Stochastic Differential Equations

For a more technical approach to the formulas introduced here, take a look at stochastic integration

Consider a differential equation with changes at exponentially distributed intervals. For example, regular interest rates in an account (jumps with monthly interest), the level of a body of water (jumps with rain), stock prices (jumps at opening), or neuron Firing-Rate Models (jumps with action potential)

Jump Stochastic Differential Equations (driven by Poisson processes) are continuous-state space and continuous time stochastic processes

• $(x(t){)}_{t\ge 0}$ with discontinuities (jumps) at random times, where the duration between jumps are independent exponential random variables $\lambda$ (a Poisson process with times $N_{\lambda }(t)$)
• Between jumps, $\frac{dx(t)}{dt}=f(x(t))$
• At a jump, $x(t)=x(t^{-})+\beta (x(t^{-}))$, where $x(t^{-})$ is the value “right before” the jump

We use a bit of an abuse of notation,
$dx(t)=f(x(t))\cdot dt+\beta (x(t))\cdot d{N}_{\lambda }(t)$

Solutions of SDEs with jumps satisfy the Markov property, $\mathbb{P}[x(t)\in A\mid x(s),0\le s\le \tau ]=\mathbb{P}[x(t)\in A\mid x(\tau )]$

Itô’s Lemma: If $\Phi$ is a differentiable function and $x$ is the solution of an SDE with jumps, then $\Phi (x(t))$ is a solution of the SDE $d\Phi (x(t))={\Phi }^{\prime }(x(t))f(x(t))\cdot dt+\left[\Phi \left(x(t)+\beta (x(t))\right)-\Phi (x(t))\right]\cdot d{N}_{\lambda }(t)$

Theorem: If $dx(t)=f(x(t))dt+\beta (x(t))d{N}_{\lambda }(t)$, then $\frac{d\mathbb{E}[x(t)]}{dt}=\mathbb{E}[f(x(t))]+\lambda \mathbb{E}[\beta (x(t))]$

We can use this and Itô’s Lemma to solve a lot of problems
For example, we can often use Itô with a clever $\Phi$ (like $\log (x)$) and then integrate for a direct solution for $x$

Examples:
$dx(t)=\alpha \cdot dt+\beta \cdot d{N}_{\lambda }(t)$
$x(t)=x(0)+\alpha t+\beta N_{\lambda }(t)$

$dx(t)=-\alpha x(t)\cdot dt+\beta x(t)\cdot d{N}_{\lambda }(t)$
$x(t)=x(0)e^{-\alpha t}(1+\beta {)}^{N_{\lambda }(t)}$

Definition: A compound Poisson process is a Poisson process with i.i.d. jumps. We can drive an SDE with this process instead of $N_{\lambda }(t)$. In this case, the amplitude of the jump is $\zeta \beta (x(t^{-}))$ where $\zeta$ is a random variable.

If $Y$ denotes our compound Poisson process with rate $\lambda$ and jump size law $\zeta$, then we write the associated SDE as $dx(t)=f(x(t))\cdot dt+\beta (x(t))\cdot d{Y}_t$

We can write an Itô formula where the jump of $\Phi (x(t))$ is $\Phi (x+\zeta \beta (x(t)))-\Phi (x(t))$

$\frac{d\mathbb{E}[x(t)]}{dt}=\mathbb{E}[f(x(t))]+\lambda \cdot \mathbb{E}[\zeta ]\cdot \mathbb{E}[\beta (x(t))]$
$\frac{d\mathbb{E}[\Phi (x(t))]}{dt}=\mathbb{E}[{\Phi }^{\prime }(x(t))f(x(t))]+\lambda \mathbb{E}[\Phi (x(t)+\zeta \beta (x(t)))-\Phi (x(t))]$