Existence and Comparisons for BSDEs in general spaces Samuel N. Cohen (joint work with Robert J. Elliott) University of Oxford (University of Adelaide) 25 October 2010 S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 1 / 24
Outline BSDEs in general spaces 1 Comparison results 2 Nonlinear Expectations 3 Conclusions 4 S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 2 / 24
Classical BSDEs Classically, a BSDE is an equation of the form � � Y t − F ( ω, u , Y u , Z u ) du + Z u dW u = Q ] t , T ] ] t , T ] where the solution pair ( Y , Z ) is adapted, Z is predictable and Q is some F T -measurable random variable. My interest is on generalising these equations to allow for different types of filtrations and randomness. Various generalisations of the filtration have been done (eg Jump processes, Markov chains) Various generalisations of this structure are possible (eg delay equations, general semimartingale decompositions) I seek to retain the structure, but work in a general filtration. S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 3 / 24
BSDEs in Discrete time My recent work has considered BSDEs in discrete time, finite state systems � � Y t − F ( ω, u , Y u , Z u ) + Z u ∆ M u + 1 = Q . t ≤ u < T t ≤ u < T where M is a R N -valued martingale defining the filtration Existence and comparison results can be obtained for these equations These equations form a complete representation of time-consistent nonlinear expectations on L 0 ( F T ) . Is there a way to unite this discrete time theory with the classical one? S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 4 / 24
BSDEs in general spaces Today we will consider BSDEs where both the martingale and driver terms can jump. This will include, as special cases, both the discrete time and continuous time theory of BSDEs Very few assumptions are needed on the underlying probability space. Our first step is to state a general form of the Martingale representation theorem... S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 5 / 24
Theorem (Davis & Varaiya 1974) Let (Ω , F T , {F t } t ∈ [ 0 , T ] , P ) be a filtered probability space. Suppose L 2 ( F T ) is separable. Then there exists a sequence of martingales M 1 , M 2 ... such that any martingale N can be written as ∞ � � Z i u dM i N t = N 0 + u ] 0 , t ] i = 1 for some predictable processes Z i , and � M 1 � ≻ � M 2 � ≻ ... as measures on Ω × [ 0 , T ] . i.e. � M i � ( A ) = E [ � [ 0 , T ] I A d � M i � ] for A ⊆ Ω × [ 0 , T ] . S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 6 / 24
We need an appropriate norm for { Z i } i ∈ N under which to consider continuity of the driver F . Definition Let µ be a fixed deterministic nonnegative Stieltjes measure on [ 0 , T ] . For each i ∈ N , let � M i � t = m i , 1 + m i , 2 t t where m i , 1 (resp. m i , 2 ) is absolutely continuous (resp. singular) with respect to P × µ , as measures on P . Then define � · � M t , the stochastic seminorm on infinite R K -valued sequences, by ∞ dm i , 1 � ( z 1 , z 2 , ... ) � 2 � � z i � 2 t M t = d ( P × µ t ) . i = 1 S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 7 / 24
This norm has some useful properties: dm i , 1 If µ t = t and M i t = W t , then d ( P × µ t ) = 1, and so � z � M t ≡ � z � ℓ 2 . t If the filtration has finite multiplicity, then all but finitely many of the M i t are zero, and this all degenerates to the Euclidean norm. If � � ˜ � Z i t dM i � Z i t dM i t = t , i i then � Z t − ˜ Z t � M t = 0, µ -a.e. No matter our choice of µ , �� � � � � � � Z u � 2 � Z i t � 2 d � M i � t E d µ ≤ E M u ] t , T ] ] t , T ] i S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 8 / 24
BSDEs in general spaces Consider an equation of the form: ∞ � � � Z i u dM i Y t − F ( ω, u , Y u − , Z u ) d µ + u = Q ] t , T ] ] t , T ] i = 1 where Q ∈ L 2 ( F T ) , Y ∈ R K is adapted and sup t ∈ [ 0 , T ] {� Y t � 2 } < ∞ , Z t ≡ ( Z 1 , Z 2 , ... ) is a sequence of predictable R K -valued processes such that Z ∈ H 2 M , that is �� � � � Z i t � 2 d � M i � t E < ∞ ] 0 , T ] i S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 9 / 24
BSDEs in general spaces ∞ � � � Z i u dM i Y t − F ( ω, u , Y u − , Z u ) d µ + u = Q ] t , T ] ] t , T ] i = 1 Also, µ is a deterministic Stieltjes measure on [ 0 , T ] . For simplicity, assume µ is nonnegative. F is a progressively measurable function such that F ( ω, t , 0 , 0 ) is µ -square-integrable. S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 10 / 24
Existence result Theorem Suppose F is firmly Lipschitz, that is, there exists a constant c and a map c ( · ) : [ 0 , T ] → [ 0 , c ] such that � F ( ω, t , y , z ) − F ( ω, t , y ′ , z ′ ) � 2 ≤ c t � y − y ′ � 2 + c � z − z ′ � 2 M t and c t (∆ µ t ) 2 < 1 . Then the BSDE has a unique solution, (up to indistinguishability if d µ ≻ dt). S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 11 / 24
As the discrete time BSDE can be embedded in continuous time, and the necessary and sufficient condition for existence in discrete time is that y �→ y − F ( ω, t , y , z ) is a bijection, the classical requirement of Lipschitz continuity is clearly insufficient. On the other hand, if µ is continuous, then these assumptions are simply classical Lipschitz continuity. By the use of the Radon-Nikodym theorem for measures on Ω × [ 0 , T ] , the requirement that µ is deterministic and nonnegative is somewhat flexible, as exceptions can be instead incorporated into F . S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 12 / 24
From a mathematical perspective, this unites the theory of BSDEs in discrete and continuous time. From a modelling perspective, it allows us to build models without quasi-left-continuity. For interest rate modelling, when central bank decisions are announced on certain dates. For evaluating contracts where some counterparty decisions must be made on a certain date. Allowing these discontinuities is one step closer to a general semimartingale theory of BSDEs. We now proceed to the proof of existence and uniqueness. S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 13 / 24
Definition (Stieltjes-Doleans-Dade Exponentials) For any cadlag function of finite variation ν , let E ( ν ; t ) = e ν t � ( 1 + ∆ ν s ) e − ∆ ν s . 0 ≤ s ≤ t and if ∆ ν s < 1 a.s. (∆ ν s ) 2 � ν ; t ) − 1 . ν t = ν t + ˜ and E ( − ν ; t ) = E (˜ 1 − ∆ ν s 0 ≤ s ≤ t Lemma (Backwards Grönwall inequality with jumps) For semimartingales u, w, a finite-variation process ν with ∆ ν s < 1 a.s., if du t ≥ − u t d ν t + dw t then ν ; t )) ≥ ( 1 − ∆ ν t ) − 1 E (˜ d ( u t E (˜ ν ; t − ) dw t . S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 14 / 24
Lemma (Bound on BSDE solutions) Let Y be a solution to a BSDE with firm Lipschitz driver, and let Z ∈ H 2 M . Then E [ sup t ∈ [ 0 , T ] {� Y t � 2 } ] < ∞ if and only if � E [ � Y t − � 2 ] d µ < ∞ . ] 0 , T ] Lemma (BSDEs, no dependence on Y , Z ) Let F : Ω × [ 0 , T ] → R K . Then a BSDE with driver F has a solution. Proof. Simple application of martingale representation theorem. S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 15 / 24
Bound on solutions Assume µ T ≤ 1 and c t ∆ µ t < 1. We have the following bound: Lemma For two BSDEs with solutions Y , Y ′ , etc. let δ Y := Y − Y ′ , δ Z := Z − Z ′ , δ 2 f t = F ( ω, t , Y ′ t − , Z ′ t ) − F ′ ( ω, t , Y ′ t − , Z ′ t ) . For meas. x , w : [ 0 , T ] → [ 0 , ∞ ] with ∆ µ t ≤ x − 1 , any A ∈ B ([ 0 , T ]) , t � � � dE [ � δ Y t � 2 ] ≥ − E [ � δ Y t � 2 ] d υ t − E [ � δ 2 f t � 2 ]( 1 − ∆ υ t ) d π t A A A �� � � � δ Z it � 2 ( 1 − ∆ υ t ) d ρ i + E . t A i d υ t = [( x − 1 − ∆ µ t )( 1 + w t ) c t + x t ] d µ t t d π t = [( x − 1 − ∆ µ t )( 1 + w − 1 )]( 1 − ∆ υ t ) − 1 d µ t t t d ρ i t = [ 1 − ( x − 1 − ∆ µ t )( 1 + w t ) c ]( 1 − ∆ υ t ) − 1 d � M i � t t S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 16 / 24
Sketch proof of existence theorem Under the assumption µ T ≤ 1, and c t (∆ µ t ) < 1, Note that as c t (∆ µ t ) is summable and strictly bounded by 1, it is bounded by 1 − ǫ Use Picard iteration on Z , (easy, convergence in equivalent norm at rate 1 / 2) Then iterate on Y , (harder, convergence rate 1 − ǫ 2 / 8) Use a measure-change argument to separate [ 0 , T ] into a finite sequence of pieces of size < 1, use backward induction to establish result. This also relaxes to assuming c t (∆ µ t ) 2 < 1. S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 17 / 24
Comparison results With our existence theory, we now wish to be able to compare solutions to BSDEs. As our martingales can jump, we need to be careful. A comparison result is closely related to a nonlinear no-arbitrage result, so similar language may be helpful. For simplicity, we shall consider the scalar case only. S.N. Cohen (Oxford) BSDEs in general spaces 25 October 2010 18 / 24
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