Space-time fractional equations

European Congress of Mathematics, Sevilla

David Gómez-Castro

Universidad Autónoma de Madrid

July 16, 2024

Co-authors

Juan Luis Vázquez

U. Autónoma de Madrid, Real Academia de Ciencias

Nicola Abatangelo

U. Bologna

Hardy Chan

U. Basel

\[ \newcommand{\dRL}{{\color{blue} \prescript{RL}{}\partial_\alpha}} \newcommand{\dC}{{\color{blue} \prescript{C}{}\partial_\alpha}} \newcommand{\iRL}{{\color{blue} I_\alpha}} \newcommand{\Rd}{{\mathbb R^d}} \]

Aim

\[ \text{``}{\color{blue}\partial_t^\alpha}\text{''} u + \text{``}{\color{red} (-\Delta)^s} \text{''} u = f \]

Fractional in time: “\({\color{blue} \partial_t}\)

Fractional in time

Let \({\color{blue} I}: C([0,T]) \to C([0,T])\) the Riemann integral \[ {\color{blue} I} f(t) = \int_0^t f(s) ds. \] Then \[\begin{align*} {\color{blue} I}^2 f (t) &= \int_0^t I_1f (s) ds = \int_0^t \int_0^s f(\sigma) d\sigma ds = \int_0^t f(\sigma) \int_s^t ds d \sigma \\ &= \int_0^t f(\sigma) (t-s) d \sigma, \\ {\color{blue} I}^n f (t) &= \int_0^t f(\sigma) K_n(t-s) d \sigma, \qquad K_n(t) = \frac{t^{n-1}}{(n-1)!}. \end{align*}\]

We define the Riemann-Liouville integral for \(\alpha > 0\) \[ \iRL f (t) = \int_0^t f(\sigma) K_\alpha(t-s) d \sigma, \qquad {\color{blue} K_\alpha}(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}. \]

Key properties

\[ \iRL f (t) = \int_0^t f(\sigma) K_\alpha(t-s) d \sigma, \qquad K_\alpha(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}. \]

  • Semigroup: \(\iRL {\color{blue} I_\beta} = {\color{blue} I_{\alpha + \beta}}\)

  • Monomial, for \(\beta > -1\) \[ \iRL t^\beta =\frac{\Gamma(\beta+1)}{\Gamma(\beta+1+\alpha)} t^{\beta+\alpha}. \]

  • Laplace transform symbol \[ {\mathfrak L}[\iRL f](s) = s^{-\alpha} {\mathfrak L}[f](s). \]

Fractional derivatives

A fractional derivative should have symbol \(s^\alpha\). Thus we get two choices:

  • Riemann-Liouville derivative \[ \dRL u = \partial_t {\color{blue} I_{1-\alpha}} u \]

  • Caputo derivative \[ \dC u = {\color{blue} I_{1-\alpha}} \partial_t u \]

Using the properties of \(I_\alpha\) and \(\partial_t\) it follows directly that \[\begin{align*} \dC t^\beta &= \dRL t^\beta = C(\beta, \alpha) t^{\beta - \alpha} \qquad \text{if } \beta > 0. \\ \dC 1 &= 0 \\ \dRL 1 &= \frac{t^{\alpha - 1}}{\Gamma(1-\alpha)} \ne 0 \\ \dRL t^{\alpha - 1} &= 0. \end{align*}\]

Integral formulation

\[\begin{align*} \dC u = f &\iff u = u(0) + \iRL f \\ \dRL v = f &\iff v = C_\alpha {\color{blue} I_{1-\alpha}} v(0) t^{\alpha-1} + \iRL f \end{align*}\]

Who cares? Stochastic volatility model for asset pricing.

Let \(S_t\) be the discounted spot price of an asset. We model \[ dS_t = S_t \sqrt{V_t} dW_t, % \qquad \text{i.e. } S_t = S_0 + \int_0^t S_s \sqrt{V_s} dW_s, \] where \(W_t\) be a Brownian motion in risk-neutral probability measure.

Consider \(B_t\) a second Brownian motion with \(\langle dW_t, dB_t \rangle = \rho\).

  • Black-Scholes (1973): \(V_t = \sigma^2\) constant.
  • Heston (1993): \[ V_t = V_0 + \int_0^t \kappa( \theta - V_s ) ds + \int_0^t \sigma \sqrt{V_t} d B_s, \]

  • Rough Heston (2014) model: \[ V_t = V_0 + \int_0^t \kappa( \theta - V_s ) {\color{blue} K_\alpha (t-s)} ds + \int_0^t \sigma \sqrt{V_t}{\color{blue} K_\alpha (t-s)} d B_s. \] Several problem related to RH involve fractional ODEs.

Does \(\alpha\) matter?

Arbitrage-free price of European call option: \(e^{-rT} \mathbb E[(S_T - K)_+]\)