arXiv subjects
Publications and source records attributed to Gilles Pagès.
We design a fully implementable scheme to compute the invariant distribution of ergodic McKean-Vlasov SDE satisfying a uniform confluence property. Under natural conditions, we prove various convergence results notably we obtain rates for the Wasserstein distance in quadratic mean and almost sure sense.
We introduce a new class of neural networks designed to be convex functions of their inputs, leveraging the principle that any convex function can be represented as the supremum of the affine functions it dominates. These neural networks, inherently convex with respect to their inputs, are particularly well-suited for approximating the prices of options with convex payoffs. We detail the architecture of this, and establish theoretical convergence bounds that validate its approximation capabilities. We also introduce a \emph{scrambling} phase to improve the training of these networks. Finally, we demonstrate numerically the effectiveness of these networks in estimating prices for three types of options with convex payoffs: Basket, Bermudan, and Swing options.
Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order between a Gaussian distribution and a Gaussian mixture. We give and discuss intrinsic necessary and sufficient conditions for convex ordering. On the examples that we have worked out, the two conditions appear to be closely related.
We propose two parametric approaches to evaluate swing contracts with firm constraints. Our objective is to define approximations for the optimal control, which represents the amounts of energy purchased throughout the contract. The first approach involves approximating the optimal control by means of an explicit parametric function, where the parameters are determined using stochastic gradient descent based algorithms. The second approach builds on the first one, where we replace parameters in the first approach by the output of a neural network. Our numerical experiments demonstrate that by using Langevin based algorithms, both parameterizations provide, in a short computation time, better prices compared to state-of-the-art methods.
In this paper, we are interested in the propagation of convexity by the strong solution to a one-dimensional Brownian stochastic differential equation with coefficients Lipschitz in the spatial variable uniformly in the time variable and in the convex ordering between the solutions of two such equations. We prove that while these properties hold without further assumptions for convex functions of the processes at one instant only, an assumption almost amounting to spatial convexity of the diffusion coefficient is needed for the extension to convex functions at two instants. Under this spatial convexity of the diffusion coefficients, the two properties even hold for convex functionals of the whole path. For directionally convex functionals, the spatial convexity of the diffusion coefficient is no longer needed. Our method of proof consists in first establishing the results for time discretization schemes of Euler type and then transferring them to their limiting Brownian diffusions. We thus exhibit approximations which avoid {\em convexity arbitrages} by preserving convexity propagation and comparison and can be computed by Monte Carlo simulation.
We propose a new algorithm for variance reduction when estimating $f(X_T)$ where $X$ is the solution to some stochastic differential equation and $f$ is a test function. The new estimator is $(f(X^1_T) + f(X^2_T))/2$, where $X^1$ and $X^2$ have same marginal law as $X$ but are pathwise correlated so that to reduce the variance. The optimal correlation function $ρ$ is approximated by a deep neural network and is calibrated along the trajectories of $(X^1, X^2)$ by policy gradient and reinforcement learning techniques. Finding an optimal coupling given marginal laws has links with maximum optimal transport.
Stochastic Gradient Descent Langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numerical resolution of stochastic control problems through gradient descent, where the control is parametrized by a neural network. If the control is applied at many discretization times then solving the stochastic control problem reduces to minimizing the loss of a very deep neural network. We numerically show that Langevin algorithms improve the training on various stochastic control problems like hedging and resource management, and for different choices of gradient descent methods.
We give bounds for the total variation distance between the solutions to two stochastic differential equations starting at the same point and with close coefficients, which applies in particular to the distance between an exact solution and its Euler-Maruyama scheme in small time. We show that for small $t$, the total variation distance is of order $t^{r/(2r+1)}$ if the noise coefficient $σ$ of the SDE is elliptic and $\mathcal{C}^{2r}_b$, $r\in \mathbb{N}$ and if the drift is $C^1$ with bounded derivatives, using multi-step Richardson-Romberg extrapolation. We do not require the drift to be bounded. Then we prove with a counterexample that we cannot achieve a bound better than $t^{1/2}$ in general.
In this paper, we are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous $\R^d$-valued paths and for the monotonic convex order when $d=1$. Even if in general these solutions are neither semi-martingales nor Markov processes, we are able to exhibit conditions on their coefficients enabling the comparison. Our approach consists in first comparing their Euler schemes and then taking the limit as the time step vanishes. We consider two types of Euler schemes depending on the way the Volterra kernels are discretized. The conditions ensuring the comparison are slightly weaker for the first scheme than for the second one and this is the other way round for convergence. Moreover, we extend the integrability needed on the starting values in the existence and convergence results in the literature to be able to only assume finite first order moments, which is the natural framework for convex ordering.
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic differential equation $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. Allowing $σ$ to depend on the position brings faster convergence in comparison with the classical Langevin equation $dY_t = -\nabla V(Y_t)dt + σdW_t$. In a previous paper we established the convergence in $L^1$-Wasserstein distance of $Y_t$ and of its associated Euler scheme $\bar{Y}_t$ to $\text{argmin}(V)$ with the classical schedule $a(t) = A\log^{-1/2}(t)$. In the present paper we prove the convergence in total variation distance. The total variation case appears more demanding to deal with and requires regularization lemmas.
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic equation $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. This setting can be applied to optimization problems arising in Machine Learning. The case where $σ$ is a constant matrix has been extensively studied however little attention has been paid to the general case. We prove the convergence for the $L^1$-Wasserstein distance of $Y_t$ and of the associated Euler-scheme $\bar{Y}_t$ to some measure $ν^\star$ which is supported by $\text{argmin}(V)$ and give rates of convergence to the instantaneous Gibbs measure $ν_{a(t)}$ of density $\propto \exp(-2V(x)/a(t)^2)$. To do so, we first consider the case where $a$ is a piecewise constant function. We find again the classical schedule $a(t) = A\log^{-1/2}(t)$. We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.
We establish the functional convex order results for two scaled McKean-Vlasov processes $X=(X_{t})_{t\in[0, T]}$ and $Y=(Y_{t})_{t\in[0, T]}$ defined on a filtered probability space $(Ω, \mathcal{F}, (\mathcal{F}_{t})_{t\geq0}, \mathbb{P})$ by \[\begin{cases} dX_{t}= b(t, X_{t}, μ_{t})dt+σ(t, X_{t}, μ_{t})dB_{t}, \;\;X_{0}\in L^{p}(\mathbb{P}),\\ dY_{t}\,= b(t, \,Y_{t}\,,\, ν_{t})dt+θ(t, \,Y_{t}\,,\, ν_{t})dB_{t}, \;\;Y_{0}\in L^{p}(\mathbb{P}), \end{cases}\] where $p\geq2$, for every $ t\in[0, T]$, $μ_t$, $ν_t$ denote the probability distribution of $X_t$, $Y_t$ respectively and the drift coefficient $b(t, x, μ)$ is affine in $x$ (scaled). If we make the convexity and monotony assumption (only) on $σ$ and if $σ\preceqθ$ with respect to the partial matrix order, the convex order for the initial random variable $X_0 \preceq_{\,cv} Y_0$ can be propagated to the whole path of process $X$ and $Y$. That is, if we consider a convex functional $F$ defined on the path space with polynomial growth, we have $\mathbb{E}F(X)\leq\mathbb{E}F(Y)$; for a convex functional $G$ defined on the product space involving the path space and its marginal distribution space, we have $\mathbb{E}\,G\big(X, (μ_t)_{t\in[0, T]}\big)\leq \mathbb{E}\,G\big(Y, (ν_t)_{t\in[0, T]}\big)$ under appropriate conditions. The symmetric setting is also valid, that is, if $θ\preceq σ$ and $Y_0 \leq X_0$ with respect to the convex order, then $\mathbb{E}\,F(Y) \leq \mathbb{E}\,F(X)$ and $\mathbb{E}\,G\big(Y, (ν_t)_{t\in[0, T]}\big)\leq \mathbb{E}\,G(X, (μ_t)_{t\in[0, T]})$. The proof is based on several forward and backward dynamic programming principles and the convergence of the Euler scheme of the McKean-Vlasov equation.
Fishing quotas are unpleasant but efficient to control the productivity of a fishing site. A popular model has a stochastic differential equation for the biomass on which a stochastic dynamic programming or a Hamilton-Jacobi-Bellman algorithm can be used to find the stochastic control -- the fishing quota. We compare the solutions obtained by dynamic programming against those obtained with a neural network which preserves the Markov property of the solution. The method is extended to a similar multi species model to check its robustness in high dimension.
We propose a novel approach in the assessment of a random risk variable $X$ by introducing magnitude-propensity risk measures $(m_X,p_X)$. This bivariate measure intends to account for the dual aspect of risk, where the magnitudes $x$ of $X$ tell how hign are the losses incurred, whereas the probabilities $P(X=x)$ reveal how often one has to expect to suffer such losses. The basic idea is to simultaneously quantify both the severity $m_X$ and the propensity $p_X$ of the real-valued risk $X$. This is to be contrasted with traditional univariate risk measures, like VaR or Expected shortfall, which typically conflate both effects. In its simplest form, $(m_X,p_X)$ is obtained by mass transportation in Wasserstein metric of the law $P^X$ of $X$ to a two-points $\{0, m_X\}$ discrete distribution with mass $p_X$ at $m_X$. The approach can also be formulated as a constrained optimal quantization problem. This allows for an informative comparison of risks on both the magnitude and propensity scales. Several examples illustrate the proposed approach.
We establish upper bounds for the $L^p$-quantization error, p in (1, 2+d), induced by the recursive Markovian quantization of a d-dimensional diffusion discretized via the Euler scheme. We introduce a hybrid recursive quantization scheme, easier to implement in the high-dimensional framework, and establish upper bounds to the corresponding $L^p$-quantization error. To take advantage of these extensions, we propose a time discretization scheme and a recursive quantization-based discretization scheme associated to a reflected Backward Stochastic Differential Equation and estimate $L^p$-error bounds induced by the space approximation. We will explain how to numerically compute the solution of the reflected BSDE relying on the recursive quantization and compare it to other types of quantization.
In this paper, we establish the monotone convex order between two $\mathbb{R}$-valued McKean-Vlasov processes $X=(X_t)_{t\in [0, T]}$ and $Y=(Y_t)_{t\in [0, T]}$ defined on a filtered probability space $(Ω, \mathcal{F}, (\mathcal{F}_{t})_{t\geq0}, \mathbb{P})$ by \begin{align} &dX_{t}=b(t, X_{t}, μ_{t})dt+σ(t, X_{t}, μ_{t})dB_{t}, \quad X_{0}\in L^{p}(\mathbb{P})\; \text{with}\; p\geq 2,\nonumber\\ &dY_{t}=β(t, Y_{t}, ν_{t})dt+θ(t, \,Y_{t}, ν_{t})\,dB_{t}, \,\quad Y_{0}\in L^{p}(\mathbb{P}), \nonumber \end{align} where $\forall\, t\in [0, T],\: μ_{t}=\mathbb{P}\circ X_{t}^{-1}, \:ν_{t}=\mathbb{P}\circ Y_{t}^{-1}. $ If we make the convexity and monotony assumption (only) on $b$ and $|σ|$ and if $b\leq β$ and $|σ|\leq |θ|$, then the monotone convex order for the initial random variable $X_0\preceq_{\,\text{mcv}} Y_0$ can be propagated to the whole path of processes $X$ and $Y$. That is, if we consider a non-decreasing convex functional $F$ defined on the path space with polynomial growth, we have $\mathbb{E}\, F(X)\leq \mathbb{E}\, F(Y)$; for a non-decreasing convex functional $G$ defined on the product space involving the path space and its marginal distribution space, we have $\mathbb{E}\, G(X, (μ_{t})_{t\in [0, T]})\leq \mathbb{E}\, G(Y, (ν_{t})_{t\in [0, T]})$ under appropriate conditions. The symmetric setting is also valid, that is, if $Y_0\preceq_{\,\text{mcv}} X_0$ and $|θ|\leq |σ|$, then $\mathbb{E}\, F(Y)\leq \mathbb{E}\, F(X)$ and $\mathbb{E}\, G(Y, (ν_{t})_{t\in [0, T]})\leq \mathbb{E}\, G(X, (μ_{t})_{t\in [0, T]})$. The proof is based on several forward and backward dynamic programming principle and the convergence of the truncated Euler scheme of the McKean-Vlasov equation.
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its optimal) quadratic primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension $1$ so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to $\infty$.
We are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order. We propose to alternate transitions according to a martingale Markov kernel mapping a probability measure in the sequence to the next and dual quantization steps. In the case of ARCH models and in particular of the Euler scheme of a driftless Brownian diffusion, the noise has to be truncated to enable the dual quantization step. We analyze the error between the original ARCH model and its approximation with truncated noise and exhibit conditions under which the latter is dominated by the former in the convex order at the level of sample-paths. Last, we analyse the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.