BMO Martingales Research Articles

Representation of solutions to quadratic 2BSDEs with unbounded terminal values

Second order backward stochastic differential equations (2BSDEs, for short) are one of useful tools in solving stochastic control problems with model uncertainty. In this paper, we prove a representation formula for quadratic 2BSDEs with an unbounded terminal value under a convex assumption on the generator. Because of the unboundedness of the terminal value, we are unable to use some fine properties of BMO martingales, which are often employed in the literature to deal with bounded solutions to quadratic backward stochastic differential equations. Instead, we utilize the θ-technique. We also prove an existence result under an additional assumption that the terminal value is of uniformly continuous.

Constrained mean-variance investment-reinsurance under the Cramér–Lundberg model with random coefficients

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under the Cramér–Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively.

Quadratic BSDEs with mean reflection driven by G-brownian motion

In this paper, the well-posedness of a kind of backward stochastic differential equation driven by [Formula: see text]-Brownian motions ([Formula: see text]-BSDEs for short) with mean reflection is investigated. In particular, the generator is allowed having quadratic growth in [Formula: see text]. Combining a representation of the solution with [Formula: see text]-BMO martingale techniques, fixed point argument, and [Formula: see text]-method, existence and uniqueness results for such [Formula: see text]-BSDEs are provided under bounded and unbounded terminal condition.

Mean-field type quadratic BSDEs

<p style='text-indent:20px;'>In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean. The general case can be treated by a fixed point argument.</p>

A Modified Method of Successive Approximations for Stochastic Recursive Optimal Control Problems

Based on the stochastic maximum principle for the partially coupled forward-backward stochastic control system (FBSCS for short), a modified method of successive approximations (MSA for short) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step since the control domain is not necessarily convex. Thanks to the theory of bounded mean oscillation martingales (BMO martingales for short), we give a delicate proof of the error estimate and then prove the convergence of the modified MSA algorithm. In a special case, we obtain a logarithmic convergence rate. When the control domain is convex and compact, a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear FBSCSs.

Sharp Lorentz-norm estimates for BMO martingales

Let X be a BMO martingale with continuous paths and let 2≤q≤p<∞ be given parameters. The paper contains the proof of the Lorentz-norm inequality ‖X∞‖p,q≤2−1∕pp∕q(q+1)∕qΓ(q+1)1∕q‖X‖BMO,and the constant is shown to be the best possible.

BMO martingale method for backward stochastic differential equations driven by general càdlàg local martingales

BMO martingale method for backward stochastic differential equations driven by general càdlàg local martingales

An equivalent characterization of weak BMO martingale spaces

In this paper, we give an equivalent characterization of weak BMO martingale spaces due to Ferenc Weisz 1998.

Some new properties concerning BLO martingales

Some new properties concerning BLO martingales are given. The BMO-BLO boundedness of martingale maximal functions and Bennett type characterization of BLO martingales are shown. Also, a non-negative BMO martingale that is not in BLO is constructed.

The predual and John-Nirenberg inequalities on generalized BMO martingale spaces

In this paper we introduce the generalized BMO martingale spaces by stopping time sequences, which enable us to characterize the dual spaces of martingale Hardy-Lorentz spaces H s p,q for 0 < p ≤ 1,1 < q < ∞. Moreover, by duality we obtain a John-Nirenberg theorem for the generalized BMO martingale spaces when the stochastic basis is regular. We also extend the boundedness of fractional integrals to martingale Hardy-Lorentz spaces.

Numerical simulation of quadratic BSDEs

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to $z$ and bounded terminal conditions. We first study a slight modification of the classical dynamic programming equation arising from the time-discretization of BSDEs. By using a linearization argument and BMO martingales tools, we obtain a comparison theorem, a priori estimates and stability results for the solution of this scheme. Then we provide a control on the time-discretization error of order $\frac{1}{2}-\varepsilon$ for all $\varepsilon>0$. In the last part, we give a fully implementable algorithm for quadratic BSDEs based on quantization and illustrate our convergence results with numerical examples.

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose ith component depends only on the ith row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John–Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in El Karoui and Hamadène [Stochastic Process. Appl. 107 (2003), page 164].

Sharp maximal estimates for BMO martingales

We introduce a method which can be used to study maximal inequalities for martingales of bounded mean oscillation. As an application, we establish sharp $\Phi$-inequalities and tail inequalities for the one-sided maximal function of a BMO martingale. The results can be regarded as BMO counterparts of the classical maximal estimates of Doob.

BMO martingales and positive solutions of heat equations

In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds.More precisely, we give some estimatesof the gradient of logarithm of a positive solution via the uniform bound ofthe logarithm of the solution. Moreover, we give a generalized version ofLi-Yau's estimate. Our proof is based on the link between PDE and quadraticBSDE. Our method might be useful to study some (nonlinear) PDEs.

Optimal investment, consumption and proportional reinsurance for an insurer with option type payoff

This paper is devoted to the study of optimization of investment, consumption and proportional reinsurance for an insurer with option type payoff at the terminal time under the criterion of exponential utility maximization. The surplus process of the insurer and the financial risky asset process are assumed to be diffusion processes driven by Brownian motions which are non-Markovian in general. Very general constraints are imposed on the investment and the proportional reinsurance processes. Based on the martingale optimization principle, we use BSDE and BMO martingale techniques to derive the optimal strategy and the optimal value function. Some interesting particular cases are studied in which the explicit expressions for the optimal strategy are given by using the Malliavin calculus.

A simple constructive approach to quadratic BSDEs with or without delay

This paper provides a simple approach for the consideration of quadratic BSDEs with bounded terminal conditions. Using solely probabilistic arguments, we retrieve the existence and uniqueness result derived via PDE-based methods by Kobylanski (2000) [14]. This approach is related to the study of quadratic BSDEs presented by Tevzadze (2008) [19]. Our argumentation, as in Tevzadze (2008) [19], highly relies on the theory of BMO martingales which was used for the first time for BSDEs by Hu et al. (2005) [12]. However, we avoid in our method any fixed point argument and use Malliavin calculus to overcome the difficulty. Our new scheme of proof allows also to extend the class of quadratic BSDEs, for which there exists a unique solution: we incorporate delayed quadratic BSDEs, whose driver depends on the recent past of the Y component of the solution. When the delay vanishes, we verify that the solution of a delayed quadratic BSDE converges to the solution of the corresponding classical non-delayed quadratic BSDE.

Backward stochastic differential equations with time delayed generators—results and counterexamples

We deal with backward stochastic differential equations with time delayed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lipschitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a BSDE with time delayed generator, including the inheritance of boundedness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.

Harmonic analysis of stochastic equations and backward stochastic differential equations

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in $${\mathcal{R}^p}$$ ( $${p\in [1,\infty)}$$ ) and backward stochastic differential equations (BSDEs) in $${\mathcal{R}^p\times \mathcal{H}^p}$$ ( $${p\in (1, \infty)}$$ ) and in $${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$$ , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Holder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.

Carleson measures and vector-valued BMO martingales

We study the relationship between vector-valued BMO martingales and Carleson measures. Let $${(\Omega,\mathcal {F} ,P)}$$ be a probability space and 2 ≤ q < ∞. Let X be a Banach space. Given a stopping time τ, let $${\widehat{\tau}}$$ denote the tent over τ: $$\widehat{\tau}=\{(w,k)\in \Omega\times \mathbb {N}: \tau(w)\leq k, \tau(w) < \infty\}.$$ We prove that there exists a positive constant c such that $$\sup_{\tau}\frac{1}{P(\tau < \infty)}\int \limits_{{\widehat{\tau}}}\|df_k\|^qdP\otimes dm\leq c^q\|f\|_{BMO(X)}^q$$ for any finite martingale with values in X iff X admits an equivalent norm which is q-uniformly convex. The validity of the converse inequality is equivalent to the existence of an equivalent p-uniformly smooth norm. And then we also give a characterization of UMD Banach lattices.

Time consistent dynamic risk processes

A crucial property for dynamic risk measures is the time consistency. In this paper, a characterization of time consistency in terms of a “cocycle condition” for the minimal penalty function is proved for general dynamic risk measures continuous from above. Then the question of the regularity of paths is addressed. It is shown that, for a time consistent dynamic risk measure normalized and non-degenerate, the process associated with any bounded random variable has a càdlàg modification, under a mild condition always satisfied in the case of continuity from below. When normalization is not assumed, a right continuity condition on the penalty has to be added. Applying these results and using right continuous BMO martingales, families of not necessarily normalized dynamic risk measures leading to càdlàg paths, and allowing for jumps, are exhibited.