Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs

Kazi-Tani, Nabil; Possamaï, Dylan; Zhou, Chao

doi:10.1214/ejp.v20-3569

Cited by 14 publications

(16 citation statements)

References 34 publications

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“…In Hollender [13] the results of [20] are generalized to upper expectations over state-dependent Lévy triplets, see also Kühn [16] for existence results on the respective integro-differential equations under fairly general conditions. A related concept to nonlinear Lévy processes are second order backward stochastic differential equation with jumps, see Kazi-Tani et al [14], [15] and also Soner et al [28]. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

A semigroup approach to nonlinear Lévy processes

Denk

Kupper

Nendel

2020

Stochastic Processes and their Applications

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We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-toone relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators (A λ ) λ∈Λ of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE ∂tu = sup λ∈Λ A λ u. The results are illustrated with several examples.Proof. As J π is a sublinear Markovian convolution for all π ∈ P t , the same holds for S (t). For f ∈ D, x ∈ G and ε > 0 there exists π 0 ∈ P t such thatBy Lemma 5.1 it follows thatFrom this, we obtain that lim tց0 S (t)f − f ∞ = 0 for f ∈ BUC(G) with the same density argument as in the proof of Lemma 5.2. 18ROBERT DENK, MICHAEL KUPPER, AND MAX NENDEL Lemma 5.4. For t ≥ 0, let (π n ) n∈N be a sequence in P t such that π n ⊂ π n+1 for all n ∈ N, and lim n→∞ |π n | ∞ = 0.ThenProof. Fix f ∈ BUC(G). For t = 0, the statement is trivial. For t > 0 and x ∈ G we defineThen, J ∞ is a sublinear Markovian convolution. Since π n ⊂ π n+1 , it follows from (5.3) that J πn f ր J ∞ f, as n → ∞. By definition of S (t), it clearly holds J ∞ f ≤ S (t)f . As for the other inequality, let π = {t 0 , t 1 , . . . , t m } ∈ P t with m ∈ N and 0 = t 0 < t 1 < . . . < t m = t. Since |π n | ∞ ց 0 as n → ∞, we may w.l.o.g. assume that #π n ≥ m + 1 for all n ∈ N. Moreover, we can choose 0 = t n 0 < t n 1 < . . . < t n m = t with π ′ n := {t n 0 , t n 1 , . . . , t n m } ⊂ π n and lim n→∞ t n i = t i for all i = 1, . . . , m − 1. Then, by Lemma 5.2, we have thatTaking the supremum over all π ∈ P t , we thus get thatCorollary 5.5. Let t ≥ 0. Then, there exists a sequence (π n ) in P t such thatMoreover, S (t)f = sup n∈N J t n n f = sup n∈N J 2 −n t 2 n f = lim n→∞ J 2 −n t 2 n f, (5.6) for all f ∈ BUC(G), where the supremum is understood pointwise.Proof. Choose π n := kt 2 n : k ∈ {0, . . . , 2 n } in Lemma 5.4 to obtain the first statement. In particular, S (t)f = S (t)f = sup n∈N J πn f = sup n∈N J 2 −n t 2 n

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Section: Introductionmentioning

confidence: 99%

A semigroup approach to nonlinear Lévy processes

Denk

Kupper

Nendel

2020

Stochastic Processes and their Applications

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“…Again, these regularity assumptions are made to obtain the continuity of the value function a priori, which allows to avoid completely the use of the measurable selection theorem. Since then, the 2BSDE theory has been extended by allowing more general generators, filtrations and constraints (see [53,54,63,65,79,81]), but no progress has been made concerning the regularity assumptions. However, the 2BSDEs (see for instance [66]) have proved to provide a particularly nice framework to study the so-called robust problems in finance, which were introduced by [2,61] and in a more rigorous setting by [27].…”

Section: Introductionmentioning

confidence: 99%

Stochastic control for a class of nonlinear kernels and applications

Possamaï¹,

Tan²,

Zhou³

2018

Ann. Probab.

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127

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We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).1 generally based on the stability of the controls with respect to conditioning and concatenation, together with a measurable selection argument, which, roughly speaking, allow to prove the measurability of the associated value function, as well as constructing almost optimal controls through "pasting". This is exactly the approach followed by Bertsekas and Shreve [6], and Dellacherie [25] for discrete time stochastic control problems. In continuous time, a comprehensive study of the dynamic programming principle remained more elusive. Thus, El Karoui,in [34], established the dynamic programming principle for the optimal stopping problem in a continuous time setting, using crucially the strong stability properties of stopping times, as well as the fact that the measurable selection argument can be avoided in this context, since an essential supremum over stopping times can be approximated by a supremum over a countable family of random variables. Later, for general controlled Markov processes (in continuous time) problems, El Karoui, Huu Nguyen and Jeanblanc [36] provided a framework to derive the dynamic programming principle using the measurable selection theorem, by interpreting the controls as probability measures on the canonical trajectory space (see e.g. Theorems 6.2, 6.3 and 6.4 of [36]). Another commonly used approach to derive the DPP was to bypass the measurable selection argument by proving, under additional assumptions, a priori regularity of the value function. This was the strategy adopted, among others, by Fleming and Soner [43], and in the so-called weak DPP of Bouchard and Touzi [15], which has then been extended by Bouchard and Nutz [10,12] and Bouchard, Moreau and Nutz [9] to optimal control problems wit...

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“…This would be very unfortunate from the point of view of applications, since, if we look, for instance, at classical problems of portfolio optimization in finance, the process Z is usually related to the corresponding optimal investment strategy. Therefore, in a context of uncertainty, one will definitely need an optimal strategy which works for every possible model, that is to say for every measure P. Nonetheless, we prove that the solution of a 2BSDEJ, (Y, Z, U ), can still be constructed in such a way that it is defined for all ω, independently of probability measures; we refer the reader to our companion paper [21] for more details.…”

Section: A Primer On 2bsdejs and Main Difficultiesmentioning

confidence: 82%

“…In this context, the 2BSDEJs are the natural candidates for a probabilistic solution of fully nonlinear integro-differential equations. This is the purpose of our accompanying paper [21].…”

Section: Second-order Bsdes With Jumps: Formulation and Uniquenessmentioning

confidence: 97%

Second-order BSDEs with jumps: Formulation and uniqueness

Kazi-Tani¹,

Possamaï²,

Zhou³

2015

Ann. Appl. Probab.

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In this paper, we define a notion of second-order backward stochastic differential equations with jumps (2BSDEJs for short), which generalizes the continuous case considered by Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our setting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measures. Then in our formulation of 2BSDEJs, the generator of the 2BSDEJs depends also on the underlying probability measures through the compensator. But the solution to the 2BSDEJs can still be defined universally. Moreover, we obtain a representation of the $Y$ component of a solution of a 2BSDEJ as a supremum of solutions of standard backward SDEs with jumps, which ensures the uniqueness of the solution.Comment: Published at http://dx.doi.org/10.1214/14-AAP1063 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs

Abstract: International audienc

Cited by 14 publications

References 34 publications

A semigroup approach to nonlinear Lévy processes

A semigroup approach to nonlinear Lévy processes

Stochastic control for a class of nonlinear kernels and applications

Second-order BSDEs with jumps: Formulation and uniqueness

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