# Space-time fractional equations

European Congress of Mathematics, Sevilla

David Gómez-Castro

July 16, 2024

## Co-authors

Juan Luis Vázquez

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: $$e^{-rT} \mathbb E[(S_T - K)_+]$$