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Thomas Kruse

Thomas Kruse appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
math.PRmath.OCmath.NANumerical Analysismath.APq-fin.TRMachine LearningMethodologyq-fin.RM

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Published work

14 published item(s)

preprint2026arXiv

Matrix Riccati BSDEs with singular terminal condition and stochastic LQ control with linear terminal constraint

We analyze a class of multidimensional linear-quadratic stochastic control problems with random coefficients, motivated by multi-asset optimal trade execution. The problems feature non-diffusive controlled state dynamics and a terminal constraint that restricts the terminal state to a prescribed random linear subspace. We derive the associated Riccati backward stochastic differential equation (BSDE) and identify a suitable formalization of its singular terminal condition. Via a penalization approach, we establish existence of a minimal supersolution of the Riccati BSDE and use it to characterize both the value function and the optimal control. We analyze the asymptotic behavior of the supersolution near terminal time and discuss special cases where closed-form solutions can be obtained.

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preprint2026arXiv

Optimal Asymptotic Rates for (Stochastic) Gradient Descent under the Local PL-Condition: A Geometric Approach

Stochastic gradient descent (SGD) has been studied extensively over the past decades due to its simplicity and broad applicability in machine learning. In this work, we analyze the local behavior of gradient descent and stochastic gradient descent for minimizing $C^2$-functions that satisfy the Polyak-Lojasiewicz (PL) inequality and under a multiplicative gradient noise model motivated by overparameterized neural networks. Using a geometric interpretation of the PL-condition, we prove a simple yet surprising fact: in this possibly non-convex setting, the asymptotic convergence rate of (S)GD matches the rate obtained for strongly convex quadratics.

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preprint2022arXiv

On the speed of convergence of Picard iterations of backward stochastic differential equations

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is $z$-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

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preprint2022arXiv

Self-exciting price impact via negative resilience in stochastic order books

Most of the existing literature on optimal trade execution in limit order book models assumes that resilience is positive. But negative resilience also has a natural interpretation, as it models self-exciting behaviour of the price impact, where trading activities of the large investor stimulate other market participants to trade in the same direction. In the paper we discuss several new qualitative effects on optimal trade execution that arise when we allow resilience to take negative values. We do this in a framework where both market depth and resilience are stochastic processes.

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preprint2021arXiv

Multilevel Picard approximations for McKean-Vlasov stochastic differential equations

In the literatur there exist approximation methods for McKean-Vlasov stochastic differential equations which have a computational effort of order $3$. In this article we introduce full-history recursive multilevel Picard (MLP) approximations for McKean-Vlasov stochastic differential equations. We prove that these MLP approximations have computational effort of order $2+$ which is essentially optimal in high dimensions.

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preprint2020arXiv

A functional limit theorem for coin tossing Markov chains

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process in natural scale can be approximated with such Markov chains arbitrarily well. The functional limit theorem applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. Our results allow to practically approximate such processes with irregular behavior; we illustrate this with Markov processes exhibiting sticky features, e.g., sticky Brownian motion and a Brownian motion slowed down on the Cantor set.

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preprint2020arXiv

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows polynomially both in the dimension and in the reciprocal of the required accuracy.

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preprint2020arXiv

Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of coin tossing Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of $1/4$ with respect to every $p$-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than $1/4$. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.

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preprint2019arXiv

Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen-Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

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preprint2017arXiv

Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated under suitable assumptions with computational complexity that grows polynomially both in the dimension and the reciprocal of the accuracy.

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preprint2016arXiv

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.

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preprint2015arXiv

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 $1/N$.

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preprint2015arXiv

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.

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preprint2013arXiv

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.

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