Large deviation principle for a backward stochastic differential equation driven by $G$-Brownian motion with subdifferential operator
Abdoulaye Soumana Hima, Ibrahim Dakaou
该文针对带次微分算子的$G$-布朗运动驱动倒向随机微分方程，开展其解的大偏差原理研究。
本文研究了一类带次微分算子、由$G$-布朗运动驱动的倒向随机微分方程解的大偏差原理。
In this paper, we study a large deviation principle for the solution of a backward stochastic differential equation driven by $G$-Brownian motion with subdifferential operator.
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Our aim is to study the existence and uniqueness of the $L^{p}$ - variational solution, with $p>1,$ of the following multivalued backward stochastic differential equation with $p$-integrable data: \[ \left\{ \begin{align*} &-dY_{t}+\partial_{y}Ψ\left( t,Y_{t}\right) dQ_{t} \ni H\left( t,Y_{t},Z_{t}\right) dQ_{t}-Z_{t}dB_{t},\;t\in\left[ 0,T\right] ,\\ &Y_{T} =η, \end{align*} \right. \] where $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}Ψ$ is the subdifferential of the convex lower semicontinuous function $y\mapstoΨ(t,y)$. In the framework $p\geq2$…
Backward Stochastic Differential Equations (BSDEs) Using Infinite-dimensional Martingales with Subdifferential Operator
本文研究由含正定核算子 Q 的无限维鞅驱动、带次微分算子的 BSDEs，利用 Yosida 逼近与不动点定理证明其解存在且唯一，并将结果应用于倒向随机偏微分方程。
In this paper, we deal with a class of reflected backward stochastic differential equations associated to the subdifferential operator of a lower semi-continuous convex function driven by Teugels martingales associated with Lévy process. We obtain the existence and uniqueness of solutions to these equations by means of the penalization method. As its application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.
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In this paper, we prove that there exists at least one solution for the reflected forward-backward stochastic differential equation driven by G-Brownian motion satisfying the obstacle constraint with monotone coefficients.
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We study the anticipative backward stochastic differential equations (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H greater than 1/2. The stochastic integral used throughout the paper is the divergence operator type integral. We obtain the existence and uniqueness of solutions to these equations. A comparison theorem for this type of anticipative BSDEs is also established.
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In this paper, we consider forward-backward stochastic differential equation driven by $G$-Brownian motion ($G$-FBSDEs in short) with small parameter $\varepsilon > 0$. We study the asymptotic behavior of the solution of the backward equation and establish a large deviation principle for the corresponding process.
Backward Stochastic Volterra integral equations driven by G-Brownian motion
本文研究了由G-布朗运动驱动的反向随机Volterra积分方程，通过新的反向迭代方法构建近似序列，并利用随机分析技术证明了解的存在性、唯一性和连续性，同时推导了比较定理。
Stochastic averaging for a class of backward stochastic differential equations driven by both standard and fractional Brownian motions (SFrBSDEs in short), is investigated. An averaged SFrBSDEs for the original SFrBSDEs is proposed, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square and also in probability.
In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is also presented.
In this paper, we obtain the existence and uniqueness theorem of $L^{p}$-solution for coupled forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with arbitrary $T$ under weakly coupling condition. Specially, the result for $p\in(1,2)$ is completely different from the one for $p\geq2$. Furthermore, by considering the dual linear FBSDE under a suitable reference probability, we establish the comparison theorem for G-FBSDEs under weakly coupling condition.
In this paper, we focus on the mean-field backward stochastic differential equations (BSDEs) driven by a fractional Brownian motion with Hurst parameter H greater then 1/2. First, the existence and uniqueness of these equations are established under Lipschitz condition. Then, a comparison theorem for such mean-field BSDEs is obtained. Finally, as an application, we connect this mean-field BSDE with a nonlocal partial differential equation (PDE).
Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations
该研究探讨布朗滤波初始放大对BSDEs和RBSDEs的影响，建立相关表示性质与恒等式，通过反例说明奇异方程解的唯一性问题，发展正则BSDE的条件参数化并讨论扩展方向。
Stochastic PDEs driven by G-Brownian motion and the associated Backward Doubly Stochastic Differential Equations
本文研究了由G-布朗运动驱动的随机偏微分方程的well-posedness，并通过分析方法证明了其弱解的存在性和唯一性，同时建立了与反向双重随机微分方程的关系。
In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic…
In this paper, we consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in [10.11], we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in [28]. Then we obtain a generalized dynamic programming principle and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.
We first state a special type of Itô formula involving stochastic integrals of both standard and fractional Brownian motions. Then we use Doss-Sussman transformation to establish the link between backward doubly stochastic differential equations, driven by both standard and fractional Brownian motions, and backward stochastic differential equations, driven only by standard Brownian motions. Following the same technique, we further study associated nonlinear stochastic partial differential equations driven by fractional Brownian motions and partial differential equations with stochastic coeffic…
We study the existence and uniqueness of the solution for the following backward stochastic variational inequality with oblique reflection (for short, $BSVI\left(H(t,y),φ,F\right)$), written under differential form \[ \left\{\begin{array} [c]{l}% -dY_{t}+H\left(t,Y_{t}\right) \partialφ\left(Y_{t}\right) \left(dt\right) \ni F\left(t,Y_{t},Z_{t}\right) dt-Z_{t}dB_{t},\quad t\in\left[ 0,T\right] ,\smallskip\\ Y_{T}=η, \end{array} \right. \] where $H$ is a bounded symmetric smooth matrix and $φ$ is a proper convex lower semicontinuous function, with $\partialφ$ being its subdifferential operator. …
In this paper, we study backward stochastic differential equations driven by a G-Brownian motion. The solution of such new type of BSDE is a triple (Y,Z,K) where K is a decreasing G-martingale. Under a Lipschitz condition for generator f and g in Y and Z. The existence and uniqueness of the solution (Y,Z,K) is proved. Although the methods used in the proof and the related estimates are quite different from the classical proof for BSDEs, stochastic calculus in G-framework plays a central role.
In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.
Infinite horizon quadratic backward stochastic differential equations driven by $G$-Brownian motion
该研究证明了带二次生成元的$G$-布朗运动驱动无限时域$G$-BSDEs的适定性，构造了无界系数线性$G$-BSDEs显式解并采用二次假设下线性化方法，同时建立了两类时域下的比较定理。
In this note, we consider Jensen's inequality for the nonlinear expectation associated with backward SDEs driven by $G$-Brownian motion ($G$-BSDEs for short). At first, we give a necessary and sufficient condition for $G$-BSDEs under which one-dimensional Jensen inequality holds. Second, we prove that for $n>1$, the $n$-dimensional Jensen inequality holds for any nonlinear expectation if and only if the nonlinear expectation is linear, which is essentially due to Jia (Arch. Math. 94 (2010), 489-499). As a consequence, we give a necessary and sufficient condition for $G$-BSDEs under which the $…
Mean Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Constraints
本文研究带双重均值反射的G-布朗运动驱动的倒向随机微分方程，通过倒向Skorokhod问题与不动点理论证明解的存在唯一性，拓展非Lipschitz系数情形并得到相关性质。
Doubly Reflected Backward SDEs Driven by G-Brownian Motions and Fully Nonlinear PDEs with Double Obstacles
本文提出通过单调递减的惩罚反射G-BSDE逼近解的新方法，在已知最弱条件下证明了双重反射G-BSDE解的适定性，并首次揭示其与带双重障碍的完全非线性偏微分方程的关系。
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The aim of the paper is to prove the existence and uniqueness of the $L^{p}$--variational solution, with $p>1,$ of the following multivalued backward stochastic differential equation with $p$--integrable data: \begin{equation*} \left\{ \begin{array}[c]{l} -dY_{t}+\partial_{y}Ψ(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<τ,\\[0.1cm] Y_τ=η, \end{array} \right. \end{equation*} where $τ$ is a stopping time, $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}Ψ$ is the subdifferential of the convex lower semicontinuous function $y\mapstoΨ(t…
A Moving Boundary Problem for Brownian Particles with Singular Forward-Backward Interactions
该研究提出一类带奇异正反向相互作用的布朗粒子系统，将其转化为新型分层运动边界偏微分方程问题，证明了该问题的经典适定性，并据此求解原正反向系统且证明其唯一性。
Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator
本文研究由$G$-布朗运动驱动的双反射反向随机微分方程在二次生成函数下的性质，证明了解的存在性和唯一性，并通过罚函数反射建立其与具有双障碍的非线性偏微分方程的关系。
Our aim is to study the following new type of multivalued backward stochastic differential equation: \[ \left\{\begin{array} [c]{r}-dY\left(t\right) +\partialφ\left(Y\left(t\right)\right) dt\ni F\left(t,Y\left(t\right),Z\left(t\right),Y_{t},Z_{t}\right) dt+Z\left(t\right) dW\left(t\right),\;0\leq t\leq T,\medskip\\ \multicolumn{1}{l}{Y\left(T\right) =ξ,}\end{array} \right. \] where $\partialφ$ is the subdifferential of a convex function and $\left(Y_{t},Z_{t}\right):=(Y(t+θ),Z(t+θ))_{θ\in\lbrack-T,0]}$ represent the past values of the solution over the interval $\left[ 0,t\right] $. Our result…
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We introduce a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian motion are in fact path dependent PDEs in the corresponding Sobolev spaces under G-norms. For the linear case of G corresponding the classical Wiener probability space with Wiener measure P, we have established a 1-1 correspondence between BSDE and such new type of quasilinear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space a…
Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing random walks corresponding to color genealogies. Asympotics for large time and space on the lattice are described via a continuum space-time voter model whose boundary motion is expressed by the {\it Brownian web} (BW) of coalescing forward Brownian motions. In this paper, we study how small noise in the voter model, corresponding to the nucleation of rando…
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Quadratic BSDEs with double constraints driven by G-Brownian motion
本文研究由G-布朗运动驱动的带双重均值反射的二次倒向随机微分方程的适定性，利用G-BMO鞅技术、不动点论证、Skorokhod问题及θ-方法，在有界和无界终端条件下建立了解的存在唯一性。
In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators, the $z$ parts of whose $l$-th components only depend on the $l$-th columns. The existence and uniqueness of solutions are obtained via a contraction argument for $Y$ component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional $G$-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differ…
In this paper, we study the existence and uniqueness of solutions to the fully coupled nonlinear forward-backward stochastic differential equations driven by G-Brownian motion. Assuming that the diffusion coefficient $σ$ is uniformly elliptic and all coefficients are differentiable, combining the results of fully nonlinear PDEs, we prove the existence and uniqueness of solutions to these equations.
Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator
本文研究由分数布朗运动驱动且生成子含时滞的预期倒向随机微分方程，利用不动点定理证明了解的存在唯一性，并建立了解的比较定理。
In this paper, we study the reflected backward stochastic differential equation driven by G-Brownian motion (reflected G-BSDE for short) with an upper obstacle. The existence is proved by approximation via penalization. By using a variant comparison theorem, we show that the solution we constructed is the largest one.
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We present an explicit solution triplet $(Y, Z, K)$ to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ and $K$ are expressed by Hida-Malliavin derivatives involving $Y$.
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In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected GBSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.
The aim of this paper is twofold. Firstly, we derive upper and lower non-Gaussian bounds for the densities of the marginal laws of the solutions to backward stochastic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilising a relationship between fractional BSDEs and quasilinear partial differential equations of mixed type, together with the profound Nourdin-Viens formula. In the linear case, upper and lower Gaussian bounds for the densities and the tail probabilities of solutions are obtained with simple arguments by their explicit expressions …
We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate- approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poi…
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Rough backward stochastic differential equations
本文提出一种直接基于粗糙半鞅的粗糙倒向随机微分方程理论，在正则性条件下建立适定性，并建立与倒向双随机微分方程的对应关系。
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This paper introduces a backward stochastic differential equation driven by both Brownian motion and a Markov chain (BSDEBM). Regime-switching is also incorporated through its driver. The existence and uniqueness of the solution of the BSDEBM are proved. A comparison theorem is also derived. Filtration consistent sublinear expectations are defined and characterized as solutions to the BSDEBM. The bid and ask prices are then represented using sublinear expectations.
The present paper is devoted to investigating the existence and uniqueness of solutions to a class of non-Lipschitz scalar valued backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs). In fact, when the generators are Lipschitz continuous in $y$ and uniformly continuous in $z$, we construct the unique solution to such equations by monotone convergence argument. The comparison theorem and related Feynman-Kac formula are stated as well.
Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional brownian motion and the multifractional Ornstein-Uhlenbeck process. By an Itô formula, proven in the context of Malliavin calculus, the BSDE is associated to a linear second order partial differential equation with terminal condition whose solution is given by a Feynman-Kac type formula. An application to self-financing trading strategies is discussed.
We study solutions of a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and we derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.
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