Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

Backward Stochastic Differential Equations in Finance

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

https://www.pnas.org/content/pnas/115/34/8505.full.pdf

Solving high-dimensional partial differential equations using deep learning

We provide existence, comparison and stability results for one- dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) $F(t,Y, Z)$ is continuous and has a quadratic growth in $Z$ and the terminal condition is boundede also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations

/pdf/backward-stochastic-differential-equations-and-partial-5f6lw1axlk.pdf

Backward stochastic differential equations and partial differential equations with quadratic growth

On resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de Skorokhod

Stochastic differential equations with reflecting boundary conditions

We study reflected solutions of one-dimensional backward stochastic differential equations. The “reflection” keeps the solution above a given stochastic process. We prove uniqueness and existence both by a fixed point argument and by approximation via penalization. We show that when the coefficient has a special form, then the solution of our problem is the value function of a mixed optimal stopping–optimal stochastic control problem. We finally show that, when put in a Markovian framework, the solution of our reflected BSDE provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation.

/pdf/reflected-solutions-of-backward-sde-s-and-related-obstacle-1pq9hqlb8r.pdf

Reflected solutions of backward SDE's, and related obstacle problems for PDE's

This paper examines the continuous-time portfolio-consumption problem of an agent with a recursive utility in the presence of nonlinear constraints on the wealth.Using backward stochastic differential equations, we state a dynamic maximum principle which generalizes the characterization of optimal policies obtained by Duffie and Skiadas [J.Math Econ. 23, 107 –131 (1994)] in the case of a linear wealth. From this property, we derive a characterization of optimal wealth and utility processes as the unique solution of a forward-backward system. Existence of an optimal policy is also established via a penalization method.

/pdf/a-dynamic-maximum-principle-for-the-optimization-of-7rd0yg8eb8.pdf

A dynamic maximum principle for the optimization of recursive utilities under constraints

Non-linear pricing theory and backward stochastic differential equations

In this paper, martingale measures, introduced by J.B. Walsh, are investigated. We prove, with techniques of stochastic calculus, that each continuous orthogonal martingale measure is the time-changed image martingale measure of a white noise. We also exhibit a representation theorem for certain vector martingale measures as stochastic integrals of orthogonal martingale measures. Thus we can study the following martingale problem:
 
$$f(X_t ) - f(X_0 ) - \int\limits_0^t {\int\limits_E {Lf(s,X_s ,x)q_s (dx)ds} } isaP - martingale,$$

 whereL is a second order differential operator andq a predictable random measure-valued process. We prove that this problem is bound to a stochastic differential equation with a term integral with respect to a martingale measure.

N. El Karoui

Papers

Backward Stochastic Differential Equations in Finance

Reflected solutions of backward SDE's, and related obstacle problems for PDE's

A dynamic maximum principle for the optimization of recursive utilities under constraints

Non-linear pricing theory and backward stochastic differential equations

Martingale measures and stochastic calculus