Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.

We establish some well-posedness and comparison results for BSDEs driven by one-and multi-dimensional martingales. On the one hand, our approach is largely motivated by results and methods developed in Carbone et al. [3] and El Karoui and Huang [7]. On the other hand, our results are also motivated by the recent developments in arbitrage pricing theory under funding costs and collateralization. A new version of the comparison theorem for BSDEs driven by a multi-dimensional martingale is established and applied to the pricing and hedging BSDEs studied in Bielecki and Rutkowski [1] and Nie and Rutkowski [25]. This allows us to obtain the existence and uniqueness results for unilateral prices and to demonstrate the existence of no-arbitrage bounds for a collateralized contract when both agents have non-negative initial endowments.

We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets Γ k of R m . The decentralized strategies for individual agents and consistency condition system are represented in an unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on Γ k . The well-posedness of consistency system is established in both the local and global cases by the contraction mapping and discounting method respectively. Related ε−Nash equilibrium property is also verified.

We examine the pricing and hedging of general contracts in an extension of the market model proposed by [B-1995]. We study both problems from the perspectives of the hedger and the counterparty with arbitrary initial endowments. We derive inequalities satisfied by unilateral prices of a contract and we give the range for its fair bilateral prices. Our study hinges on results for backward stochastic differential equations (BSDEs) driven by multi-dimensional continuous martingales obtained in Nie & Rutkowski (2014b). We also derive the pricing partial differential equations (PDEs) for path-independent contingent claims of European style in a Markovian framework.

Results from Nie and Rutkowski [12,14] are extended to the case of the margin account, which may depend on the contract's value for the hedger and/or the counterparty (recall that the collateral was given exogenously in [12,14]). The present work generalizes also the papers by Bergman [1], Mercurio [11] and Piterbarg [16]. Using the comparison theorems for BSDEs, we derive inequalities for the unilateral prices and we give the range for its fair bilateral prices. We also establish results yielding the link to the market model with a single interest rate. In the case where the collateral amount is negotiated between the counterparties, so that it depends on their respective unilateral values, the backward stochastic viability property studied by Buckdahn et al. [4] is used to derive the bounds on fair bilateral prices.

In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as:where η is a stochastic process given by η(t) = η(0) + t 0 σ(s)δB H (s), t ∈ [0, T ], and B H is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng's paper, BDSEs driven by fBm, SIAM J Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equationwhere ∂ϕ is a multivalued operator of subdifferential type associated with the convex function ϕ. (2010): 60H10, 60G22, 47J20, 60H05 (Lucian Maticiuc), nietianyang@163.com (Tianyang Nie) Mathematics Subject ClassificationHere, in our manuscript, we work without such a condition. Let us mention that in [12], it is assumed thatIn our paper, we adopt the formT ], since we will use the proof of Theorem 3.8 from [12], especially the quasi-conditional expectation formulâ. Based on the above-described framework, we prove the existence and the uniqueness for BSDE (1). This approach includes, in particular, first a discussion of the equationAfter, the existence for BSDE (1) is proved by using a fixed point theorem over an appropriate Banach space. Based on our results on BSDE driven by a fBm and on Pardoux and Rȃşcanu [19] on BSVI governed by a standard Brownian motion, we consider the following fractional BSVIwhere ∂ϕ is the subdifferential of a convex lower semicontinuous (l.s.c.) function ϕ : R → (−∞, +∞]. The existence of the solution will be proved. Now, we give the outline of our paper: In Section 2 we recall some definitions and results about fractional stochastic integrals and the related Itô formula. We present the assumptions and some auxiliary results including the Itô formula w.r.t. the divergence-type integral in Section 3. Section 4 is devoted to prove the existence and the uniqueness result for BSDE driven by a fBm. In Section 5, we study the existence for fractional BSVI governed by a fBm. Finally, in the Appendix, we prove a more general Itô formula based on Theorem 8 [3] and an auxiliary lemma. Preliminaries: Fractional stochastic calculusIn this section, we shall recall some important definitions and results concerning the Malliavin calculus, the stochastic integral with respect to a fBm, and Itô's formula. For a deeper discussion, we refer the reader to [2, 3, 7, 9, 10] and [16].Throughout our paper, we assume that the Hurst parameter H always satisfies H > 1/2. Define φ(x) = H(2H − 1)|x| 2H−2 , x ∈ R. Let us denote by |H| the Banach space of measurable functions f : [0, T ] → R such that f 2 |H| := T 0 T 0 φ(u − v)|f (u)||f (v)|dudv < +∞.

We consider a kind of stochastic exit time optimal control problems, in which the cost function is defined through a nonlinear backward stochastic differential equation. We study the regularity of the value function for such a control problem. Then extending Peng's backward semigroup method, we show the dynamic programming principle. Moreover, we prove that the value function is a viscosity solution to the following generalized Hamilton-Jacobi-Bellman equation with Dirichlet boundary:  

This paper deals with a stochastic recursive optimal control problem, where the diffusion coefficient depends on the control variable and the control domain is not necessarily convex. We focus on the connection between the general maximum principle and the dynamic programming principle for such control problem without the assumption that the value is smooth enough, the set inclusions among the sub-and super-jets of the value function and the first-order and secondorder adjoint processes as well as the generalized Hamiltonian function are established. Moreover, by comparing these results with the classical ones in Yong and Zhou [Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999], it is natural to obtain the first-and second-order adjoint equations of Hu [Direct method on stochastic maximum principle for optimization with recursive utilities, arXiv:1507.03567v1 [math.OC], 13 Jul. 2015].1. Introduction. There are usually two ways to study optimal control problems: Pontryagin's maximum principle (MP) and Bellman's dynamic programming principle (DPP), involving the adjoint variable ψ, the Hamiltonian function H, and the value function V , respectively. The classical result on the connection between the MP and the DPP for the deterministic optimal control problem can be seen in Fleming and Rishel [10], which is known as ψ(t) = −V x (t,x(t)) and V t (t,x(t)) = H(t,x(t),ū(t), ψ(t)), whereū is the optimal control andx is the optimal state. Since the value function V is not always smooth, some non-smooth versions of the classical result were studied by using non-smooth analysis and generalized derivatives. An attempt to relate the MP and the DPP without assuming the smoothness of the value function was first made by Barron and Jessen [1], where the viscosity solution was used to derive the MP from the DPP. Within the framework of viscosity solution, Zhou [35] showed that( 1.1) where D 1,− x V, D 1,+ x V denote the first-order sub-and super-jets of V in the x-variable, and D − t V, D + t V denote the sub-and super-jets of V in the t-variable, respectively.