Stochastic equations in infinite dimensions

1. Introduction


1.1. Searching the Mechanism of Evaluations of Risky Assets


1.2. Axiomatic Assumptions for Evaluations of Derivatives


1.3. Organization of the Lecture






2. Brownian Filtration Consistent Evaluations and Expectations


2.1. Main Notations and Definitions


2.2. Open image in new window -Consistent Nonlinear Expectations


2.3. Open image in new window -Consistent Nonlinear Evaluations






3. Backward Stochastic Differential Equations: g-Evaluations and g-Expectations


3.1. BSDE: Existence, Uniqueness and Basic Estimates


3.2. 1-Dimensional BSDE


3.3. A Monotonic Limit Theorem of BSDE


3.4. g-Martingales and (Nonlinear) g-Supermartingale Decomposition Theorem






4. Finding the Mechanism: Is an Open image in new window -Expectation a g-Expectation?


4.1. Open image in new window -Dominated Open image in new window -Expectations


4.2. Open image in new window -Consistent Martingales


4.3. BSDE under Open image in new window -Consistent Nonlinear Expectations


4.4. Decomposition Theorem for Open image in new window -Supermartingales


4.5. Representation Theorem of an Open image in new window -Expectation by a g-Expectation


4.6. How to Test and Find g?


4.7. A General Situation: Open image in new window -Evaluation Representation Theorem






5. Dynamic Risk Measures


6. Numerical Solution of BSDEs: Euler’s Approximation


7. Appendix


7.1. Martingale Representation Theorem


7.2. A Monotonic Limit Theorem of Ito’s Processes


7.3. Optional Stopping Theorem for Open image in new window -Supermartingale






References


References on BSDE and Nonlinear Expectations

/pdf/nonlinear-expectations-nonlinear-evaluations-and-risk-1df84s74m8.pdf

Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

We examine a market in which long-lived firms face a short-term incentive to exert low effort, but could earn higher profits if it were possible to commit to high effort. There are two types of firms, "inept" firms who can only exert low effort, and "competent" firms who have a choice between high and low effort. There is occasional exit, and competent and inept potential entrants compete for the right to inherit the departing firm's reputation. Consumers receive noisy signals of effort choice, and so competent firms choose high effort in an attempt to distinguish themselves from inept firms. A competent firm is most likely to enter the market by purchasing an average reputation, in the hopes of building it into a good reputation, than either a very low reputation or a very high reputation. Inept firms, in contrast, find it more profitable to either buy high reputations and deplete them or buy low reputations.

Who wants a good reputation

The paper studies the convergence, as N tends to infinity, of a system of N coupled Hamilton-Jacobi equations, the Nash system. This system arises in differential game theory. We describe the limit problem in terms of the so-called master equation " , a kind of second order partial differential equation stated on the space of probability measures. Our first main result is the well-posedness of the master equation. To do so, we first show the existence and uniqueness of a solution to the " mean field game system with common noise " , which consists in a coupled system made of a backward stochastic Hamilton-Jacobi equation and a forward stochastic Kolmogorov equation and which plays the role of characteristics for the master equation. Our second main result is the convergence, in average, of the solution of the Nash system and a propagation of chaos property for the associated " optimal trajectories " ."

The Master Equation and the Convergence Problem in Mean Field Games

Handbook of Game Theory with Economic Applications

In this paper, we consider a mean-field stochastic differential equation, also called the McKean–Vlasov equation, with initial data (t,x)∈[0,T]×Rd(t,x)∈[0,T]×Rd, whose coefficients depend on both the solution Xt,xsXst,x and its law. By considering square integrable random variables ξξ as initial condition for this equation, we can easily show the flow property of the solution Xt,ξsXst,ξ of this new equation. Associating it with a process Xt,x,PξsXst,x,Pξ which coincides with Xt,ξsXst,ξ, when one substitutes ξξ for xx, but which has the advantage to depend on ξξ only through its law PξPξ, we characterize the function V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)]V(t,x,Pξ)=E[Φ(XTt,x,Pξ,PXTt,ξ)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean-field type, involving the first- and the second-order derivatives of VV with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first- and second-order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Ito formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au College de France: Theorie des jeu a champs moyens (2013)], and we extend it in a direct way to the second-order derivatives.

/pdf/mean-field-stochastic-differential-equations-and-associated-2c785o2s6d.pdf

Mean-field stochastic differential equations and associated PDEs

In this paper we consider a mean-field stochastic differential equation, also called Mc Kean-Vlasov equation, with initial data $(t,x)\in[0,T]\times R^d,$ which coefficients depend on both the solution $X^{t,x}_s$ but also its law. By considering square integrable random variables $\xi$ as initial condition for this equation, we can easily show the flow property of the solution $X^{t,\xi}_s$ of this new equation. Associating it with a process $X^{t,x,P_\xi}_s$ which coincides with $X^{t,\xi}_s$, when one substitutes $\xi$ for $x$, but which has the advantage to depend only on the law $P_\xi$ of $\xi$, we characterise the function $V(t,x,P_\xi)=E[\Phi(X^{t,x,P_\xi}_T,P_{X^{t,\xi}_T})]$ under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a non local PDE of mean-field type, involving the first and second order derivatives of $V$ with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first and second order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Ito formula. In our approach we use the notion of derivative with respect to a square integrable probability measure introduced in \cite{PL} and we extend it in a direct way to second order derivatives.

Existence and characterization of Nash equilibrium payoffs are proved for stochastic nonzero-sum differential games

Nash Equilibrium Payoffs for Nonzero-Sum Stochastic Differential Games

In this Note, we give a necessary and sufficient condition for the existence of control that keeps the corresponding trajectory of the related stochastic control system within a prescribed closed subset of the state space. The problem of existence of stochastic control under a state-constraint is also called the viability property of the underlying control system. Our result is: the square of the distance function of this constraint is a viscosity supersolution of a Hamilton-Jacobi-Bellman equation if and only if the system enjoys the viability property.

Existence of stochastic control under state constraints

We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual viscosity solutions of some second order Hamilton-Jacobi equation.

Catherine Rainer

Papers

Mean-field stochastic differential equations and associated PDEs

Mean-field stochastic differential equations and associated PDEs

Nash Equilibrium Payoffs for Nonzero-Sum Stochastic Differential Games

Existence of stochastic control under state constraints

Stochastic Differential Games with Asymmetric Information