Jump-Diffusion Processes

Abstract

In this chapter we introduce jump-diffusion processes and provide a theoretical framework that justifies the nonparametric (data-based) extraction of the parameters and functions controlling the arrival of a jump and the distribution of the jump size from the estimated conditional Kramers–Moyal moments. The method and the results are applicable to both stationary and nonstationary time series in the presence of discontinuous jump components; see Chap. 17.

Problems

12.1

Jump-drift process

Consider the following jump-drift process

$$ dx(t) = x(t)~ [ a(t) dt + b(t) dJ ] $$

with initial condition $x(0) = x_0 > 0$. Here a(t) and b(t) are drift and jump-amplitude coefficients. Also J(t) is Poisson jump with jump rate $\lambda (t)$.

Transform x(t) to y(t) via $y(t)=\ln (x(t))$ and show that,

(a) $\langle dy(t) \rangle = [ a(t) + \lambda (t) \ln (1+b(t))] dt$

and infinitesimal variance is

(b) $var [dy(t)] = \lambda (t) \ln ^2(1+b(t)) dt$ .

For constant coefficients $a(t)=a_0$, $b(t)=b_0$ and $\lambda (t)=\lambda _0$, show that the stochastic solution is,

(c) $x(t) = x(t_0) \exp [ a_0 (t-t_0)] ( 1+ b_0) ^{(J(t) - J(t_0))}$

and its mean and variance are given by,

(d) $\langle x(t) \rangle = x_0 \exp [ (a_0 + \lambda _0 b_0)(t-t_0)]$

(e) $var [x(t)] = \langle x(t) \rangle ^2 \left( \exp [{\lambda _0 b_0^2(t-t_0)}] - 1\right)$.

12.2

Jump-drift process

Solve the following jump-drift equation for x(t) and mean:

$$ dx(t) = a ~\sqrt{x(t)} dt + b ~( b+ 2 \sqrt{x(t)} )~dJ(t) $$

where $\langle J(t) \rangle = \lambda _0 t$ and $x(0) = x_0 > 0$, $\lambda _0 \ge 0$ and a, b are real constants.

12.3

Jump-drift process

Solve the following jump-drift equation for x(t), mean $\langle x(t) \rangle$ and var[x(t)];

$$ dx(t) = -a~ x^2(t) dt - \frac{cx^2(t)}{1+cx(t)} ~dJ(t) $$

where $\langle J(t) \rangle = \lambda _0 t$ and $x(0) = x_0 > 0$ and a, c and $\lambda _0$ are positive constants.

12.4

Jump-diffusion process, Kramers’ escape time

Consider the jump-diffusion dynamical equation (12.1) with $N(x,t) = -\partial U(x) / \partial x$ which is a force to the potential U(x). Assume that U(x) has a single non-degenerate minimum at $x=a$ and maximum at $x=b$ and that $U(x) \rightarrow \infty$ as $x \rightarrow -\infty$.

Show that mean exit time from interval $x\in (-\infty ,b]$ satisfies the following equation with boundary conditions,

$$ \mathcal L_{KM}^{\dagger } \tau (x) = -1, ~~~~~ ~~\tau (b)=0, ~~ \tau _x(-\infty ) =0, $$

where adjoint Kramers–Moyal operator is given by,

$$ \mathcal L_{KM}^{\dagger } = \sum _{n=1} ^{\infty } \frac{1}{n!} M^{(n)}(x) \frac{\partial ^n }{\partial x^n} $$

with Kramers–Moyal coefficients

$$ M^{(1)}(x) = -\frac{\partial U(x)}{ \partial x} + \lambda \langle \xi \rangle , ~M^{(2)}(x) = D(x) + \lambda \langle \xi ^2 \rangle , $$

and

$$ ~M^{(j)}(x) = \lambda \langle \xi ^j \rangle ,~~ j\ge 3. $$

Here $\lambda$ and $\langle \xi ^j \rangle$ are the jump rate and statistical moments of jump amplitude, respectively. Hint. See Problem 7.4 and [12].

12.5

Compound Poisson process

Suppose $J\equiv \{J(t): t \ge 0\}$ be a Poisson process with jump rate $\lambda$, so that $\langle J(t) \rangle = \lambda t$ for $t \ge 0$. Let $y_1,\ldots , y_n$ identically independent distributed random variables of J. A compound Poisson process is defined as:

$$ Q(t) = \sum _{i=1} ^{J(t)} y_i. $$

for $t\ge 0$. The process J(t) and the random sequence $y_i$ are independent.

(a) Show that the mean and the variance of compound Poisson process and its increments are:

$$\begin{aligned} \langle Q(t) \rangle= & {} \langle y_i \rangle ~\lambda ~t \\\\ var(Q(t))= & {} \langle y_i^2 \rangle ~\lambda ~t \\\\ \langle (Q(s)-Q(t)) \rangle= & {} \langle y_i \rangle ~\lambda ~(s-t) \\\\ var(Q(s)-Q(t))= & {} \langle y_i^2 \rangle ~\lambda ~(s-t). \end{aligned}$$

(b) Verify that, if $s< t < u$, the increments $Q(u)-Q(t)$ and $Q(t) - Q(s)$ are independent.

12.6

Itô-Taylor formula for jump-diffusion processes

Let x be a diffusion process with jumps, defined as the sum of a drift term, a Brownian (Wiener) stochastic integral and a compound Poisson process [5]:

$$ x(t) = x_0 + \int _0 ^t a(s) ds + \int _0 ^t b(s) dW(s) + \sum _{i=1} ^{J(t)} \varDelta x_i $$

where a(s) and $b^2(s)$ are nonanticipating time dependent drift and diffusion coefficients. The term $\sum _{i=1} ^{J(t)} \varDelta x_i$ is the compound Poisson jump process and J(t) is Poisson jump process with a rate $\lambda >0$.

Prove the Itô-Taylor formula for this jump-diffusion process for any differentiable function f(x, t) (first-order and second-order differentiable with respect to t and x, respectively)

$$\begin{aligned} f(x(t),t)-f(x_0,0)= & {} \int _0 ^t \left[ \frac{\partial f(x(s),s)}{\partial s} + a(s) \frac{\partial f(x(s),s)}{\partial x} \right] ds \\\\+ & {} \frac{1}{2}\int _0 ^t b^2(s) \frac{\partial ^2 f(x(s),s)}{\partial x^2} ds \\\\+ & {} \int _0 ^t b(s) \frac{\partial f(x(s),s)}{\partial x} dW(s) \\\\+ & {} \sum _{\{i \ge 1, T_i \le t \}} [ f(x_{T_i -} + \varDelta x_i) - f(x_{T_i -})]. \end{aligned}$$

In differential notation

$$\begin{aligned} df(x(t),t)= & {} \frac{\partial f(x(t),t)}{\partial t} dt + a(t) \frac{\partial f(x(t),t)}{\partial x} dt + \frac{b^2(t)}{2} \frac{\partial ^2 f(x(t),t)}{\partial x^2} dt \\\nonumber \\+ & {} b(t) \frac{\partial f(x(t),t)}{\partial x} dW(t) + [ f(x_{t-} + \varDelta x(t)) - f(x_{t-})]. \end{aligned}$$