Kenneth Kuttler

848 CHAPTER 31. OPTIONAL SAMPLING THEOREMS

Theorem 31.4.4 Let {X (t)} be a right continuous sub-martingale adapted to thenormal filtration Ft for t ∈ [0,T ] and X∗ (t) defined as in Theorem 31.4.3

X∗ (t)≡ sup{X (s) : 0 < s < t} , X∗ (0)≡ 0,

P([X∗ (T )> λ ])≤ 1λ

E (|X (T )|) (31.4)

For t > 0, letX∗ (t) = inf{X (s) : s < t} .

ThenP([X∗ (T )<−λ ])≤ 1

λE (|X (T )|+ |X (0)|) (31.5)

AlsoP([sup{|X (s)| : s < T}> λ ])

≤ 2λ

E (|X (T )|+ |X (0)|) (31.6)

Proof: The function f (r)= r+≡ 12 (|r|+ r) is convex and increasing. Therefore, X+ (t)

is also a sub-martingale but this one is nonnegative. Also

[X∗ (T )> λ ] =[(

X+)∗(T )> λ

]and so from Theorem 31.4.3,

P([X∗ (T )> λ ]) = P([(

X+)∗(T )> λ

])≤ 1

λE(X+ (T )

)≤ 1

λE (|X (T )|) .

Next letτ = min(inf{t : X (t)<−λ} ,T )

then as before, X (0) ,X (τ) ,X (T ) is a sub-martingale and so∫[τ<T ]

X (τ)dP+∫[τ=T ]

X (τ)dP =∫

Ω

X (τ)dP≥∫

Ω

X (0)dP

Now for ω ∈ [τ < T ] ,X (t)(ω)<−λ for some t < T and so it follows that by right conti-nuity, X (τ)(ω)≤−λ . therefore,

−λ

∫[τ<T ]

dP≥−∫[τ=T ]

X (T )dP+∫

Ω

X (0)dP

If X∗ (T ) < −λ , then from the definition given above, there exists t < T such that X (t) <−λ and so τ < T. If τ < T, then by definition, there exists t < T such that X (t)<−λ andso X∗ (T )<−λ . Hence [τ < T ] = [X∗ (T )<−λ ] . It follows that

P([X∗ (T )<−λ ]) = P([τ < T ])

≤ 1λ

∫[τ=T ]

X (T )dP− 1λ

∫Ω

X (0)dP≤ 1λ

E (|X (T )|+ |X (0)|)

and this proves 31.5.Finally, combining the above two inequalities,

P([sup{|X (s)| : s < T}> λ ]) = P([X∗ (T )<−λ ])+P([X∗ (T )> λ ])

≤ 2λ

E (|X (T )|+ |X (0)|) . ■

848 CHAPTER 31. OPTIONAL SAMPLING THEOREMSTheorem 31.4.4 Let {X (t)} be a right continuous sub-martingale adapted to thenormal filtration F, for t € |0,T] and X* (t) defined as in Theorem 31.4.3X* (t) =sup {X (s) :0<s<t}, X*(0) =0,1P(X" (T) > A]) < TE (IX (T)I) (31.4)For t > 0, letX, (t) = inf {X (s) :5 <r}.Then 1P(X. (T) <—A]) s 7 E (|X (T)| + |X (0)1) (31.5)AlsoP([sup {|X (s)|:s<T}>A])< FB (\K(T)|+1X O)) G16)Proof: The function f(r) = r+ = 5 (|r| +r) is convex and increasing. Therefore, X* (r)is also a sub-martingale but this one is nonnegative. Also[X*(T) >A] = [(X*)*(T) >A]and so from Theorem 31.4.3,1P([X*(T) >A]) =P([(X*)* (7) >A]) < ye (X* (T)) < —E (|X (T)|).Next letT = min (inf {t: X (t) < -A},T)then as before, X (0) ,X (7) ,X (T) is a sub-martingale and soX (t)dP+ x(r)aP= | x(e)aP> | x@ar[t<T] [t=7] JQ QNow for @ € [t < T],X (t)(@) < —A for some t < T and so it follows that by right conti-nuity, X (t)(@) < —A. therefore,A ap>— | x(r)dP+ | x (0)aP[t<T] [t=T] QIf X, (7) < —A, then from the definition given above, there exists t < T such that X (t) <—A and so t < T. If t < T, then by definition, there exists t < T such that X (t) << —A andso X,(T) < —A. Hence [t < T] = [X. (T) < —A]. It follows thatP([X.(T) < —A]) = P([t < T])1 1 1<q fx (Mar—z [x War < BUX TI + 1X (O)Xrand this proves 31.5.Finally, combining the above two inequalities,P([sup {|X (s)| 8 < T} > A]) = P([X. (7) < —A]) + P(X" (7) > A)< ££ (\X(T)|+|X(0)))