Stefan Ankirchner

University of Bonn | Uni Bonn

We consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and dif...

We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best [Formula: see text] of all states receive a fixed prize. Within the mean field limit version of the game, we compute...

We consider a stochastic control problem with time-inhomogeneous linear dynamics and a long-term average quadratic cost functional. We provide sufficient conditions for the problem to be well-posed. We describe an explicit optimal control in terms of a bounded and non-negative solution of a Riccati equation on $$[0, \infty )$$ [ 0 , ∞ ) , without a...

We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density...

We consider the problem of controlling the diffusion coefficient of a diffusion with constant negative drift rate such that the probability of hitting a given lower barrier up to some finite time horizon is minimized. We assume that the diffusion rate can be chosen in a progressively measurable way with values in the interval [0, 1]. We prove that...

We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly recalled). Special cases include irregular stochastic differential equations and processes with sticky features. In particular, we highlight differences from t...

We consider fully coupled forward–backward stochastic differential equations (FBSDEs), where all function parameters are Lipschitz continuous, the terminal condition is monotone and the diffusion coefficient of the forward part depends monotonically on z, the control process component of the backward part. We show that there exists a class of linea...

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 certain 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 approxim...

We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly recalled). Special cases include irregular stochastic differential equations and processes with sticky features. In particular, we highlight differences from t...

We study a stopping problem arising from a sequential testing of two simple hypotheses H₀ and H₁ on the drift rate of a Brownian motion. We impose an expectation constraint on the stopping rules allowed and show that an optimal stopping rule satisfying the constraint can be found among the rules of the following type: stop if the posterior probabil...

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to t...

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 app...

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 f...

We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form $d A_t =\mu (t, A_t) d t + \sigma(t, A_t) d W_t$. We provide sufficient conditions guaranteeing that for a given probability measure $\nu$ on $\mathbb{R}$ there exists a bounded stopping time $\...

We set up a game theoretical model to analyze the optimal attacking intensity of sports teams during a game. We suppose that two teams can dynamically choose among more or less offensive actions and that the scoring probability of each team depends on both teams’ actions. We assume a zero sum setting and characterize a Nash equilibrium in terms of...

We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between (Formula presented.) and (Formula presented.), where (Formula presented.). The optimal control...

We provide a weak law of large numbers for arrays of nonnegative and pairwise negatively associated (in each row) random variables under a rather weak domination condition.

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regul...

We consider the problem of how to optimally close a large asset position in a market with a linear temporary price impact. We take the perspective of an agent who obtains a signal about the future price evolvement. By means of classical stochastic control we derive explicit formulas for the closing strategy that minimizes the expected execution cos...

We provide a new method for approximating the law of a diffusion $M$ solving a stochastic differential equation with coefficients satisfying the Engelbert-Schmidt conditions. To this end we construct Markov chains whose law can be embedded into the diffusion $M$ with a sequence of stopping times that have expectation $1/N$, where $N \in \mathbb{N}$...

We reveal pitfalls in the hedging of insurance contracts with a minimum return guarantee on the underlying investment, e.g. an external mutual fund. We analyze basis risk entailed by hedging the guarantee with a dynamic portfolio of proxy assets for the funds. We also take account of liquidity risk which arises since the insurer may need to advance...

We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion X: given a distribution \rho, we construct a stopping time T such that the stopped process X_T has the distribution \rho? Our solution method makes use of martingale representations (in a similar way to Bass [3] who solves the SEP for Brownian motion) and draws...

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 resu...

Consider an agent with a forward position of an illiquid asset (e.g. a commodity) that has to be closed before delivery. Suppose that the liquidity of the asset increases as the delivery date approaches. Assume further that the agent has two possibilities for hedging the risk inherent in the forward position: first, he can enter customized forward...

When managing risk, frequently only imperfect hedging instruments are at hand. We show how to optimally cross-hedge risk when the spread between the hedging instrument and the risk is stationary. For linear risk positions we derive explicit formulas for the hedge error, and for nonlinear positions we show how to obtain numerically efficient estimat...

Frequently, dynamic hedging strategies minimizing risk exposure are not given in closed form, but need to be approximated numerically. This makes it difficult to estimate residual hedging risk, also called basis risk, when only imperfect hedging instruments are at hand. We propose an easy to implement and computationally efficient least-squares Mon...

We consider the stochastic control problem of how to optimally close a large asset position in an illiquid market with price impact. We assume that the risk attributed to an open position depends on the price evolvement since the beginning of the trading period. Within a continuous-time model with a linear temporary price impact we show how to obta...

This paper is concerned with the study of quadratic hedging of contingent claims with basis risk. We extend existing results by allowing the correlation between the hedging instrument and the underlying of the contingent claim to be random itself. We assume that the correlation process ρ evolves according to a stochastic differential equation with...

We consider the problem of how to optimally close a large asset po-sition in a market with a linear temporary price impact. We take the perspective of an agent with a market opinion that translates into a (lin-ear) drift in asset price dynamics. By appealing to classical stochastic control we derive explicit formulas for the closing strategy that m...

We present a discrete time model that allows for a price-sensitive closure of a large asset position, providing thus a device to introduce skewness in the proceeds/costs. By appealing to dynamic programming we derive semi-explicit formulas for the optimal execution strategies. We then present a numerical algorithm for approximating optimal executio...

This article deals with the Skorokhod embedding problem in bounded time for the Brownian motion with drift Xt = κt+Wt: Given a probability measure µ we aim at finding a stopping time τ such that the law of Xτ is µ, and τ is almost surely smaller than some given fixed time horizon T > 0. We provide necessary and sufficient conditions on the distribu...

Let μ be a Poisson random measure, let \mathbbF\mathbb{F} be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let \mathbbG\mathbb{G} be the initial enlargement of \mathbbF\mathbb{F} with the σ-field generated by a random variableG. In this paper, we first show that the mutual information betwe...

When managing energy or weather related risk often only imperfect hedging instruments are available. In the first part we illustrate problems arising with imperfect hedging by studying a toy model. We consider an airline’s problem with covering income risk due to fluctuating kerosine prices by investing into futures written on heating oil with clos...

This paper is concerned with the study of insurance related derivatives on financial markets that are based on nontradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forwar...

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the oper...

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs n...

Let M be a purely discontinuous martingale relative to a filtration (Ft). Given an arbitrary extension (Gt) of the filtration (Ft), we will provide sufficient conditions for M to be a semimartingale relative to (Gt). Moreover we describe methods of how to find the Doob–Meyer decomposition with respect to the enlarged filtration. To this end we prov...

We solve Skorokhod's embedding problem for Brownian motion with linear drift (Wt + κ t)t≥0 by means of techniques of stochastic control theory. The search for a stopping time T such that the law of WT + κ T coincides with a prescribed law μ possessing the first moment is based on solutions of backward stochastic differential equations of quadratic...

We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These r...

We consider insurance derivatives depending on an external physical risk process, for example a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, an...

We review a general mathematical link between utility and information theory appearing in a simple financial market model with two kinds of small investors: insiders, whose extra information is stored in an enlargement of the less informed agents'filtration. The insider's expected logarithmic utility increment is described in terms of the informati...

We consider Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. Thes...

We show that the maximal expected utility satisfies a monotone continuity property with respect to increasing information. Let (G_tⁿ) be a sequence of increasing filtrations converging to (G_t^∞), and let uⁿ(x) and u^∞(x) be the maximal expected utilities when investing in a financial ma...

The utility maximisation problem is considered for investors with anticipative additional information. We distinguish between models with conditional measures and models with enlarged filtrations. The dual functions of the maximal expected utility are determined with the help of f-divergences. We assume that our measures are absolutely continuous w...

The background for the general mathematical link between utility and information theory investigated in this paper is a simple financial market model with two kinds of small traders: less informed traders and insiders, whose extra information is represented by an enlargement of the other agents’ filtration. The expected logarithmic utility incremen...

The background for the general mathematical link between utility and information theory investigated in this paper is a simple financial market model with two kinds of small traders: less informed traders and insiders, whose extra information is represented by an enlargement of the other agents’ filtration. The expected logarithmic utility incremen...

We consider financial markets with two kinds of small traders: regular traders who perceive the (continuous) asset price process S through its natural filtration, and insiders who possess some information advantage which makes the filtrations through which they experience the evolution of the market richer. We discuss the link between (NFLVR), the...

Let (Gt) be an enlargement of the filtration (Ft). Jeulin and Jacod discussed a sufficient criterion for the inheritance of the semimartingale property when passing to the larger filtration. We provide alternative proofs of their results in a more general setting by using decoupling measures and Girsanov’s changes of measure. We derive necessary an...

The extent to which catastrophic weather events occur strongly depends on global climate conditions such as average sea surface tem- peratures (SST) or sea level pressures. Some of the factors can be predicted up to a year in advance, and should therefore be taken into account in any reasonable management of weather related risk. In this paper we f...

We consider Þnancial markets with two kinds of small traders: regular traders who perceive the asset price process S through its natural Þltration, and insid- ers who possess some information advantage which makes the Þltrations through which they perceive the evolution of the market richer. The basic question we dis- cuss is the link between (NFLV...