Front Matter Pages Basics of Stochastic Calculus. Stochastic Differential Equations. Backward Stochastic Differential Equations. Reflected Backward SDEs. Forward-Backward SDEs. Stochastic Calculus Under Weak Formulation. Nonlinear Expectation. Only g is assumed to be satisfying H1 and g 0 s, Ys0 , Zs0 has to be just an element of H RT Using the same argument as in the proof of Theorem 1.
The proof is based on regularization by inf-convolution techniques. Lemme 1. Proof of Theorem 1. By the Comparison Theorem 1. Now if Y 0 , Z 0 is another solution for the BSDE 3 , given that the coefficients gn are Lipschitz pro- cesses, we can apply the comparison theorem 1.
Just as previously, if we had used a non-increasing scheme we would have constructed the maximal solution of the BSDE 3. The way in which the minimal maximal solution for 3 has been constructed allows us to deduce a comparison result. Then, with the Lipschitz continuous coefficients gn defined in Lemma 1. The same property holds for the limit process Y. In the following, the set of admissible self-financing portfolios is denoted by A. Moreover Y is the minimum value of all admissible portfolios, i.
Proof: By Theorem 1. As in Proposition 1. In addition, it is invertible and its inverse is bounded note that this latter condition can be relaxed.
The set of those controls is called admissible and is denoted by U. The function f is assumed to be measurable and bounded. For the sake of simplicity, we assume the following strong properties: f and h are bounded, continuous with respect to u. Then, by Comparison Theorem 1. It is easy to represent J u.
Comparison Theorem 1. There are different ways to prove this result. With our strong assumptions, we give below a complete answer to this point. For the use of ess supremum, see for instance [22]. Therefore, thanks to Theorem 1. It remains to show that we have the converse inequality. We do this by providing an optimal control. We summarize these results in the following theorem: Theorem 2. Note that the same type of results is used under a different set of assumptions in the same book of the paper by Barrieu and El Karoui when defining g-conditional risk measures as the maximal solution of a BSDE associated with a convex coefficient g.
In particular, the paper of Quenez provides a link between martingale methods in stochastic control in [16] or [26] and BSDEs. To control the system the agent c1 resp. The set of admissible controls for c1 resp. As in the control setting, the interventions of the controllers generate a drift in the dynamics of the evolution of the system. Between the agents c1 and c2 , depending on the used pair of strategies u, v , there is a payoff J u, v , which is a reward for c1 and a cost for c2.
In addition, it is continuous with respect to u, v. For an admissible pair of strategies u, v , let us define the the Hamiltonian of the zero-sum game H t, x. Note that H is an affine function, uni- formly Lipschitz continuous with respect to z. We have the following result related to the existence of a saddle-point for the zero-sum stochastic differential game.
Theorem 2. Short proof : For the sake of simplicity, we only consider here the case of g bounded. The function which associates H t, x. On the other hand, both the Comparison The- orem 1. In this section, we will show this relationship, starting from very general results to end up with fine regularity results. Moreover, in order to have good estimates of the solution, we assume that the conditions M1f are satisfied, where the symbol f is used to refer to forward equation.
The problem is then much more complicated and does not admit a solution in all cases See [44] for a complete review on this problem. We will now show that the solution Yst,x , Zst,x is Markovian in the sense that these processes can be expressed through deterministic functions of s and Xst,x.
More precisely, Theorem 3. The result can be obtained from Theorem 6. By applying Lemma 3. Hence, u b may be substituted to u in the coefficient g. Using approximation by inf- convolution as in Theorem 1. Let us denote by un and dn the functions associated with the solution of the Lipschitz BSDEs as in b above.
Let us first introduce the definition of a viscosity solution: Definition 3. The polynomial growth of u follows from the Lp -estimates for Y. For the sake of simplicity, we consider only the one-dimensional case, i. By continuity, 32 still holds for a neighbourhood of the point t, x. We will study in details the regularity of the functions of the BSDE solution with respect to the initial conditions and provide a probabilistic method for studying solutions of the semilinear PDE The objective is to find probabilistic estimates with respect to the initial conditions in time and space , in order to show that this is a regular solution of the PDE.
The proof of these results is technical, so we do not present it here and refer to Pardoux and Peng [49], Lemma 2. This problem is related to a zero-sum switching game. Then we tackle the problem of the uniqueness of the solution.
Finally, we apply the obtained results and prove that, without the usual monotonicity condition, the associated PDE system has a unique solution in. As explained below, equation 24 is the first- order version of the second-order equation associated with For example, we saw in Section 3 that the emission process did not see the trap in the uniformly elliptic setting.
In the current framework, the diffusion coefficient vanishes in a linear way as time tends to the maturity: it decays too fast to prevent almost every realization of the process from falling into the trap. In order to obtain the representation property a third martingale is considered. The convergence of this approximation relies on the fact that the first random walk converges to the driving Brownian motion, and the second to the driving compensated Poisson process.
Let us define these two random walks. Backward stochastic differential equations driven by Gaussian Volterra processes tifractional Ornstein-Uhlenbeck processes.
In the first part we study multidimensional BSDE with generators that are linear functions of the solution. Under an integrability condition on a functional of the second moment of the Volterra process in a neighbourhood of the terminal time, we solve the associated PDE ex- plicitely and deduce the solution of the linear BSDE. We discuss an application in the context of self-financing trading stategies.
The main results are the existence and uniqueness of the solution in a space of regular functionals of the Volterra process , and a comparison theorem for the solutions of BSDE. We give two proofs for the existence and uniqueness of the solution, one is based on the associated PDE and a second one without making reference to this PDE, but with probabilistic and functional theoretic methods.
Especially this second proof is technically quite complex, and, due to the absence of martingale properties in the context of Volterra pro- cesses, requires to work with different norms on the underlying Hilbert space that is defined by the kernel of the Volterra process. Contrary to the more classical cases of BSDE driven by Brownian or fractional Brow- nian motion, an assumption on the behaviour of the kernel of the driving Volterra process is in general necessary for the wellposedness of the BSDE.
For multifractional Brownian motion this assumption is closely related to the behaviour of the Hurst function. A cubature based algorithm to solve decoupled McKean-Vlasov Forward Backward Stochastic Differential Equations The main idea of the cubature method consists in replacing the Brownian motion by choosing ran- domly a path among an a priori finite set 1 of continuous functions from [0, T ] to R d with bounded variations such that the expectation of the iterated integrals against both the Brownian and such paths are the same, up to a given order m.
We give the main idea to construct a cubature based approximation scheme for 1. This dependence breaks the Markov property considered only on R d of the process so that it is not possible to apply, a priori, many classical analysis tools.
In order to handle this problem, the idea consists in taking benefit on the following observation: given the law of the solution of the system, 1.
Similarly, we do not take energy dependence into account. As a result, we consider here the simplified but realistic case of monokinetic particle, that is a particle that has a constant speed and that can not give birth to other subparticles. For this model of a monokinetic particle, the set of the successive collision point positions constitute thus a Markov chain. Furthermore, with probability one, the monokinetic particle is absorbed after a finite number of collisions.
The small probability we are interested in, in this work, is thus the probability that a Markov chain, that is almost surely stationary, in finite time, "pass" through a shielding system and reach a domain of interest before absorption. Reflected backward stochastic differential equations with jumps and partial integro-differential variational inequalities We illustrate these results on an optimal stopping problem for dynamic risk measures induced by BSDEs with jumps.
By [15], the value function, which represents the minimal risk measure, coincides with the solution of an RBSDE with jumps. Neutronics is the study of neutron population in fissile media that can be modeled using the linear Boltzmann equation, also known as the transport equation. On the one hand, criticality studies aim at understanding the neutron population dynamics due to the branching process that mimics fission reaction see for instance [ 71 ] for a recent survey on branching processes in neutronics.
On the other hand, when neutrons are propagated through media where fission reactions do not occur, or can safely be neglected, their transport can be modeled by simple exponential flights [ 72 ] : indeed, between each collisions, neutrons travel along straight path distributed exponentially.
Among this last category, shielding studies allow to size shielding structures so as to protect humans from ionizing particles, and imply, by definition, the attenuation of initial neutron flux typically by several decades. For instance, the vessel structure of a nuclear reactor core attenuates the thermal neutron flux inside the code by a factor roughly equal to 10
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