Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free rate, $\beta>0$ is the payout rate, and $\sigma$ is the earnings volatility. Let $X_0 = x$ be the initial value. Now assume that the firm has perpetual debt with coupon $C$ (which is taken to be fixed) and assume there are no taxes. Assume that shareholders can choose a stopping time at which the firm defaults, i.e., $\tau = \inf\{t: X_t \le x_B \}$ (for now assume that $x_B$ is fixed). Let $E_t = E(X_t)$ denote the value of equity at time $t$, then the value of equity at time $t =0$ is given by $$E_0 = E(X_0) = E(x)= \mathbb{E}\left[\int_0^{\tau} e^{-rt}(X_t - C)dt + e^{-r\tau} E(X_{\tau}) \right] \ \ \cdots (1)$$.

Applying the Feyman-Kac formula, we get $$rE(x) = x - C + (r-\beta) xE'(x) + \frac{\sigma^2 x^2}{2}E''(x) \ \ \cdots (2)$$

My question is: how do we get from Equation $(1)$ to $(2)$? What I've done so far is this.

Consider the value of the equity at time $t$, which is given by $$E_t = E(X_t) = \mathbb{E}\left[\int_t^{\tau} e^{-r(s-t)}(X_s - C)ds + e^{-r(\tau-t)} E(X_{\tau}) \right] \ \ \cdots (3)$$ Now use Ito's Lemma on Equation $(3)$ and we get \begin{align} dE_t & = E'(X_t) dX_t + \frac{1}{2}E''(X_t) (dX_t)^2 \\ & = (E'(X_t) X_t (r-\beta) + \frac{1}{2}E''(X_t) \sigma^2 X_t^2) dt+ E'(X_t)\sigma dZ_t \end{align} This is where I am stuck. I can see relevant terms in the above equation that appear in (2) but can't seem to go any further.


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