arXiv subjects

Thomas Kruse

Publications and source records attributed to Thomas Kruse.

42 records · Page 3Linked to original sources

arXiv2017

An inverse optimal stopping problem for diffusion processes

Let $X$ be a one-dimensional diffusion and let $g\colon[0,T]\times\mathbb{R}\to\mathbb{R}$ be a payoff function depending on time and the value of $X$. The paper analyzes the inverse optimal stopping problem of finding a time-dependent function $π:[0,T]\to\mathbb{R}$ such that a given stopping time $τ^{\star}$ is a solution of the stopping problem $\sup_τ\mathbb{E}\left[g(τ,X_τ)+π(τ)\right]\,.$ Under regularity and monotonicity conditions, there exists a solution $π$ if and only if $τ^{\star}$ is the first time when $X$ exceeds a time-dependent barrier $b$, i.e. $τ^{\star}=\inf\left\{ t\ge0\,|\,X_{t}\ge b(t)\right\} \,.$ We prove uniqueness of the solution $π$ and derive a closed form representation. The representation is based on an auxiliary process which is a version of the original diffusion $X$ reflected at $b$ towards the continuation region. The results lead to a new integral equation characterizing the stopping boundary $b$ of the stopping problem $\sup_τ\mathbb{E}\left[g(τ,X_τ)\right]$.

math.OC↗

arXiv2017

L^p -solution for BSDEs with jumps in the case p \textless{} 2. Corrections to the paper "BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration"

In [8] we established existence and uniqueness of solutions of backward stochastic differential equations in L^p under a monotonicity condition on the generator and in a general filtration. There was a mistake in the case 1 \textless{} p \textless{} 2. Here we give a corrected proof. Moreover the quasi-left continuity condition on the filtration is removed.

math.PR↗

arXiv2016

Backward Stochastic Differential Equations with Nonmarkovian Singular Terminal Values

We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving its drift and with the terminal boundary condition $ ξ= \infty \cdot \mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ ξ= \infty \cdot \mathbf{1}_{B(m,r)}$, where $B(m,r)$ is the ball in the path space $C([0,T])$ of the underlying Brownian motion centered at the constant function $m$ and radius $r$. The solution involves the derivation and solution of a related heat equation in which $f$ serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain the BSDE has continuous sample paths with the prescribed terminal value.

math.PR↗

arXiv2015

What's in a ball? Constructing and characterizing uncertainty sets

In the presence of model risk, it is well-established to replace classical expected values by worst-case expectations over all models within a fixed radius from a given reference model. This is the "robustness" approach. We show that previous methods for measuring this radius, e.g. relative entropy or polynomial divergences, are inadequate for reference models which are moderately heavy-tailed such as lognormal models. Worst cases are either infinitely pessimistic, or they rule out the possibility of fat-tailed "power law" models as plausible alternatives. We introduce a new family of divergence measures which captures intermediate levels of pessimism.

q-fin.RM↗

arXiv2015

A functional limit theorem for irregular SDEs

Let $X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: \mathbb R \to \mathbb R_+$. We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{\lfloor N t \rfloor})_{t \in \mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t \in \mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N \to \infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.

math.PR↗

arXiv2013

BSDEs with singular terminal condition and control problems with constraints

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [7] and [8]. We perform a verification and discuss special cases for which the control problem has explicit solutions.

math.OC↗