Second-order BSDEs with jumps: Formulation and uniqueness

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

doi:10.1214/14-aap1063

Cited by 25 publications

(29 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our result below extends all the results for continuous processes to markets with nonlinear portfolio dynamics. Of course, the same proof would go through for the more general jump case, provided that a 2BSDE theory, extending that of [53,54], is obtained in such a setting.…”

Section: A Super-hedging Duality In Uncertain Incomplete and Nonlinementioning

confidence: 79%

“…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%

See 1 more Smart Citation

Stochastic control for a class of nonlinear kernels and applications

Possamaï¹,

Tan²,

Zhou³

2018

Ann. Probab.

Self Cite

127

View full text Add to dashboard Cite

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...

show abstract

Section: A Super-hedging Duality In Uncertain Incomplete and Nonlinementioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Stochastic control for a class of nonlinear kernels and applications

Possamaï¹,

Tan²,

Zhou³

2018

Ann. Probab.

Self Cite

127

View full text Add to dashboard Cite

show abstract

“…which is the same as (3.6). ‡ ‡ This family could also be characterized by considering all the choices of control u for which there exists at least one strong solution to the SDE for M (see Soner, Touzi and Zhang (2011)), or, equivalenntly, to a certain martingale problem (see Kazi-Tani, Possamaï and Zhou (2013) and Neufeld and Nutz (2014)).…”

Section: Discussionmentioning

confidence: 99%

Moral Hazard in Dynamic Risk Management

2017

Self Cite

View full text Add to dashboard Cite

We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. We identify a family of admissible contracts for which the optimal agent's action is explicitly characterized, and, using the recent theory of singular changes of measures for Itô processes, we study how restrictive this family is. In particular, in the special case of the standard Homlstrom-Milgrom model with fixed volatility, the family includes all possible contracts. We solve the principal-agent problem in the case of CARA preferences, and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the quadratic variation component of the optimal contract.

show abstract

“…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

View full text Add to dashboard Cite

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

show abstract

Second-order BSDEs with jumps: Formulation and uniqueness

Cited by 25 publications

References 37 publications

Stochastic control for a class of nonlinear kernels and applications

Stochastic control for a class of nonlinear kernels and applications

Moral Hazard in Dynamic Risk Management

A semigroup approach to nonlinear Lévy processes

Contact Info

Product

Resources

About