Christoph Belak


We derive convergence rates for a deep solver for semilinear partial differential equations which is based on a Feynman-Kac representation in terms of a forward-backward stochastic differential equation and a discretization in time. We show that the error of the deep solver is bounded in terms of its loss functional, hence yielding a direct measure to judge the quality in numerical applications, and that the loss functional converges sufficiently fast to zero to guarantee that the approximation error vanishes in the limit. As a consequence of these results, we show that the deep solver has a strong convergence rate of order 1/2.

Preprint, 2021. Available at

We study the problem of pricing contingent claims in the presence of uncertainty about the timing and the size of a jump in the price of the underlying. We characterize the price of the claim as the minimal solution of a constrained BSDE and derive a pricing PDE in the special case of a Markovian market model. In a Black-Scholes market, explicit solutions are obtained.

Preprint, 2017. Available at


We consider an optimal investment problem for an investor facing fixed and proportional transaction costs and study the limit as the fixed cost tends to zero. Combining the stochastic Perron’s method with stability arguments for viscosity solutions, we show that the value function converges to the value function of the problem with purely proportional costs. Moreover, using a Komlos-type argument, we show that forward-convex combinations of the optimal strategies in the problem with fixed costs converge to an optimal strategy without a fixed cost.

SIAM Journal on Control and Optimization, Vol. 60, No. 5, pp. 2712-2736, 2022. Available at

This paper considers a utility maximization and optimal asset allocation problem in the presence of a stochastic endowment that cannot be fully hedged through trading in the financial market. After studying continuity properties of the value function for general utility functions, we rely on the dynamic programming approach to solve the optimization problem for power utility investors including the empirically relevant and mathematically challenging case of relative risk aversion larger than one. For this, we argue that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. The homogeneity of the value function is then used to reduce the HJB equation by one dimension, which allows us to prove that the value function is even a classical solution thereof. Using this, an optimal strategy is derived and its asymptotic behavior in the large wealth regime is discussed.

SIAM Journal on Financial Mathematics, Vol. 13, No. 3, pp. 969-1003, 2022. Available at

We study optimal portfolio decisions for a retail investor that faces a strictly positive transaction cost in a classical Black-Scholes market. We provide a construction of optimal trading strategies and characterize the value function as the unique viscosity solution of the associated quasi-variational inequalities. Moreover, we numerically investigate the optimal trading regions for a variety of real-world cost structures faced by retail investors. We find that the cost structure has a strong effect on the qualitative shape of the no-trading region and optimal strategies.

Mathematical Finance, Vol. 32, No. 2, pp. 555-594, 2022. Available at

We extend the branching diffusion Monte Carlo method of Henry-Labordère et al to the case of parabolic partial differential equations with mixed local-nonlocal analytic nonlinearities. We investigate branching diffusion representations of classical solutions, and we provide sufficient conditions under which the branching diffusion representation solves the partial differential equation in the viscosity sense. Our theoretical setup directly leads to a Monte Carlo algorithm, whose applicability is showcased in the valuation of financial positions with defaultable, systemically important counterparties and a high-dimensional underlying.

Journal of Computational Finance, Vol. 25, No. 3, pp. 51-86, 2021. Available at

We formulate and analyze a mathematical framework for continuous-time mean field games with finitely many states and common noise, including a rigorous probabilistic construction of the state process and existence and uniqueness results for the resulting equilibrium system. The key insight is that we can circumvent the master equation and reduce the mean field equilibrium to a system of forward-backward systems of (random) ordinary differential equations by conditioning on common noise events. In the absence of common noise, our setup reduces to that of Gomes, Mohr and Souza (Appl Math Optim 68(1): 99–143, 2013) and Cecchin and Fischer (Appl Math Optim 81(2):253–300, 2020).

Applied Mathematics & Optimization, Vol. 84, pp. 3173-3216, 2021. Available at

We study optimal liquidation in “target zone models” – asset prices with a reflecting boundary enforced by regulatory interventions. This can be treated as a special case of an Almgren-Chriss model with running and terminal inventory costs and general predictive signals about price changes. The optimal liquidation rate in target-zone models can in turn be characterized as the “theta” of a lookback option, leading to explicit formulas for Bachelier or Black-Scholes dynamics.

Market Microstructure and Liquidity, Vol. 4, No. 03n04, 1950010 (12 pages), 2020. Available at

We revisit the problem of maximising expected utility of terminal wealth in a Black-Scholes market with proportional transaction costs. While it is known that the value function of this problem is the unique viscosity solution of the HJB equation and that the HJB equation admits a classical solution on a reduced state space, it has been an open problem to verify that these two coincide. We establish this result by devising a verification procedure based on superharmonic functions. In the process, we construct optimal strategies and provide a detailed analysis of the regularity of the value function.

Finance and Stochastics, Vol. 23, No. 4, pp. 861-888, 2019. Available at

We study the problem of maximising expected utility of terminal wealth under constant and proportional transaction costs in a multidimensional market with prices driven by a factor process. We show that the value function is the unique viscosity solution of the associated quasi-variational inequalities and construct optimal strategies. While the value function turns out to be truly discontinuous, we are able to establish a comparison principle for discontinuous viscosity solutions which is strong enough to argue that the value function is unique, globally upper semicontinuous, and continuous if restricted to either borrowing or non-borrowing portfolios.

Finance and Stochastics, Vol. 23, No. 1, pp. 29-96, 2019. Available at

Building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations of the form

\qquad X_t = \mathcal{E}\Bigl[\int_t^T g(s,X)\mu(ds) + \xi\Bigr],\qquad t \in [0,T].

We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. As an application, we construct continuous-time recursive utilities under ambiguity and identify the corresponding utility processes as limits of discrete-time recursive utilities.

Mathematics and Financial Economics, Vol. 12, No. 1, pp. 111-134, 2018. Available at

This paper establishes existence of optimal controls for a general stochastic impulse control problem. For this, the value function is characterized as the pointwise minimum of a set of superharmonic functions, as the unique continuous viscosity solution of the quasi-variational inequalities (QVIs), and as the limit of a sequence of iterated optimal stopping problems. A combination of these characterizations is used to construct optimal controls without relying on any regularity of the value function beyond continuity. Our approach is based on the stochastic Perron method and the assumption that the associated QVIs satisfy a comparison principle.

SIAM Journal on Control and Optimization, Vol. 55, No. 2, pp. 627-649, 2017. Available at

We investigate a utility maximization problem in the presence of asset price bubbles. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime-switching model for the warnings and we make no assumptions about the distribution of the timing and the size of the crashes. Instead, we assume that the investor takes a worst-case perspective towards their impacts, i.e. the investor maximizes her expected utility under the worst-case crash scenario. We characterize the value function by a system of Hamilton-Jacobi-Bellman equations and derive a coupled system of ordinary differential equations for the optimal strategies. Numerical examples are provided.

International Journal of Theoretical and Applied Finance, Vol. 19, No. 2, 1650009 (36 pages), 2016. Available at

We study the uniqueness of viscosity solutions of a Hamilton-Jacobi-Bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs. Our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility.

SIAM Journal on Control and Optimization, Vol. 53, No. 5, pp. 2878-2897, 2015. Available at

We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be either in a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario. We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.

Stochastics, Vol. 87, No. 4, pp. 623-663, 2015. Available at

We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. We then verify that this strategy outperforms every other trading strategy using a direct comparison approach. We conclude with numerical examples and calculating the costs of hedging against crashes.

Statistics & Probability Letters, Vol. 90, pp. 140-148, 2014. Available at

PhD Thesis

In this thesis we extend the worst-case modeling approach as first introduced by Hua and Wilmott (1997) (option pricing in discrete time) and Korn and Wilmott (2002) (portfolio optimization in continuous time) in various directions. In the continuous-time worst-case portfolio optimization model (as first introduced by Korn and Wilmott (2002)), the financial market is assumed to be under the threat of a crash in the sense that the stock price may crash by an unknown fraction at an unknown time. It is assumed that only an upper bound on the size of the crash is known and that the investor prepares for the worst-possible crash scenario. That is, the investor aims to find the strategy maximizing her objective function in the worst-case crash scenario. In the first part of this thesis, we consider the model of Korn and Wilmott (2002) in the presence of proportional transaction costs. First, we treat the problem without crashes and show that the value function is the unique viscosity solution of a dynamic programming equation (DPE) and then construct the optimal strategies. We then consider the problem in the presence of crash threats, derive the corresponding DPE and characterize the value function as the unique viscosity solution of this DPE. In the last part, we consider the worst-case problem with a random number of crashes by proposing a regime switching model in which each state corresponds to a different crash regime. We interpret each of the crash-threatened regimes of the market as states in which a financial bubble has formed which may lead to a crash. In this model, we prove that the value function is a classical solution of a system of DPEs and derive the optimal strategies.

Technische Universität Kaiserslautern, 2015. Available at