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In this paper, we investigate the robust utility maximization problems under both preferences of extremely ambiguity loving and ambiguity aversion. By a fundamental martingale characterization technique on nonlinear expectations, optimal investment strategies are explicitly solved in a general non-Markovian framework via backward stochastic differential equations. Different with previous works in the literature assuming the convexity of the set of prior probability measures , our analysis are independent of the cardinality of . Our results show that extremely ambiguity-loving (resp. -aversion) investors will adopt the extremely aggressive (resp. conservative) investment strategy.

We provide a verification and characterization result of optimal maximal sub-solutions of BSDEs in terms of fully coupled forward backward stochastic differential equations. We illustrate the application thereof in utility optimization with random endowment under probability and discounting uncertainty. We show with explicit examples how to quantify the costs of incompleteness when using utility indifference pricing, as well as a way to find optimal solutions for recursive utilities.

Inspired by the α -maxmin expected utility, we propose a new class of mean-variance criterion, called α -maxmin mean-variance criterion, and apply it to the reinsurance-investment problem. Our model allows the insurer to have different levels of ambiguity aversion (rather than only consider the extremely ambiguity-averse attitude as in the literature). The insurer can purchase proportional reinsurance and also invest the surplus in a financial market consisting of a risk-free asset and a risky asset, whose dynamics is correlated with the insurance surplus. Closed-form equilibrium reinsurance-investment strategy is derived by solving the extended Hamilton–Jacobi–Bellman equation. Our results show that the equilibrium reinsurance strategy is always more conservative if the insurer is more ambiguity-averse. When the dependence between insurance and financial risks are weak, the equilibrium investment strategy is also more conservative if the insurer is more ambiguity-averse. However, in order to diversify the portfolio, a more ambiguity-averse insurer may adopt a more aggressive investment strategy if the insurance market is very ambiguous. For an ambiguity-neutral insurer, the investment strategy is identical to the non-robust investment strategy.

In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp -utility and the power utility, when g(t, z) = βt|z|2 + γtz.

In this study, we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium.

We consider an incomplete market with two information structures, complete and partial information and , respectively. The dynamics of the market are given by a risky asset driven by a m-dimensional Brownian motion W = (W1, …, Wm)′ as well as an integer-valued random measure μ(du, dy). To study the values with respect to the different information filtrations, we introduce the concept of dynamic -utility indifference value (UIV) of with respect to denoted by Ct and the concept of dynamic -UIV of the contingent claim H denoted by Ct(H), and we describe the dynamics of Ct and Ct(H) by BSDEs.

Our main topic in this article is the forward utility field, which is a quite a new concept introduced by Musiela and Zariphopoulou. Different from most article in this field discussing forward utility in a continuous market, we extend this concept to jump market case. We first provide a generalized Ito-Ventzell formula, which can be applied in a general jump semimartingale driven by Brownian motion and Poisson random measure. Then three special forward utility models are discussed by exploiting this generalized Ito-Ventzell formula.

We consider the forward CDS in the framework of stochastic interest rates whose term structures are modeled in the sense of the Heath–Jarrow–Morton model with jumps adapted to a filtration 𝔽 (see [2]). Under the assumption that the density process of the default is a bounded 𝔽-predictable process, we obtain a quadratic-exponential type system of BSDEs similar to [2], which always has a unique solution (X, θ, ϑ). By the solution of such a system of BSDEs, we will describe the dynamics of the the pre-default values of the defaultable bond, the defaultable forward Libor rates and the restricted defaultable forward measure (see in [6]) explicitly. Then we introduce another quadratic-exponential type system of BSDEs (called adjoint system of BSDEs), which also always has a unique solution, and, using this solution, we describe the dynamic of the fair spread of the forward CDS with the tenor structure 𝕋 = {a = T 0

We construct a model for the term structure in a market of defaultable bonds with jumps {p d (t, T); t ≤ T}, T ∈ (0, T*]. We derive the instantaneous defaultable forward rate f d (t, T) defined by in the real world probability. We are also given default-free bonds {p(t, T); t ≤ T}, T ∈ (0, T*] and we establish the market consisting of both the defaultable and the nondefaultable bonds. In this market, we study the common equivalent martingale measure and in this arbitrage free market we derive the relationship between the forward rates f(t, T) and f d (t, T) associated with the two sorts of bonds. Especially, it is proved that in a parameterized market with common equivalent martingale measure where f(t, T) can be described by (1.1) the defaultable forward rate f d (t, T) can be reconstructed from the special form of the default-free forward rate f(t, T) if a certain system of BSDEs has a solution. Finally, we extend the results to a market with recovery rate and give examples where the system of BSDEs has...

We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market. We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks [Formula: see text], we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market [Formula: see text]. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim [Formula: see text] is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks [Formula: see text] is computed.

In this paper, we discuss the dynamics of credit default risk for bilateral collateralized credit valuation adjusted (BCCVA) for counterparty credit risk with two positive collateral accounts by assuming default times for both investor and counterparty satisfying the density hypothesis, and the contagion risk between them being reflected by the density process. We first split the price process into three key parts, and then describe the dynamics of each part by using backward stochastic differential equations (BSDEs). As applications, we introduce the “double” Cox model, in which the BSDEs have their specific forms and thus results in this paper generalize and improve corresponding theoretic results in the existing literature. We also would like to point out that, in this paper, we do not pay attention to explore how the introduction of default contagion impacts with true market data for the BCCVA predicted by for a model without contagion versus a model with contagion, but will plan to conduct the study on the impact for BCCVA with or without contagion in a separate research project.

We consider the optimal exponential utility in a bond market with jumps basing on a model similar to Bjork et al. [4], which is arbitrage free. Similar to the normalized integral with respect to the cylindrical martingale first introduced in Mikulevicius and Rozovskii [13], we introduce the (𝕄, Q 0)-normalized martingale and local (𝕄, Q 0)-normalized martingale. For a given maturity T 0 ∈ [0, T*], we describe the minimal entropy martingale (MEM) based on [T 0, T*] by a backward semimartingale equation (BSE) w.r.t. the (𝕄, Q 0)-normalized martingale. Then we give an explicit form of the optimal approximate wealth to the optimal exp-utility problem by making use of the solution of the BSE. Finally, we describe the dynamics of the exp utility indifference valuation of a bounded contingent claim H ∈ L ∞(ℱ T 0 ) by another BSE under the minimal entropy martingale measure in the incomplete market.

We construct a market of bonds with jumps driven by a general marked point process as well as by a ℝ n -valued Wiener process based on Bjork et al. [6], in which there exists at least one equivalent martingale measure Q 0. Then we consider the mean-variance hedging of a contingent claim H ∈ L 2(ℱ T 0 ) based on the self-financing portfolio based on the given maturities T 1,…, T n with T 0

We consider a financial market in which the discounted price process S is an ℝd-valued semimartingale with bounded jumps, and the variance-optimal martingale measure (VOMM) Qopt is only known to be a signed measure. We give a backward semimartingale equation (BSE) and show that the density process Zopt of Qopt with respect to P is a possibly non-positive stochastic exponential if and only if this BSE has a solution. For a general contingent claim H, we consider the following generalized version of the classical mean-variance hedging problem $$ \min_{\pi\in Adm} E\{(X^{w,\pi}_{\tilde\tau})^2 I_{\{{\tilde\tau}\leq T\}}+|H - X^{w,\pi}_T|^2 I_{\{{\tilde\tau} > T\}}\}, $$ where ${\tilde\tau} = \inf\{t > 0; Z^{\rm opt}_t=0\}$. We represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward martingale equation (BME) and an appropriate predictable process δ both with a straightforward intuitive interpretation.

We consider the dynamic convex valuation (DCV) in an incomplete market of m stocks S = (S 1,…, S m ) in the Brownian motion setting. In this framework, we continue our work in Xiong and Kohlmann [17] on S-related DCV by now considering the S-related DCV generated by a conditional g-expectation under an equivalent martingale measure Q 0 for a given function g(t, y, z) satisfying a Lipschitz condition. We give a sufficient and necessary condition for g so that ℰ g is an S-related DCV. We mainly study the dynamics of an -dominated S-related DCV C = {(C t (ξ)); ξ ∊L ∞(ℱ T )}. By applying Theorem 5 of Delbaen et al. [4], it is seen that the penalty functional α of C satisfies for a function with , where k is a positive constant. Under the assumption that is continuous with respect to l, we prove that {C t (ξ); t ∊ [0, T]} is the unique bounded solution of a BSDE generated by the function g(t, z 2) with quadratic growth in z 2. This main result generalizes Theorem 7.1 of Coquet et al. [2] about the “ℰμ-dominate...

We consider an incomplete market with general jumps, in which the discounted price process S of a risky asset is a given bounded semimartingale. We continue our work on the S-related dynamic convex valuation (DCV) initiated in Xiong and Kohlmann [23] by considering here an S-related DCV whose dynamic penalty functional is generated by a convex function . So the penalty functional takes the following form where is the density process of an equivalent martingale measure (EMM) Q for S with respect to the fundamental EMM Q 0. For a given ξ ∊L ∞(ℱ T ), we prove that is the first component of the minimal bounded solution of a backward semimartingale equation (BSE) generated by a convex, possibly non-Lipschitz g. If this BSE has a bounded solution such that θ2 is also bounded and , we prove that , Q 0-a.s., for all t ∊ [0, T]. Finally, we introduce the concept of a bounded -(super-)martingale and derive a decomposition for a -supermartingale.

We consider an incomplete market with general jumps in the given price process S of a risky asset. We define the S-related dynamic convex valuation (S-related DCV) which is time-consistent. We discuss the representation for a given S-related DCV C in terms of a ‘penalty functional’ α and give some characteristics of α, which are the sufficient conditions for a given C to be an S-related DCV. Finally, we give two special forms of α satisfying those conditions to describe the dynamics of the corresponding S-related DCV by a backward semimartingale equation.

In this article, we consider a new valuation, which we call dynamic q-valuation of a contingent claim in a semimartingale model with a general continuous filtration. We prove that this valuation has the properties of a convex risk valuation and by making use of the (p, B)-optimal martingale measure introduced in Mania et al. [8] we obtain a backward semimartingale equation (BSE) to characterize the dynamic q-valuation. We prove the convexity of this q-valuation its time-consistency property. Given q and , we consider the ∊ f-convolution of and . This new risk valuation is shown to have an explicitly stated representation as a backward semimartingale equation (BSE). Furthermore, we discuss the convergence of a sequence of ∊ f-valuation of . So, starting from the q-valuation we derive several new risk-measures which allow for an explicit representation.

We first consider the minimal entropy martingale measure in a general jump model introduced in Kohlmann and Xiong (International Journal of Pure and Applied Mathematics, 37(3):321–348) and give a description of this measure (MEMM) as the solution of a backward martingale equation (BME). To relate the (MEMM) to the p-optimal martingale measure (p-OMM) we consider the convergence of the solution of the BME associated with the (p-OMM) to the solution of the BME associated with the MEMM. Under some assumptions, we prove the convergence of the p-OMM to the MEMM both in entropy and strongly in L 1(P). As an application, we consider the exp-optimal utility of an investor with utility function U exp(x) = −exp(− k 0 x), and as q↑∞, we show that the q-optimal terminal wealth of an investor with utility converges to the exp-optimal terminal wealth of an investor with utility function U exp(x) strongly in L r (P) for a large enough r.

We consider the mean–variance hedging (MVH) problem (under measure P) of two kinds of investors for two different levels of information, described by two filtrations and such that . Under the assumption that there exists a measure such that all -martingales are -martingales, we give the variance-optimal martingale measure (VOMM) with respect to and through a couple of stochastic Riccati equation (SRE)s, which can be viewed as the same SRE with differential terminal value under . Then we derive an explicit form of the optimal mean–variance strategy and the optimal costs with respect to and . We describe the concept of -no-value-to-investment in the means of mean–variance, and for a given contingent claim , we compare their optimal costs with respect to and .

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.

In a market with a discontinuous filtration, whose price is influenced by a random factor, we study an optimization problem of an investor who is facing a sequence of losses driven by a Cox process. We give a form of variance-optimal martingale measure by changing the filtration. By using the solutions of the stochastic Riccati equation and another associated backward stochastic equation, we obtain a solution of the optimization problem of the investor.

INFORMATION AND DYNAMIC COHERENT RISK MEASURES

A utility optimization problem for continuous time financial markets is examined in the presence of additional information. Three cases, including “side information known in advance,” “information disclosure at the market-known time,” and “information disclosure at the market-unknown time,” are discussed. The martingale representation theorems for each case are obtained by using stochastic filtering technique. In the case of logarithmic utility, the analytic forms of optimal solutions are obtained and the effect of these kinds of additional information to investor's strategies are revealed.

We assume that there exist two kinds of investors in the market, the 𝔽-investors and the 𝔾-investors. The 𝔽-investors have the market information 𝔽, which is given by a d-dimensional Brownian motion W = (W1,...;Wd)' as well as an integer-valued random measure μ(du, dy). The market might default at time ˜τ, modeled by the so called the generalized Cox model, and the information of the 𝔾-investors is the by the default, progressively enlarged fitration of 𝔽. We give the explicit form of the survival process. Then we derive the dynamic CRRA-utility indifference value(UIV) Ct of the 𝔽-investors with respect to the 𝔾-investors and describe the dynamics of Ct by two BSDEs. In the end, we give an example in which we can give the explicit expression of Ct. For the generalized Cox model we typically have that Ct ≥ 1 in contrast to the standard Cox model.

We assume that the filtrationFis generated by ad-dimensional Brownian motionW=(W1,…,Wd)′as well as an integer-valued random measureμ(du,dy). The random variableτ~is the default time andLis the default loss. LetG={Gt;t≥0}be the progressive enlargement ofFby(τ~,L); that is,Gis the smallest filtration includingFsuch thatτ~is aG-stopping time andLisGτ~-measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond inGand the forward CDS with random loss explicitly by the BSDEs method.

We give the explicit form of the survival process of the default time [Formula: see text] modeled by the generalized Cox process model. Then we derive the dynamic CRRA-utility indifference value (UIV) Ct of the 𝔽-investors with respect to the 𝔾-investors and describe the dynamics of Ct by two BSDEs. Finally, we give an example in which we can give the explicit expression of Ct. For the generalized Cox process model we typically have that Ct ≥ 1 in contrast to the standard Cox process model.