BSDEs under partial information and financial applications

In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square-integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Föllmer–Schweizer decomposition of a square-integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model.     

BSDEs in utility maximization with BMO market price of risk

This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum of distinct square-integrable solutions. This feature occurs since, contrary to the classical Itô representation theorem, a representation of random variables in terms of stochastic exponentials is not unique. We study in detail when the BSDE has a bounded solution and derive a new dynamic exponential moments condition which is shown to be the minimal sufficient condition in a general filtration. The main results are complemented by several interesting examples which illustrate their sharpness as well as important properties of the utility maximization BSDE.     

BSDE with quadratic growth and unbounded terminal value

In this paper, we study the existence of solution to BSDE with quadratic growth and unbounded terminal value. The main idea consists in using a localization procedure together with a priori bounds.     

Some results on quadratic hedging with insider trading

We consider the hedging problem in an arbitrage-free incomplete financial market, where there are two kinds of investors with different levels of information about the future price evolution, described by two filtrations F and G=F∨σ(G) where G is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches.

In particular, for LRM, we show that for a large class of additional non trivial r.v.s G both investors will pursue the same locally risk minimizing portfolio strategy and the cost process of the ordinary agent is just the projection on F of that of the insider. For the MVH approach, we study also some general stochastic volatility model, including Hull and White, Heston and Stein and Stein models. In this more specific setting and for r.v.s G which are measurable with respect to the filtration generated by the volatility process, we obtain an expression for the insider optimal strategy in terms of the ordinary agent optimal strategy plus a process admitting a simple feedback-type representation.     

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$Y$ 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.     

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z$Z$. We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.     

A BSDE approach to risk-based asset allocation of pension funds with regime switching

An asset allocation problem of a member of a defined contribution (DC) pension fund is discussed in a hidden, Markov regime-switching, economy using backward stochastic differential equations, (BSDEs). A risk-based approach is considered, where the member selects an optimal asset mix with a view to minimizing the risk described by a convex risk measure of his/her terminal wealth. Firstly, filtering theory is adopted to transform the hidden, Markov regime-switching, economy into one with complete observations and to develop, (robust), filters for the hidden Markov chain. Then the optimal asset allocation problem of the member is formulated as a two-person, zero-sum stochastic differential game between the member and the market in the economy with complete observations. The BSDE approach is then used to solve the game problem and to characterize the saddle point of the game problem. An explicit expression for the optimal asset mix is obtained in the case of a convex risk measure with quadratic penalty and it can be considered a generalized version of the Merton ratio. An explicit expression for the optimal strategy of the market is also obtained, which leads to a risk-neutral wealth dynamic and may provide some insights into asset pricing in the economy with inflation risk and regime-switching risk. Numerical examples are provided to illustrate financial implications of the BSDE solution.     

Hedging electricity swaptions using partial integro-differential equations

The basic contracts traded on energy exchanges are swaps involving the delivery of electricity for fixed-rate payments over a certain period of time. The main objective of this article is to solve the quadratic hedging problem for European options on these swaps, known as electricity swaptions. We consider a general class of Hilbert space valued exponential jump-diffusion models. Since the forward curve is an infinite-dimensional object, but only a finite set of traded contracts are available for hedging, the market is inherently incomplete. We derive the optimization problem for the quadratic hedging problem under the risk neutral measure and state a representation of its solution, which is the starting point for numerical algorithms.     

Mean-Variance Hedging Under Multiple Defaults Risk

We solve a mean-variance hedging problem in an incomplete market where multiple defaults can occur. For this purpose, we use a default-density modeling approach. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of the default times is modelled using a conditional density hypothesis. We prove the quadratic form of each value process between consecutive default times and recursively solve systems of coupled quadratic backward stochastic differential equations (BSDEs). We demonstrate the existence of these solutions using BSDE techniques. Then, using a verification theorem, we prove that the solutions of each subcontrol problem are related to the solution of our global mean-variance hedging problem. As a byproduct, we obtain an explicit formula for the optimal trading strategy. Finally, we illustrate our results for certain specific cases and for a multiple defaults case in particular.     

On the orthogonal component of BSDEs in a Markovian setting

In this note we consider a quadratic growth backward stochastic differential equation (BSDE) driven by a continuous martingale M. We prove (in Theorem 3.2) that if M is a strong Markov process and if the BSDE has the form (2.2) with regular data then the unique solution (Y,Z,N) of the BSDE is reduced to (Y,Z), i.e. the orthogonal martingale N is equal to zero, showing that in a Markovian setting the “usual” solution (Y,Z) (of a BSDE with regular data) has not to be completed by a strongly orthogonal component even if M does not enjoy the martingale representation property.     

Multi-dimensional BSDE with oblique reflection and optimal switching

In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimates. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.     

基于指数效用无差异价值过程的不完备市场下的可违约期权的定价模型

本文研究不完备市场情况下的可违约期权的动态指数效用无差异定价。不同于大多数的可违约期权定价文献,本文没有假定鞅的不变性,即通常的H 假设,而是通过信息流的扩张和测度的变换,将信用风险敏感的资产转换为一个G 局部鞅,其后引入一个具体的倒向随机微分方程（BSDE）,并证明该方程解的存在性与唯一性;然后利用无差异价值过程C_t（B,α）在最小熵鞅测度下对一般的投资策略为上鞅,而在最优投资策略下为鞅的事实,证明无差异价值过程C_t（B,α）就是BSDE 的解,从而给出可违约期权的定价。     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.     

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.     

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.     

Stability of utility-maximization in incomplete markets

The effectiveness of utility-maximization techniques for portfolio management relies on our ability to estimate correctly the parameters of the dynamics of the underlying financial assets. In the setting of complete or incomplete financial markets, we investigate whether small perturbations of the market coefficient processes lead to small changes in the agent’s optimal behavior, as derived from the solution of the related utility-maximization problems. Specifically, we identify the topologies on the parameter process space and the solution space under which utility-maximization is a continuous operation, and we provide a counterexample showing that our results are best possible, in a certain sense. A novel result about the structure of the solution of the utility-maximization problem, where prices are modeled by continuous semimartingales, is established as an offshoot of the proof of our central theorem.     

On measure solutions of backward stochastic differential equations

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions.     

Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process

In this paper, we study the optimal investment and proportional reinsurance strategy when an insurance company wishes to maximize the expected exponential utility of the terminal wealth. It is assumed that the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process. Using stochastic control theory and Hamilton-Jacobi-Bellman equations, explicit expressions for the optimal strategy and value function are derived not only for the compound Poisson risk model but also for the Brownian motion risk model. Further, we investigate the partially observable optimization problem, and also obtain explicit expressions for the optimal results.     

Robust utility maximization with limited downside risk in incomplete markets

In this article we consider the portfolio selection problem of an agent with robust preferences in the sense of Gilboa and Schmeidler [Itzhak Gilboa, David Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics 18 (1989) 141–153] in an incomplete market. Downside risk is constrained by a robust version of utility-based shortfall risk. We derive an explicit representation of the optimal terminal wealth in terms of certain worst case measures which can be characterized as minimizers of a dual problem. This dual problem involves a three-dimensional analogue of f$f$-divergences which generalize the notion of relative entropy.