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The classical Method of Successive Approximations (MSA) is an iterative method for solving stochastic control problems and is derived from Pontryagin’s optimality principle. It is known that the MSA may fail to converge. Using careful estimates for the backward stochastic differential equation (BSDE) this paper suggests a modification to the MSA algorithm. This modified MSA is shown to converge for general stochastic control problems with control in both the drift and diffusion coefficients. Under some additional assumptions the rate of convergence is shown. The results are valid without restrictions on the time horizon of the control problem, in contrast to iterative methods based on the theory of forward-backward stochastic differential equations.

Semilinear stochastic partial differential equations on bounded domains$${\mathscr {D}}$$Dare considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are$$L^p$$Lp-estimates for such equations. The$$L^p$$Lp-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space$$H^2({\mathscr {D}}')$$H2(D′)and$$\ell ^2$$ℓ2-integrable with values in$$H^3({\mathscr {D}}')$$H3(D′), for any compact$${\mathscr {D}}' \subset {\mathscr {D}}$$D′⊂D. Using results from$$L^p$$Lp-theory of SPDEs obtained by Kim (Stoch Proc Appl 112:261–283, 2004) we get analogous results in weighted Sobolev spaces on the whole$${\mathscr {D}}$$D. Finally it is shown that the solution is Hölder continuous in time of order$$\frac{1}{2} - \frac{2}{q}$$12-2qas a process with values in a weighted$$L^q$$Lq-space, whereqarises from the integrability assumptions imposed on the initial condition and forcing terms.

We prove the existence of weak solutions to McKean–Vlasov SDEs defined on a domain D⊆Rd with continuous and unbounded coefficients and degenerate diffusion coefficient. Using differential calculus for the flow of probability measures due to Lions, we introduce a novel integrated condition for Lyapunov functions in an infinite dimensional space D×P(D), where P(D) is a space of probability measures on D. Consequently we show existence of solutions to the McKean–Vlasov SDEs on [0,∞). This leads to a probabilistic proof of the existence of a stationary solution to the nonlinear Fokker–Planck–Kolmogorov equation under very general conditions. Finally, we prove uniqueness under an integrated condition based on a Lyapunov function. This extends the standard monotone-type condition for uniqueness.

This paper concerns the McKean–Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its role in connecting weak solutions to McKean–Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the existence of a weak solution assuming boundedness and joint continuity of the coefficients (allowing for degenerate diffusions). Weak uniqueness is established when the private (idiosyncratic) noise’s diffusion coefficient is nondegenerate and the drift is regular in the total variation distance. This seems sharp when one considers using finite-dimensional noise to regularise an infinite dimensional problem. The proof relies on a suitably tailored cost function in the Monge–Kantorovich problem and representation of weak solutions via Girsanov transformations.

A key problem for order book exchanges is how to attract liquidity providers and retain their support in all market conditions. This is commonly approached through individual business agreements with market makers whereby a bespoke contract is negotiated for specific obligations and rewards. Such approaches require a central intermediary that profits from liquidity provision to administer, and typically fail to align the incentives of exchanges and liquidity providers as markets grow. This is costly, slow, and scalability is limited by the exchange’s resources, contacts, and expertise.

Motivated by applications to SPDEs we extend the It\^o formula for the square of the norm of a semimartingale $y(t)$ from Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the case \begin{equation*} \sum_{i=1}^m \int_{(0,t]} v_i^{\ast}(s)\,dA(s) + h(t)=:y(t)\in V \quad \text{$dA\times \mathbb{P}$-a.e.}, \end{equation*} where $A$ is an increasing right-continuous adapted process, $v_i^{\ast}$ is a progressively measurable process with values in $V_i^{\ast}$, the dual of a Banach space $V_i$, $h$ is a cadlag martingale with values in a Hilbert space $H$, identified with its dual $H^{\ast}$, and $V:=V_1\cap V_2 \cap \ldots \cap V_m$ is continuously and densely embedded in $H$. The formula is proved under the condition that $\|y\|_{V_i}^{p_i}$ and $\|v_i^\ast\|_{V_i^\ast}^{q_i}$ are almost surely locally integrable with respect to $dA$ for some conjugate exponents $p_i, q_i$. This condition is essentially weaker than the one which would arise in application of the results in Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the semimartingale above., Comment: Updated to the version published in Stochastics and Partial Differential Equations: Analysis and Computations

Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Any given model provides only a crude approximation to reality and the risk of using an inadequate model is hard to detect and quantify. By contrast, modern data science techniques are opening the door to more robust and data-driven model selection mechanisms. However, most machine learning models are "black-boxes" as individual parameters do not have meaningful interpretation. The aim of this paper is to combine the above approaches achieving the best of both worlds. Combining neural networks with risk models based on classical stochastic differential equations (SDEs), we find robust bounds for prices of derivatives and the corresponding hedging strategies while incorporating relevant market data. The resulting model called neural SDE is an instantiation of generative models and is closely linked with the theory of causal optimal transport. Neural SDEs allow consistent calibration under both the risk-neutral and the real-world measures. Thus the model can be used to simulate market scenarios needed for assessing risk profiles and hedging strategies. We develop and analyse novel algorithms needed for efficient use of neural SDEs. We validate our approach with numerical experiments using both local and stochastic volatility models.

Exchanges and other trading venues must be able to reliably offer sufficient liquidity to meet traders' demand, and, in turn, benefit from higher revenue and faster growth. Traders enjoy lower volatility, tighter spreads, and more efficient pricing in liquid markets. A key problem for exchanges is how to attract liquidity providers and retain their support in all market conditions. This is commonly approached through individual business agreements with market makers whereby a bespoke contract is negotiated for specific obligations and rewards. Such approaches require a central intermediary that profits from liquidity provision to administer, and typically fail to align the incentives of exchanges and liquidity providers as markets grow. This is costly, slow, and scalability is limited by the exchange’s resources, contacts, and expertise. This paper develops mechanisms for creating open, automated and scalable liquidity markets. We describe formal methods to quantify liquidity and discuss various approaches to determine its price. In so doing, we introduce a novel way to structure liquidity commitments, along with a mechanism based on a financial bond with penalties for under-provision to maximize market makers’ adherence to their obligations. We also investigate mechanisms to allocate rewards derived from trading fees between market makers, so as to incentivise desirable-but-risky behaviors such as market creation and early commitment of liquidity. We complement this work with several agent based simulations exploring the proposed mechanisms.

We consider the problem of risk model calibration that is faced by all decentralized derivative exchanges. Financial model calibration is hard for two reasons: firstly it relies on data inputs that can be unreliable, incorrect and in general needing manual cleaning. Secondly, even if perfectly correct data is available the problem typically involves non-convex minimization resulting in local minima that are moreover highly dependent on small perturbations to input. Thus even honest parties won't necessarily produce the same parameters. On a decentralized exchange multiple parties need to agree on the correct calibration. Moreover, malicious actors may benefit in providing calibration parameters that benefit their trading, in case they can convince others that their calibration is the right one. Effectively we have a problem of trying to achieve consensus in continuum. We propose a phenomenological model for the problem. We analyse this in the framework of stochastic differential games and we show that a Nash equilibrium exists. We present empirical results for simple situations that arise when the risk model is assumed to be a linear function of calibration parameters.

Our work is motivated by a desire to study the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of neural networks. The key insight, already observed in the works of Mei, Montanari and Nguyen (2018), Chizat and Bach (2018) as well as Rotskoff and Vanden-Eijnden (2018), is that a certain class of the finite-dimensional non-convex problems becomes convex when lifted to infinite-dimensional space of measures. We leverage this observation and show that the corresponding energy functional defined on the space of probability measures has a unique minimiser which can be characterised by a first-order condition using the notion of linear functional derivative. Next, we study the corresponding gradient flow structure in 2-Wasserstein metric, which we call Mean-Field Langevin Dynamics (MFLD), and show that the flow of marginal laws induced by the gradient flow converges to a stationary distribution, which is exactly the minimiser of the energy functional. We observe that this convergence is exponential under conditions that are satisfied for highly regularised learning tasks. Our proof of convergence to stationary probability measure is novel and it relies on a generalisation of LaSalle's invariance principle combined with HWI inequality. Importantly, we assume neither that interaction potential of MFLD is of convolution type nor that it has any particular symmetric structure. Furthermore, we allow for the general convex objective function, unlike, most papers in the literature that focus on quadratic loss. Finally, we show that the error between finite-dimensional optimisation problem and its infinite-dimensional limit is of order one over the number of parameters., 31 pages

Higher moment a priori estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.

We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simulatenously. Having approximated the gradient of the solution one can combine it with a Monte-Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative price). This is achieved by leveraging the Martingale Representation Theorem and combining the Monte Carlo simulation with the neural network. The resulting algorithm is robust with respect to quality of the neural network approximation and consequently can be used as a black-box in case only limited a priori information about the underlying problem is available. We believe this is important as neural network based algorithms often require fair amount of tuning to produce satisfactory results. The methods are empirically shown to work for high-dimensional problems (e.g. 100 dimensions). We provide diagnostics that shed light on appropriate network architectures.

Global existence of solutions for a class of second-order evolution equations with damping is shown by proving convergence of a full discretization. The discretization combines a fully implicit time stepping with a Galerkin scheme. The operator acting on the zero-order term is assumed to be a potential operator where the potential may be nonconvex. A linear, symmetric operator is assumed to be acting on the first-order term. Applications arise in nonlinear viscoelasticity and elastodynamics. (C) 2013 Elsevier Inc. All rights reserved.

Di erential equations of the type @ttur a(r@tu) u = f are studied, where the nonlinear mapping a is continuous, monotone and satisfies a coercivity condition in terms of a (generalised) N -function. The class of problems thus includes the case of anisotropic and nonpolynomial growth. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation.

Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.

SKA ‡ Abstract. The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H 1. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated for the piecewise constant finite element approximation of the porous medium equation with the�-distribution as initial value. As a byproduct, L p -stability of theH 1 -orthogonal projection onto the space of piecewise constant functions is shown. ut − �(juj p−2 u) = f; p >1; u= u(x;t); u(�;0) = u0; (x;t) 2 � (0;T); whereR d is a "nice" bounded domain, −� is the Laplace operator acting on the spatial variables, u0 are given initial data, f is a given right-hand side and we assume some boundary condition on the boundary of . We immediately see that for p = 2 this is the classical heat equation. For p > 2, it is referred to as the porous medium equation. In this case u is the density of the gas at a given point and time. For 1 2, in the one dimensional case, the solution is given by (5.2). We will later use this in numerical experiments. The solution is also known in higher dimensions, see again Vazquez (22, Chapter 4). Throughout the article, we focus on the following generalization of the above

Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results. 2010 Mathematical subject classification: 47J35; 65M12; 47H05; 47G20; 74K15; 74B20.

Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. Uniqueness is proved as well., Comment: This is the version of the article accepted for publication. The final publication is available at http://link.springer.com

Convergence of a full discretisation method is studied for a class of nonlinear second order in time evolution equations, where the nonlinear operator acting on the first-order time derivative of the solution is supposed to be hemicontinuous, monotone, coercive and to satisfy a certain growth condition, and the operator acting on the solution is assumed to be linear, bounded, symmetric, and strongly positive. The numerical approximation combines a Galerkin spatial discretisation with a novel time discretisation obtained from a reformulation of the second-order evolution equation as a first-order system and an application of the two-step backward differentiation formula with constant time stepsizes. Convergence towards the weak solution is shown for suitably chosen piecewise polynomial in time prolongations of the resulting fully discrete solutions, and an a priori error estimate ensures convergence of second order in time provided that the exact solution to the problem fulfils certain regularity requirements. A numerical example for a model problem describing the displacement of a vibrating membrane in a viscous medium illustrates the favourable error behaviour of the proposed full discretisation method in situations where regular solutions exist.

A class of stochastic optimal control problems involving optimal stopping is considered. Methods of Krylov are adapted to investigate the numerical solutions of the corresponding normalized Bellman equations and to estimate the rate of convergence of finite difference approximations for the optimal reward functions., Comment: 36 pages, ArXiv version updated to the version accepted in Appl. Math. Optim

A general result on the method of randomized stopping is proved. It is applied to optimal stopping of controlled diffusion processes with unbounded coefficients to reduce it to an optimal control problem without stopping. This is motivated by recent results of Krylov on numerical solutions to the Bellman equation., Comment: Published in at http://dx.doi.org/10.3150/07-BEJ108 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)