1,358 research outputs found
Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method
We study semi-linear elliptic PDEs with polynomial non-linearity and provide
a probabilistic representation of their solution using branching diffusion
processes. When the non-linearity involves the unknown function but not its
derivatives, we extend previous results in the literature by showing that our
probabilistic representation provides a solution to the PDE without assuming
its existence. In the general case, we derive a new representation of the
solution by using marked branching diffusion processes and automatic
differentiation formulas to account for the non-linear gradient term. In both
cases, we develop new theoretical tools to provide explicit sufficient
conditions under which our probabilistic representations hold. As an
application, we consider several examples including multi-dimensional
semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo
method
Some Results on Skorokhod Embedding and Robust Hedging with Local Time
In this paper, we provide some results on Skorokhod embedding with local time
and its applications to the robust hedging problem in finance. First we
investigate the robust hedging of options depending on the local time by using
the recently introduced stochastic control approach, in order to identify the
optimal hedging strategies, as well as the market models that realize the
extremal no-arbitrage prices. As a by-product, the optimality of Vallois'
Skorokhod embeddings is recovered. In addition, under appropriate conditions,
we derive a new solution to the two-marginal Skorokhod embedding as a
generalization of the Vallois solution. It turns out from our analysis that one
needs to relax the monotonicity assumption on the embedding functions in order
to embed a larger class of marginal distributions. Finally, in a full-marginal
setting where the stopping times given by Vallois are well-ordered, we
construct a remarkable Markov martingale which provides a new example of fake
Brownian motion
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