In the seminal paper of Duffie, et al.[undefined], the class of affine processes is extensively explored and motivated with examples of applications to finance. From the perspective of the moment generating function, an affine process $X$ on $\bbR^d$ is characterized very succinctly—it is Markov and satsifies the following affine transform formula.

$$\rmE\Big( e^{u^\T X_T} | X_t = x \Big) = \exp\Big(\psi_0(t, T, u) + \psi(t, T, u)^\T x \Big), \quad u \in i\bbR^d $$

Here, we are adopting the slightly more general time-inhomogeneous case discussed in Filipović[undefined]. There are some great papers (for instance, [undefined], [undefined]) that subsequently study these objects—be it generalizing results, weakening hypotheses, focusing on specific cases, or studying related statistical procedures. Each of these acknowledge a very important result about affine processes: the transition semigroup from $X$ and affine transform formula above induces dynamics on $\psi_0, \psi_1$ which ultimately boil down to an integro-differential equation.

Finite activity

Let us assume that $m_0(s, \cdot), \ldots, m_d(s, \cdot)$ are finite measures for all $s \geq 0$. With such an assumption, we may define the total measures $k_i(s) = m_i(s, \bbR^d) \in (0,\infty)$ and factor out probability measures $\nu_i(s, \cdot)$ like so.

Additionally, this assumption allows us to reparameterize our vector-valued functions $b_i$

so that each $R_i$ may be rewritten more succinctly.

Stochastic differential equation

With such functions, we may consider a stochastic process $X$ solving the following differential equation for a filtration $\calF$

where $\sigma$ is some function satisfying $\sigma\sigma^\T = \alpha$, $W$ is a standard $\calF$-Brownian motion on $\bbR^d$, and $\mu$ is an integer-valued random measure with $\calF$-compensator $\mu^P$ as so.

A solution $X$ to this stochastic differential equation will be an affine process with the same parameters in the affine transform formula.

Show proof.

Proof.  Fix parameters $a_0, b_0, k_0, \nu_0, \ldots, a_d, b_d, k_d, \nu_d$ such that there exists some $T > 0$ such that the differential equation associated to the parameters $$\left\{\begin{array}{ll} \displaystyle \frac{\partial}{\partial t} \psi_0(t, T, u) = -R_0\big(t, \psi(t, T, u)\big) & \psi_0(T, T, u) = 0 \\[1em] \displaystyle \frac{\partial}{\partial t} \psi(t, T, u) = -R(t, \psi(t, T, u)\big) & \psi(T, T, u) = u \end{array}\right.$$ has a solution $\psi_0(\cdot, T, u), \psi(\cdot, T, u)$ for all $u \in i\bbR^d$. Let $X$ be a solution to the stochastic differential equation induced by the same parameters. Now define $f: \bbR_+ \times \bbR^d \rightarrow \bbR$ as the deterministic function in our affine transform formula. $$ f(t, x) = \exp\Big( \psi_0(t, T, u) + \psi(t, T, u)^\T x \Big) $$ We now evaluate the $M_t = f(t, X_t)$ according to Itô's formula. $$\begin{aligned} M_t &= M_0 + \int_0^t f_t(s, X_s) {\rm d}s + \int_0^t \nabla_xf(s, X_{s-}) {\rm d}X^c_s \\ &\qquad + \frac12 \int_0^t ({\rm d}X^c_s)^\T H_xf(s, X_{s-}) {\rm d}X^c_s + \int_{[0,t] \times \bbR^d} \big( f(s, X_{s-} + y) - f(s, X_{s-}) \big) \mu({\rm d}s \times {\rm d}y) \\[2em] &= M_0 + \int_0^t f(s, X_s) \Big( \frac{\partial}{\partial t}\psi_0(s, T, u) + \frac{\partial}{\partial t}\psi(s, T, u)^\T X_{s-} \Big) {\rm d}s + \int_0^t f(s, X_{s-}) \psi(s, T, u)^\T \Big( \beta(X_s) {\rm d}s + \sigma(X_s) {\rm d}W_s \Big) \\ &\qquad + \frac12 \int_0^t f(s, X_{s-}) \sigma(X_s)^\T \psi(s, T, u) \psi(s, T, u)^\T \sigma(X_s) {\rm d}s + \int_{[0,t]\times\bbR^d} f(s, X_{s-})\big( e^{\psi(s, T, u)^\T y} - 1 \big) \mu({\rm d}s \times {\rm d}y) \\[2em] &= M_0 + \int_0^t f(s, X_s) \psi(s, T, u)^\T \sigma(X_s) {\rm d}W_s + \int_{[0,t]\times\bbR^d} f(s, X_{s-}) \big(e^{\psi(s, T, u)^\T y} - 1 \big) (\mu - \mu^P)({\rm d}s \times {\rm d}y) \\ &\qquad + \int_0^t f(s, X_{s-}) \Bigg( \frac{\partial}{\partial t}\psi_0(s, T, u) + \frac{\partial}{\partial t}\psi(s, T, u)^\T X_{s-} + \frac12\psi(s, T, u)^\T \alpha(X_s) \psi(s, T, u) + \psi(s, T, u)^\T \beta(X_s) \Bigg) {\rm d}s \\ &\qquad + \int_{[0,t]\times\bbR^d} f(s, X_{s-}) \big( e^{\psi(s, T, u)^\T y} - 1 \big) \mu^P({\rm d}s \times {\rm d}y) \\[2em] &= M_0 + \int_0^t f(s, X_s) \psi(s, T, u)^\T \sigma(X_s) {\rm d}W_s + \int_{[0,t]\times\bbR^d} f(s, X_{s-}) \big(e^{\psi(s, T, u)^\T y} - 1 \big) (\mu - \mu^P)({\rm d}s \times {\rm d}y) \\ &\qquad + \int_0^t f(s, X_{s-}) \Bigg( \frac{\partial}{\partial t}\psi_0(s, T, u) + \frac{\partial}{\partial t}\psi(s, T, u)^\T X_s + \frac12\psi(s, T, u)^\T \alpha(X_s) \psi(s, T, u) + \psi(s, T, u)^\T \beta(X_s) + \int_{\bbR^d} \big(e^{\psi(t, T, u)^\T y} - 1 \big) \kappa(s, X_{s-}, {\rm d}y) \Big) {\rm d}s \\[2em] &= M_0 + \int_0^t f(s, X_s) \psi(s, T, u)^\T \sigma(X_s) {\rm d}W_s + \int_{[0,t]\times\bbR^d} f(s, X_{s-}) \big(e^{\psi(s, T, u)^\T y} - 1 \big) (\mu - \mu^P)({\rm d}s \times {\rm d}y) \\ &\qquad + \int_0^t f(s, X_{s-}) \Bigg( \Big( \frac{\partial}{\partial t}\psi_0(s, T, u) + R_0\big(s, \psi(s, T, u)\big) \Big) + \Big(\frac{\partial}{\partial t}\psi(s, T, u) + R\big(s, \psi(s, T, u)\big) \Big)^\T X_s \Bigg) {\rm d}s \\[2em] &= M_0 + \int_0^t f(s, X_s) \psi(s, T, u)^\T \sigma(X_s) {\rm d}W_s + \int_{[0,t]\times\bbR^d} f(s, X_{s-}) \big(e^{\psi(s, T, u)^\T y} - 1 \big) (\mu - \mu^P)({\rm d}s \times {\rm d}y) \end{aligned}$$ The local martingale structure of $W$ and $\mu - \mu^P$ is preserved through stochastic integration, so we may conclude that $M$ is a local martingale. Because $u \in i\bbR^d$, it turns out that the real part of $\psi_0(t, u) + \psi(t, u)^\T x$ is nonpositive for all $x \in \bbR^d$. This tells us that $$ |M_t| = \big| f(t, X_t) \big| \leq 1 $$ for all $t \geq 0$, and so $M$ is a martingale. We conclude with the following for each $t \geq 0$. $$ \rmE\Big( e^{u^\T X_T} | X_t = x \Big) = \rmE\Big(M_T | X_t = x \Big) = M_t = \exp\Big( \psi_0(t, T, u) + \psi(t, T, u)^\T X_t \Big) $$ This shows that the solution $X$ to the stochastic differential equation satisfies the affine transform formula up to terminal time $T$. Hide proof.

References

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