A note on forward and backward Cauchy problems

It is a note on the proof of unique solvability on graphon MFG.

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When does American and European call have the same price?

We will prove the following fact arising from finance.

American call and European call with the same strike and maturity has the same price given that their underlying assets are the same non-dividend stock.

Of course, this fact shall not be generalized to put or some other scenarios.
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Two slides on Dirichlet problems

Today, I’ve presented my recent work on the generalized solution of the Dirichlet problem at IMA, here is the slide ( pdf ). There is also a video available for the presentation ( link ). Unfortunately, it does not show my board work during the talk, but it’s still a lot fun to look back at my own performance.

One may compare this work with my previous work on strong solutions of the Dirichlet problem, see the paper here ( link ), and slides here ( pdf ).

Measurability of the infimum of a class of functions

Ref:

[1] Supremum of a class of functions

In the earlier post [1], we discussed measurability of the infimum of a class of measurable functions. In particular, for the infimum of a class of measurable functions as a function, we can show that it may not be measurable. Therefore, we shall need additional conditions to have the infimum function measurable. In this post, we show that the infimum function is

measurable if the class size is countable;
lower semicontinuous (thus measurable) if each function in the class is lower semicontinuous.

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Two representations of the fractional Laplacian operator

There are a few equivalent definitions for the fractional Laplacian operator ${(-\Delta)^{\alpha}}$ for a constant ${\alpha \in (0, 2)}$ . One is given by

$\displaystyle - (- \Delta)^{\alpha} \phi(x) = C' \int_{\mathbb R^{d}} (\phi(x+y) - \phi(x) - D \phi(x) \cdot y I_{B_{1}}(y)) \nu(y) dy,$

and the other one is given by

$\displaystyle - (- \Delta)^{\alpha} \phi(x) = C \int_{\mathbb R^{d}} (\phi(x+y) + \phi(x-y) - 2\phi(x)) \nu(y) dy,$

where ${C}$ and ${C'}$ are normalizing constants and ${\nu(y) = \frac{1}{|y|^{d + \alpha}}}$ is a symmetric Levy measure on ${\mathbb R^{d}}$ . In this below, we will show they are actually given equivalent with ${2 C = C'}$ . Continue reading →

Skorhod metric on a finite time interval

The notion of Skorohod metric provides a very useful topology in a discontinuous curve spaces, which is often arising as a sample space in stochastic analysis. Understanding Skorohod metric is also an interesting process for an undergraduate student, as a concrete example of a metric space but not a normed space. This note below is written by an undergraduate student with plenty of original examples.

PDF: here

Reference: [Bil99] ([Patrick Billingsley 1999])

Integral of a Brownian motion

We are going to show that ${M_{t} = \int_{0}^{t} B_{s} ds}$ is a normal distribution ${\mathcal N(0, \frac 1 3 t^{3})}$ for each fixed ${t\ge 0}$ , where ${B}$ is a Brownian motion.

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Ornstein-Uhlenbeck process

An important Ito process in fiance is Ornstein-Uhlenbeck process (or Mean-reverting process). In fiance, It’s called as Hull-White model to describe a stochastic interest rate.

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