Formulating an Exponential-Initial-Value Parabolic PDE Problem

Summary

The document poses a time-dependent parabolic partial differential equation with a state-dependent diffusion term, mean-reverting drift, and a linear killing or discount term. It specifies an exponential initial condition and asks how to construct an initial candidate solution and proceed toward solving the equation. The coefficients are treated as constants, with positive restrictions on several parameters and unrestricted signs for the initial exponent and killing coefficient.

No derivation, candidate solution, numerical experiment, or answer is provided, so the material does not establish a solution method or assess whether a proposed solution satisfies the initial and boundary conditions. Its value is as a mathematical problem statement that may be relevant to pricing or transform methods for diffusion models. Further work would need to select an ansatz or probabilistic representation and verify existence, parameter restrictions, and any required boundary behavior.

Key ideas

• The equation combines state-dependent diffusion, mean-reverting drift, and a term proportional to the state and solution value.
• The initial condition is exponential in the state variable.
• The document asks for a way to construct a candidate solution but does not supply one.
• Any proposed method would need to verify the initial condition and relevant boundary behavior.

Tags

# A Cauchy problem 2: How can I find the following solution?


# A Cauchy problem 2: How can I find the following solution?












Suppose that we have the following time-dependent partial differential equation:

\begin{equation} \frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\sigma^2 x\frac{\partial^2 V(t, x)}{\partial x^2}+ \theta(m-x)\frac{\partial V(t,x)}{\partial x} - wxV(t, x), \quad t> 0, x>0\\ V(0, x ) = f(x) = e^{-ux}, \quad x>0 \end{equation}

where $\theta > 0$, $m>0$, $\sigma >0$, $u\in \mathbb{R}$, and $w\in \mathbb{R}$ are constants.

How can I come up with $\tilde{V}(t, x)$ as an initial solution for the above partial differential equation system? Could you please give me the general instruction on how I should proceed to find such a solution from the above partial differential equation?

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