We investigate the maximal open domain $${\mathscr {E}}(M)$$E ( M ) on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$M ⊆ R d can be defined and study essential properties of p. We prove that if M is a $$C^1$$C 1 submanifold of $${{\mathbb {R}}}^d$$R d satisfying a Lipschitz condition on the tangent spaces, then $${\mathscr {E}}(M)$$E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is $$C^2$$C 2 or if the topological skeleton of $$M^c$$M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $$C^k$$C k -submanifold M with $$k\ge 2$$k ≥ 2 , the projection map is $$C^{k-1}$$C k - 1 on $${\mathscr {E}}(M)$$E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion $$M\subseteq {\mathscr {E}}(M)$$M ⊆ E ( M ) is that M is a $$C^1$$C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with $$M\subseteq {\mathscr {E}}(M)$$M ⊆ E ( M ) , then M must be $$C^1$$C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $${\mathscr {E}}(M)$$E ( M ) and the topological skeleton of $$M^c$$M c .

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

Project Euclid - mathematics and statistics online

We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116: 1358–1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346: 345–358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growth- and monotonicity-condition for the generator to be random and time-dependent. In the L2-case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88: 491–539, 2016). For the proof of the comparison result, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.

We consider measurable $F: \Omega \times \mathbb{R} ^d \to \mathbb{R} $ where for any $x$ the random variable $F(\cdot , x)$ belongs to the Malliavin Sobolev space $\mathbb{D} _{1,2}$ (with respect to a Lévy process) and provide sufficient conditions on $F$ and $G_1,\ldots ,G_d \in \mathbb{D} _{1,2}$ such that $F(\cdot , G_1,\ldots ,G_d) \in \mathbb{D} _{1,2}.$ The above result is applied to show Malliavin differentiability of solutions to BSDEs (backward stochastic differential equations) driven by Lévy noise where the generator is given by a progressively measurable function $f(\omega ,t,y,z).$