A study of backward stochastic differential equation on a Riemannian manifold

Abstract

Suppose N is a compact Riemannian manifold, in this paper we will introduce the definition of N -valued BSDE and L 2 (T m ; N )-valued BSDE for which the solutions are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to L 2 (T m ; N )-valued BSDE will be proved without any convexity condition on N .

g-Expectation for Conformable Backward Stochastic Differential Equations

“…Using the comparison theorem to the Equations ( 10) and (11), we see that X(t) ≤ X 3 (t). From Definition 4, one has…”

“…Subsequently, Royer [4] studied the backward stochastic differential equation driven by Brownian motion and Poisson random measure, and introduced the corresponding g-expectation and a large number of studies show that this g-expectation can be applied to financial problems (see [5][6][7][8][9]). Recently, Long et al [10] proposed a multi-step scheme on time-space grids for solving backward stochastic differential equations, and Chen and Ye [11] investigated solutions of backward stochastic differential equations in the framework of Riemannian manifold. From the paper [12], we could get the averaging principle for backward stochastic differential equations and the solutions can be approximated by the solutions to averaged stochastic systems in the sense of mean square under some appropriate assumptions.…”

Forward-backward stochastic differential equations on tensor fields and application to Navier-Stokes equations on Riemannian manifolds

“…Since F (M) is a compact Riemannian manifold, by Nash embedding theorem we can find a smooth isometric embedding Φ : F (M) → R L . Analogously to (2.5) and (2.7) in [9] (or (2.3) in [8]), we can extend…”

“…In [4,5], Blache studied BSDEs on general Riemannian manifolds whose solutions were restricted to only one local chart. Chen and Ye [9] gave a new definition of manifoldvalued BSDEs which were not necessarily situated in only one local chart, and the existence of these manifold-valued solutions was also proved without any convex condition in [9]. Moreover, through such kind of manifold-valued FBSDEs, a probabilistic representation for heat flows of harmonic map associated with time-changing Riemannian metrics was established by Chen and Ye [8].…”

“…(i) As explained in [9], the main difficulty to deal with manifold-valued BSDEs is the lack of linear structure, which is necessary to define Itô integrals in equations. For the FBSDEs on tensor fields studied in this paper, we will use stochastic parallel translations induced by forward equations, which are SDEs on orthonormal frame bundles over base manifolds, in order to define solutions of backward equations (at different times) in the same linear space, so that we can define Itô integrals by the linear structure in such space.…”