# Research

My primary area of research is in the analysis of partial differential equations (PDE). Below are some topics which I have studied as part of my dissertation.

Fluids

I have studied a generalization of the Camassa-Holm equation, called the g-$kb$CH equation which can be written as:

$\partial_t u = -u^k \partial_x u-\frac{b}{k+1}(1-\partial_x^2)^{-1} \partial_x \left( u^{k+1} \right)-\frac{3k-b}{2}(1-\partial_x^2)^{-1} \partial_x \left( u^{k-1} \partial_x u ^2 \right)-\frac{(k-1)(b-k)}{2} (1-\partial_x^2)^{-1} \left( u^{k-2} (\partial_x u )^2 \right).$

This equation also generalizes the Degasperis-Procesi equation and the Novikov equation. These three equations are integrable equations which admit peakon, or peaked traveling wave equations. These are solutions that have a peak at some point, and traveling along the $x$-axis as time progresses. Below is a picture of a peakon solution on the line corresponding to $k = 2$ and $b = 1$.

This equation is a nonlocal perturbation of Burgers equation, and can be viewed as a mathematical approximation to Euler’s equations in some sense. Thus, one may expect it shares some mathematical properties to these equations. Indeed it does share many of the same well-posedness results of these equations.

Our work has concentrated on properties of the data-to-solution map. Like Burgers equation, the data-to-solution map is not Lipshitz continuous in Sobolev spaces, denoted $H^s$. However, if we consider a weaker topology, (by which we mean, if we have a solution which is continuous in time and takes values in $H^s$, then we consider the data-to-solution map in the space of continuous functions taking values in the weaker space $H^r$, for $r) then we obtain the data-to-solution map is Holder continuous. Interestingly, the Holder index depends upon the distance $r$ is from $s$; as $s-r$ increases, the exponent increases, thus giving us a stronger result.

Mathematical Finance

The investors problem is the problem of how one decides to invest and consume their wealth after retirement. This work has been done with several collaborators including Thomas Cosimano and Alex Himonas, and we model this problem using a system of stochastic PDE’s. We make several large assumptions on the nature of stock returns based on best fit approximations from historical data. We also make assumptions in regards to how our investor values consuming their wealth each period in time as well as at the moment they die.

Using calculous and a process called dynamical programming, all of these assumptions lead us to a complicated PDE which we call the Nonlinear Financial Diffusion Equation (NFDE). This PDE is a perturbation of the heat equation, with a unique nonlinearity:

$\partial_t u = A \partial_x ^2u + L(u)+ B \frac{ (\partial_x u)^2}{u}$,

where $L(u)$ is a non-constant coefficient linear function of $u$ and $\partial_x u$. Here, the state variable $x$ is the expected excess stock return, which is calculated based on the current market conditions.

The solution to this equation yields the optimal percentage of wealth our investor should consume and invest each moment in time. It also allows the individual to calculate how much any asset is worth to them, assuming the assets future risk follows the same probabilistic relationship with the market and it’s historical risk (covariance/variance). Thus, they can calculate if the asset is over or underpriced relative to their preferences towards risk.

Since we are not able to find any economically significant solutions to this equation, we solve the PDE numerically so that we can compute the desired quantities. Previous researchers who have studied this problem have made some restrictive assumptions (both on the preferences of the investor and on the random motion of stocks) which lead to a linear PDE, so that they could find closed form solutions. When we compare our solution to their’s we find that these assumptions are really important in terms of how much our investor should consume and invest when we consider a long investment horizon.

Nonlinear Diffusion Equations

Part of my Ph.D. dissertation is on a generalization of the viscous Burgers equation:

$\partial_t u = \partial_x^2 u +u^k \partial_x u .$

This equation has been studied by several authors before me, and the case $k=1$ corresponds to the classical equation which can be transformed via the Cole-Hopf transformation into the heat equation. In our research, we show that this equation is well-posed in a class of analytic spaces (Gevrey space to be precise).

Gevrey spaces are space of function which are better than $C^\infty$, but may not yet be analytic. These functions are denoted $G^s$, where $s$ is the index. $G^1$ corresponds to analytic functions, and as the index $s$ grows, the smoothness of the functions decreases. More precisely, for a function $f$ to be $G^s$, it must be $C^\infty$ and, for every compact set $K$ in the domain there exists a constant $R$ such that

$|D^n f(x) | \le R^{n+1} (n! )^s.$

Following previous authors, we consider $L^2$ weighted spaces where the weighting is chosen particularly so that functions in these spaces are in the appropriate Gevrey class. We calculate an integral equation the solution formally satisfies, and following the techniques of many other analysts, show that this equation is a contraction on a subspace for which we can show appropriate multilinear estimates. The trick here, is to take advantage of the smoothing the diffusion portion of the equation (or equivalently the heat kernel) provides.

We also show that if the solution to the initial value problem corresponding to this equation is in the space of continuous functions from a time interval into the class $G^s$, then it is in fact in the space $G^{2s}$ in the time variable. Moreover, we show that this regularity result is sharp. In particular, we construct solutions which are not better than $G^{2s}$ in the time variable at $t = 0$.