduce a continuity concept with respect to the Hausdorff pseudometric h(., .).

Definition 1: Let 𝕐 be a Hausdorff topological space. A multifunction B: 𝕐 → P_{fc}(𝕏) is said to be

(a) h-upper semicontinuous at y_{0} ∈ 𝕐 if y → sup_{b∈B(y)} d(b, B(y_{0})) is continuous at y_{0};

i.e. for every ε > 0, there is a neighborhood U_{ε} of y_{0} such that for all y ∈ U_{ε}

sup_{b∈B(y)} d(b, B(y_{0})) < ε,

(b) h-lower semicontinuous at y_{0} ∈ 𝕐 if y → sup_{b∈B(y_{0})} d(b, B(y)) is continuous at y_{0};

i.e. for every ε > 0, there is a neighborhood U_{ε} of y_{0} such that for all y ∈ U_{ε}

sup_{b∈B(y_{0})} d(b, B(y)) < ε,

(c) h-continuous at y_{0} ∈ 𝕐 if it is both h-upper and h-lower semicontinuous at y_{0}. If (a) (resp. (b), (c)) are valid at every y_{0} ∈ 𝕐, then we say that B(.) is h-upper semicontinuous (resp. h-lower semicontinuous, h-continuous).

A 𝔉-measurable multifunction on (Ω, 𝔉, P) is called a random set. For a random set B ∈ P_{f}(𝕏), B is called 𝕃_{p}-bounded if there is g ∈ 𝕃_{p}(Ω; ℝ) such that ∥x∥ ≤ g(ω) for any x and a.a. ω ∈ Ω with x ∈ B(ω). If S_{B}^{p} ≠ ∅, then by theorem 2.2. of [3] sup_{b∈S_{B}^{p}} ∥b∥_{𝕃_{p}} = E\{|B|^{p}\}^{\frac{1}{p}} and the expectation of B is defined by

E{B} = {E{b}: b ∈ S_{B}^{1}}.

Definition 2: B = (B_{t})_{t≥0} is called a setvalued process with values in 𝕏 if B: ℝ^{+} × Ω → P_{fc}(𝕏) is a setvalued function with

(a) for all t ∈ ℝ^{+}, B_{t}(ω) is closed and convex in 𝕏 for a.a. ω ∈ Ω,

(b) for fixed t, B_{t}(.) is a 𝕏-valued random set, i.e. for all t ∈ ℝ^{+}, A ∈ 𝔅_{𝕏}, {ω ∈ Ω: B_{t}(ω) ⋂ A ≠ ∅} ∈ 𝔉.

We know from [3], [5] and [8] that, for any 𝕃_{p}-bounded B ∈ 𝕃_{1}(Ω; P_{f}(𝕏)) and sub-σ-field 𝔄 ⊂ 𝔉, there exists a unique set E{B|𝔄} ∈ 𝕃_{1}((Ω, 𝔄, P); P_{f}(𝕏)) such that S_{E{B|𝔄}}^{1} = cl{E{b|𝔄} : b ∈ S_{B}^{1}}. Here the closure is taken with respect to the 𝕃_{1}((Ω, 𝔄, P); 𝕏)-norm. This random set E{B|𝔄} is called the (multivalued) conditional expectation of B relative to 𝔄. Let I be an index set of the form {t_{0}, t_{1}, t_{2}, ...} or {t : t ≥ 0} and (𝔉_{t})_{t∈I} be an increasing family of sub-σ-fields of 𝔉.

Definition 3: A setvalued process B = (B_{t})_{t≥0} with values in 𝕏 and adapted to (𝔉_{t}) is called a (multivalued) martingale (resp. submartingale, supermartingale) with respect to 𝔉_{t} if

(a) for t ≥ 0, B_{t} is 𝕃_{1}-bounded,

(b) for 0 ≤ s < t, E{B_{t}|𝔉_{s}} = B_{s} (resp. E{B_{t}|𝔉_{s}} ⊇ B_{s}, E{B_{t}|𝔉_{s}} ⊆ B_{s} ). We call B 𝔉_{t}-adapted if B_{t} is measurable with respect to 𝔉_{t} for every t ≥ 0.

For the case of a discrete index set I interesting results about convergence, regularity and semimartingale selections can be found in [4], [9] and [3].

2 A maximal inequality for setvalued martingales

Now we study systematicly multivalued martingales and submartingales.

Lemma 1: Let (B_{t_{k}})_{k=0,1,...} be a setvalued (sub-)martingale with respect to a discrete filtration (𝔉_{t_{k}}) and (x_{t_{k}}) be a positive bounded process so that x_{t_{k}} is 𝔉_{t_{k−1}}-measurable for k > 1. The setvalued process (Y_{t_{k}})_{k=0,1,...} defined by Y_{t_{0}} = B_{t_{0}} and Y_{t_{k}} + x_{t_{k}}B_{t_{k−1}} = Y_{t_{k−1}} + x_{t_{k}}B_{t_{k}} is a (sub-)martingale. Let τ be a stopping time with respect to (𝔉_{t_{k}}). The stopped process (B_{t_{k}∧τ})_{k=0,1,...} is also a multivalued (sub-)martingale.

Proof: For the first statement we split the proof in two parts. ( i) Let B be a martingale. Using the theorems 5.2 and 5.3 of [3] and martingale definition we receive

E{Y_{t_{k}} + x_{t_{k}}B_{t_{k−1}}|𝔉_{t_{k−1}}} = E{Y_{t_{k}}|𝔉_{t_{k−1}}} + E{x_{t_{k}}|𝔉_{t_{k−1}}}B_{t_{k−1}}
= E{Y_{t_{k}}|𝔉_{t_{k−1}}} + x_{t_{k}}B_{t_{k−1}}  (1)

and

E{Y_{t_{k−1}} + x_{t_{k}}B_{t_{k}}|𝔉_{t_{k−1}}} = E{Y_{t_{k−1}}|𝔉_{t_{k−1}}} + x_{t_{k}}E{B_{t_{k}}|𝔉_{t_{k−1}}}
= E{Y_{t_{k−1}}|𝔉_{t_{k−1}}} + x_{t_{k}}B_{t_{k−1}}.  (2)

By equating (1) and (2) we have

E{Y_{t_{k}}|𝔉_{t_{k−1}}} = E{Y_{t_{k−1}}|𝔉_{t_{k−1}}} = Y_{t_{k−1}}.

(ii) Let B be a submartingale. Using submartingale property in the same way like above we obtain

E{Y_{t_{k−1}} + x_{t_{k}}B_{t_{k}}|𝔉_{t_{k−1}}} ⊇ E{Y_{t_{k−1}}|𝔉_{t_{k−1}}} + x_{t_{k}}B_{t_{k−1}}.  (3)

Combining (1) and (3) we have

E{Y_{t_{k}}|𝔉_{t_{k−1}}} ⊇ E{Y_{t_{k−1}}|𝔉_{t_{k−1}}} = Y_{t_{k−1}}.

To prove the second statement we set x_{t_{k}} = χ_{[t_{k}≤τ]} (𝔉_{t_{k−1}}-measurable), where χ denotes the characteristic function of a set. So we have x_{t_{k}} = 1 − χ_{[τ≤t_{k−1}]} and with the first part the assertion follows.

□

Lemma 2: Let (B_{t_{k}})_{k=0,1,...} be a submartingale (martingale) with respect to the discrete filtration (𝔉_{t_{k}}). If τ_{1}, τ_{2} ∈ [0, T] P-a.s. are stopping times with τ_{1} < τ_{2} then P-a.s.

B_{τ_{1}} ⊆ E{B_{τ_{2}}|𝔉_{τ_{1}}} (B_{τ_{1}} = E{B_{τ_{2}}|𝔉_{τ_{1}}}).

Furthermore an adapted and 𝕃_{p}-bounded setvalued random process B is a martingale if and only if E{B_{τ_{1}}} = E{B_{τ_{2}}} for every pair of stopping times.

Proof: ( i) Let B be a martingale. With x_{t_{k}} = χ_{[t_{k}≤τ_{2}]} − χ_{[t_{k}≤τ_{1}]} we receive from lemma 1 for t_{k} > T successively B_{τ_{1}} + (xB)_{t_{k}} = B_{τ_{2}} + B_{t_{0}}. From E{(xB)_{t_{k}}} = E{B_{t_{0}}} it follows directly E{B_{τ_{1}}} = E{B_{τ_{2}}}. Applying this setvalued equation to the following stopping times τ_{i}^{A} = τ_{i}χ_{A} + T χ_{Ω\A} (i = 1, 2), where A ∈ 𝔉_{τ_{1}}, we obtain by using the theorems 5.2 and 5.3 of [3] with

E{B_{τ_{1}}^{A}} = E{χ_{A}B_{τ_{1}} + χ_{Ω\A}B_{T} }
= E{χ_{A}E{B_{τ_{1}} + χ_{Ω\A}B_{T} |𝔉_{τ_{1}}}}
= E{χ_{A}B_{τ_{1}} + χ_{Ω\A}E{B_{T} |𝔉_{τ_{1}}}}
= E{(χ_{A} + χ_{Ω\A})B_{τ_{1}}}
= E{B_{τ_{1}}}

and with

E{B_{τ_{2}}^{A}} = E{χ_{A}E{B_{τ_{2}} + χ_{Ω\A}B_{T} |𝔉_{τ_{1}}}}
= E{χ_{A}E{B_{τ_{2}}|𝔉_{τ_{1}}} + χ_{Ω\A}E{E{B_{T} |𝔉_{τ_{2}}}|𝔉_{τ_{1}}}}
= E{E{(χ_{A} + χ_{Ω\A})B_{τ_{2}}|𝔉_{τ_{1}}}}
= E{E{B_{τ_{2}}|𝔉_{τ_{1}}}}

that P-a.s. E{B_{τ_{2}}|𝔉_{τ_{1}}} = B_{τ_{1}}. In this sense for every pair of bounded stopping times the equation E{B_{τ_{1}}} = E{B_{τ_{2}}} is sufficient for B to be a multivalued martingale.

(ii) Let B be a submartingale and x_{t_{k}} = χ_{[t_{k}≤τ_{2}]} −χ_{[t_{k}≤τ_{1}]}. For t_{k} > T we also receive by lemma 1 B_{τ1} + (xB)_{t_{k}} = B_{τ2} + B_{t_{0}}. Because of the submartingale property it follows from E{(xB)_{t_{k}}} ⊇ E{B_{t_{0}}} that E{B_{τ_{2}}} ⊇ E{B_{τ_{1}}}. Applying this to the stopping times τ_{1}^{A}, τ_{2}^{A}, where A ∈ 𝔉_{τ_{1}}, we obtain analogously to the first part with

E{B_{τ_{1}}^{A}} = E{χ_{A}E{B_{τ_{1}} + χ_{Ω\A}B_{T} |𝔉_{τ_{1}}}}
= E{χ_{A}B_{τ_{1}} + χ_{Ω\A}E{B_{T} |𝔉_{τ_{1}}}}
⊇ E{B_{τ_{1}}}

and with E{B_{τ_{2}}^{A}} = ... = E{E{B_{τ_{2}}|𝔉_{τ_{1}}}} that P-a.s. E{B_{τ_{2}}|𝔉_{τ_{1}}} ⊇ B_{τ_{1}}.

□

We derive therefrom the following maximal inequalities.

Lemma 3: If B = (B_{t_{k}}) is a 𝕃_{p}-bounded setvalued submartingale with a finite index set {t_{0}, t_{1}, ..., t_{N}} then for arbitrary λ > 0:

λ P(sup_{t_{k}} |B_{t_{k}}| ≥ λ) ≤ E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}|B_{t_{k}}|} .

Proof: We define a stopping time

τ = \{ \begin{matrix}inf\{t_{k} : |B_{t_{k}}| ≥ λ\} & : & \{t_{k} : |B_{t_{k}}| ≥ λ\} ≠ ∅ P − a.s. \\ t_{N} & : & otherwise \end{matrix} .

From lemma 2 we have

E{B_{t_{N}} } ⊇ E{B_{τ}}
= E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{τ}} + E{χ_{(sup_{t_{k}} |B_{t_{k}} |<λ)}B_{τ}}
= E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{τ}} + E{χ_{(sup_{t_{k}} |B_{t_{k}} |<λ)}B_{t_{N}} }

and in this way

E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{t_{N}} } ⊇ E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{τ}} .  (4)

Applying the Hausdorff pseudometric at first to the left side of inclusion (4) under consideration of theorem 4.1 of [3]

|E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{t_{N}} }| ≤ |clE{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{t_{N}} }|
≤ E{|χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{t_{N}} |}
= E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}|B_{t_{N}} |}

and afterwards to the right side of (4) under consideration of B_{τ}\𝔹_{λ}(\underline{0}) ≠ ∅ on

(sup_{t_{k}} |B_{t_{k}}| ≥ λ) P-a.s. where 𝔹_{λ}(\underline{0}) = \{x ∈ 𝕏 : ∥x∥ < λ\}

|E\{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{τ}\}| ≥ |E\{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}cl𝔹_{λ}(\underline{0})\}|
= E\{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}\} |cl𝔹_{λ}(\underline{0})|
= λ P(sup_{t_{k}} |B_{t_{k}}| ≥ λ)

we estimate

λ P(sup_{t_{k}} |B_{t_{k}}| ≥ λ) ≤ E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}|B_{t_{k}}|} .

□

Lemma 4: Let B be a setvalued martingale (submartingale) with finite index set {t_{0}, t_{1}, ..., t_{N}}. For all p ≥ 1 and arbitrary λ > 0 we have

(a) [λ P(sup_{t_{k}} |B_{t_{k}}| ≥ λ)]^{p} ≤ E{|B_{t_{N}} |^{p}}

and for p > 1

(b) E\{|B_{t_{N}} |^{p}\} ≤ E\{sup_{t_{k}} |B_{t_{k}}|^{p}\} ≤ (\frac{p}{p−1})^{p} E\{|B_{t_{N}} |^{p}\} .

Proof: (a) The transition to |.|^{p} in (4) supplies

|E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{t_{N}} }|^{p} ≥ |E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{τ}}|^{p}

and so, using Jensens inequality, for the left side

|E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{t_{N}} }|^{p} ≤ [E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}|B_{t_{N}} |}]^{p}
≤ E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}|B_{t_{N}} |^{p}}
≤ E{|B_{t_{N}} |^{p}}

and for the right side of the above inequality

|E{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}B_{τ}}|^{p} ≥ [λE{χ_{(sup_{t_{k}} |B_{t_{k}} |≥λ)}}]^{p}
= [λ P(sup_{t_{k}} |B_{t_{k}}| ≥ λ)]^{p} .

Combining both the assertion follows.

(b) The left inequality follows directly from Jensens inequality.

E{|B_{t_{N}} |^{p}} ≤ sup_{t_{k}} E{|B_{t_{k}}|^{p}} ≤ E{sup_{t_{k}} |B_{t_{k}}|^{p}}

For proving the right inequality we set β = sup_{t_{k}} |B_{t_{k}}|. For fixed j > 0 we receive by

applying lemma 3 and Hölders inequality

E{(β ∧ j)^{p}}
= E{∫_{0}^{β∧j} pλ^{p−1}dλ}  = E{∫_{0}^{j} pλ^{p−1}χ_{(β≥λ)}dλ}
= ∫_{0}^{j} pλ^{p−1}P(β ≥ λ)dλ  ≤ ∫_{0}^{j} pλ^{p−2}E{χ_{(β≥λ)}|B_{t_{N}} |}dλ
= pE\{|B_{t_{N}} | ∫_{0}^{β∧j} λ^{p−2}dλ\}  = \frac{p}{p−1}E\{|B_{t_{N}} |(β ∧ j)^{p−1}\}
≤ \frac{p}{p−1} (E\{|B_{t_{N}} |^{p}\})^{\frac{1}{p}} (E\{(β ∧ j)^{p}\})^{\frac{p−1}{p}} .

Summarizing the above estimation we obtain

E\{(β ∧ j)^{p}\} ≤ (\frac{p}{p − 1})^{p} E\{|B_{t_{N}} |^{p}\}

and with j → ∞ the assertion.

□

Now we have to transfer these results to the index set [0, T]. For the index set [0, T] we represent a countable dense subset D of [0, T] as union ⋃_{k} D_{k} of finite subsets D_{k}

of D which form an increasing sequence. By applying lemma 4 to D_{k} and by the limit process k → ∞ we obtain for p ≥ 1 and λ > 0

[λ P(sup_{t∈D} |B_{t}| ≥ λ)]^{p} ≤ sup_{t∈D} E{|B_{t}|^{p}}

and for p > 1

E\{sup_{t∈D} |B_{t}|^{p}\} ≤ (\frac{p}{p − 1})^{p} sup_{t∈D} E\{|B_{t}|^{p}\} .

Because D is a countable dense subset of [0, T] and β = sup_{t∈D} |B_{t}| from the above

considerations the following theorem is derived for rightcontinuous and so also right-h-continuous multivalued martingales and submartingales ([4], corollary 1.2.69).

Theorem: Let B be a right-h-continuous martingale (submartingal) with an index set [0, T] ⊂ ℝ_{+}^{1}. Then we have for λ > 0 and finite p ≥ 1

(a) [λ P( sup_{t∈[0,T]} |B_{t}| ≥ λ)]^{p} ≤ sup_{t∈[0,T]} E{|B_{t}|^{p}}

and for p > 1

(b) E\{ sup_{t∈[0,T]} |B_{t}|^{p}\} ≤ (\frac{p}{p−1})^{p} sup_{t∈[0,T]} E\{|B_{t}|^{p}\} .

Proof: (a) Let D_{1} ⊆ D_{2} ⊆ ... ⊆ D_{k} ⊆ ... be an increasing sequence of finite subsets of [0, T] such that for any t ∈ [0, T] we have a sequence (t_{k}), converging from the right hand side to t, with t_{k} ∈ D_{k}. Because of D = ⋃_{k} D_{k} it follows P-a.s.

sup_{t∈[0,T]} |B_{t}| = sup_{t∈D} |B_{t}| = lim_{k→∞} sup_{t∈D_{k}} |B_{t}| .

sup_{t∈D_{k}} |B_{t}| ≥ λ does not follow from sup_{t∈[0,T]} |B_{t}| ≥ λ for sufficient large k, but the events

[sup_{t∈D_{k}} |B_{t}| ≥ λ] form a nondecreasing sequence and it holds

[ sup_{t∈[0,T]} |B_{t}| > λ] ⊂ ⋃_{k=1}^{∞}[sup_{t∈D_{k}} |B_{t}| ≥ λ].

Therefore we get with lemma 4

P( sup_{t∈[0,T]} |B_{t}| > λ)

≤ P( ⋃_{k=1}^{∞}[sup_{t∈D_{k}} |B_{t}| ≥ λ]) = lim_{k→∞} P(sup_{t∈D_{k}} |B_{t}| ≥ λ)
≤ lim_{k→∞} \frac{1}{λ} sup_{t∈D_{k}} E\{|B_{t}|\} ≤ \frac{1}{λ} sup_{t∈[0,T]} E\{|B_{t}|\}

and by this

P( sup_{t∈[0,T]} |B_{t}| ≥ λ) = lim_{λ′↑λ} P( sup_{t∈[0,T]} |B_{t}| > λ′)
≤ \frac{1}{λ} sup_{t∈[0,T]} E\{|B_{t}|\} .

Applying Jensens inequality to the above estimation we have the first assertion.

(b) sup_{t∈[0,T]} |B_{t}|^{p} = sup_{t∈D} |B_{t}|^{p} P-a.s. because D is a countable dense subset of [0, T]

and B is right-h-continuous. That means a set of probability measure zero exists such that for all t ∈ [0, T] the following relation holds by using the lemma of Fatou and lemma 4.

E{ sup_{t∈[0,T]} |B_{t}|^{p}}

= E{sup_{t∈D} |B_{t}|^{p}}  = E{ lim_{k→∞} sup_{t∈D_{k}} |B_{t}|^{p}}
= E{lim inf_{k→∞} sup_{t∈D_{k}} |B_{t}|^{p}}  ≤ lim inf_{k→∞} E{sup_{t∈D_{k}} |B_{t}|^{p}}
≤ (\frac{p}{p−1})^{p} lim inf_{k→∞} sup_{t∈D_{k}} E\{|B_{t}|^{p}\} ≤ (\frac{p}{p−1})^{p} sup_{t∈[0,T]} E\{|B_{t}|^{p}\}
□

Remark: For the setvalued stochastic (Ito-)integral in the sense of Kisielewicz (see [6], [7] or for a comparison [5]) and for p = 2 we obtain the following estimation

E{ sup_{t∈[0,T]} | (K)∫_{0}^{t} B_{s} dw_{s}|^{2}} ≤ 4 ∫_{0}^{T} E{|B_{t}|^{2}}dt

where (w_{t})_{t≥0} is the usual Brownian motion. In this sense the setvalued stochastic integral is a multivalued martingale.

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