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Functional CLTs for subordinated L\'evy models in physics, finance, and econometrics
This paper develops functional CLTs for four different problems coming from statistical mechanics, econometrics, mathematical finance, and insurance mathematics. Each problem was briefly discussed in Section 2 of andreasfabrice_theorypaper, but the detailed analysis is the topic of the present paper. Hitherto, they have been treated in completely separate strands of literature and in less generality.
Our goal here is to give a simple, concise, and unified account, noting that, for each of the applications, the analysis ultimately boils down to nuances of the same fundamental problem: namely that of weak convergence for certain stochastic integrals driven by moving averages or, more generally, continuous-time random walks (CTRWs). The starting point for this analysis is the fact that these integrators satisfy functional CLTs with limits described by stable L\'evy processes, typically time-changed by a $\beta$-stable subordinator (and possibly with a tempered jump size distribution for the parent L\'evy process). Throughout, our analysis relies only on the framework developed in andreasfabrice_theorypaper and various generalizations thereof presented in andreasfabrice.
In the next subsection, we briefly cover the main definitions for moving averages and CTRWs along with the functional limit theorems that are available for such processes. The rest of the paper consists of four sections dedicated to the four applications mentioned above. Throughout, we shall work on Skorokhod space, both under the J1 and the M1 topology. For a brief introduction to these topologies, the reader is invited to consult andreasfabrice_theorypaper.
Consider i.i.d. waiting times $J_1, J_2,...$, and let
where $L_k:= J_1+\hdots+J_k$ is the time of the $k$-th jump. We can then consider
for a sequence of i.i.d. random variables $\theta_k$ with $-\infty < k < \infty$. We let the $\theta_k$ be in the normal domain of attraction of a non-degenerate $\alpha$-stable random variable $\tilde{\theta}$ with $0< \alpha< 2$, and let the $J_k$ be in the normal domain of attraction of a $\beta$-stable random variable with $\beta \in (0,1)$. Then (ref) is known as a correlated CTRW (or uncorrelated if $c_j=0$ for all $j\neq 0$).
In this paper, we restrict to non-negative constants $c_j\ge 0$ for all $j\geq 0$. We shall also assume that if $1<\alpha\le 2$, then either (i) $c_j=0$ for all but finitely many $j\ge 0$, or (ii) the sequence $(c_j)_{j\ge 0}$ is monotone and $\sum_{j=0}^\infty c_j^\rho <\infty$ for some $\rho<1$. If, in (ref), we take $L_k=k$, then the process $X^n$ is known as a moving average. For such moving averages we have $ X^n \Rightarrow (\sum_{j=0}^\infty c_j) Z$ on $({\bf D}_{\mathbb{R}^d}[0,\infty), \, \mathop{}\!\mathrm{d}_{\operatorname{M1}})$ (see whitt). This was first shown by Avram & Taqqu avram.
A CTRW (ref) is called uncoupled if the sequences $(J_i)_{i\geq 1}$ and $(\zeta_k)_{k\geq 1}$ are independent. In the uncoupled case, meerschaert2 extended the results of avram, showing that, for $0<\alpha < 2$,
where $Z$ is an $\alpha$-stable Lévy process (with $Z_1\sim \tilde \theta$) and $D^{-1}$ is the generalised inverse,
of a $\beta$-stable subordinator $D$. Moreover, meerschaert2 confirmed that J1 convergence fails as soon as $c_0>0$ and $c_1>0$. If $c_0>0$ is the only non-zero constant, then M1 convergence was first shown in Becker-Kern, but henry_straka later showed that the convergence in fact takes place in J1.
Let now $X^n$ be uncorrelated (i.e., take $c_0=1$ and $c_j=0$ for $j\geq 1$ in (ref)). If the jump sizes and jump times form an i.i.d. sequence $(\zeta_k,J_k)_{k\geq 1}$, but, for each $k\geq 1$, $\zeta_k$ and $J_k$ are allowed to be dependent, then the CTRW is said to be coupled. For $Z$ and $D$ as above, henry_straka,jurlewicz show that (uncorrelated) coupled CTRWs satisfy
where $x^-(t):=x(t-)$ and $x^+(t):=x(t+)$. Notice that, if the $X^n$ are uncoupled, then $Z$ and $D$ will be independent L\'evy processes (hence no common discontinuities a.s.), so the limit simplifies to $Z_{D^{-1}}$ henry_straka, as in (ref) with $c_0=1$ and $c_j=0$ for all $j\neq 0$.
Throughout the paper, we will work on a family of filtered probability spaces $(\Omega, \mathcal{F}^n, \mathbb{F}^n, \mathbb{P})$, for $n\geq 1$, where each filtration $\mathbb{F}^n$ is given by
Here, $\zeta^n_{N(ns)}$ is understood to be the random variable $\sum_{k=1}^\infty \zeta^n_k \operatorname{\mathbf{1}}_{\{N(ns)=k\}}$. We stress that all of the results in subsequent sections remain valid if one expands the filtrations $\mathbb{F}^n$ in any way such that, for $X^n$ as in (ref) with $c_j=0$ for $j\ge 1$, the independent increment property $X^n_t-X^n_s \perp \!\!\! \perp \mathcal{F}^n_s$ is preserved for all $0\le s<t$ and $n\geq 1$.
Similarly to the work of Scalas & Viles scalas, which was motivated by the study of damped harmonic oscillators subject to stable L\'evy noise, we are interested in the functional weak convergence
for given deterministic càdlàg functions $f_n,f:\mathbb{R}\rightarrow \mathbb{R}$ and a sequence of CTRW integrators $X^n$ converging to a subordinated stable Lévy process $X$. Taking $f_n=f$ to be a given continuous function, scalas built on Becker-Kern to establish (ref) in the M1 topology for uncorrelated, uncoupled CTRWs under the following conditions: Firstly, the i.i.d. innovations should be symmetric around zero and in the normal domain of attraction of an $\alpha$-stable law, for $\alpha \in (1,2)$. Secondly, the i.i.d. waiting times should be in the normal domain of attraction of a stable subordinator, for $\beta \in (0,1)$.
Based on andreasfabrice_theorypaper, we obtain the following generalisation of the aforementioned result.
The relevance of such extensions to larger classes of uncorrelated CTRWs (e.g. of the coupled type and without symmetry restrictions) was discussed in the conclusion of scalas and e.g. in hahn. Moreover, scalas and hahn also address the relevance of allowing for convergent sequences of adapted integrands, as opposed to deterministic functions. This is easily handled, provided the integrands satisfy the assumptions of andreasfabrice_theorypaper or andreasfabrice_theorypaper (see also the alternative criteria in andreasfabrice_theorypaper), and that they are adapted to the filtrations (ref).
While scalas relies on the M1 topology, J1 convergence in fact holds, as per Proposition (ref). If we instead work with correlated CTRWs, on the other hand, then one can only hope for M1 convergence. As shown in andreasfabrice_theorypaper, correlated CTRWs no longer admit good decompositions and the associated stochastic integrals can fail to converge. Nevertheless, one can still have convergence under suitable conditions. For example, we obtain the following result as a special case of andreasfabrice for correlated CTRWs (ref) with the sequence $\{c_k\}$ satisfying the technical condition
Notice that (ref) is e.g. satisfied if $\sum_{k=1}^\infty k \, c_k<\infty$ in the case $1<\alpha\le 2$ or $\sum_{k=1}^\infty k \, c_k^\rho <\infty$ for some $0<\rho<1$ in the case $\alpha=1$.
It can be shown that this also extends to random integrands adapted to (ref), where however, broadly speaking, we must ensure a certain regularity of the integrands to be independent at any time of the future trajectory of the integrators, as detailed in andreasfabrice.
The next result gives another example of what can be said for correlated CTRWs when we allow for more generality concerning the law of the innovations in return for additional regularity of the integrands. The result follows directly from andreasfabrice.
In fact, it suffices that the $f_n$ are Lipschitz on compacts $[0,T]$ with Lipschitz constant $C_T n^\gamma$. Also---provided AVCI is satisfied for the integrands and integrators (see andreasfabrice_theorypaper)---there is no need to restrict to deterministic functions as long as they almost surely exhibit this Lipschitz property, are adapted to (ref) and converge weakly to some càdlàg limit.
Exactly as Paulaskas & Rachev paulaskas_rachev, we shall focus on the general formulation of multivariate cointegrated time series from the classical work of Park & Phillips park_phillips. Given a $k$-dimensional time series $(Y_{t_i})_{i= 1}^n$ and an $m$-dimensional time series $(X_{t_i})_{i= 1}^n$, for some time interval $[0,t]$, we are interested in understanding if we have the structure \[ X^n_{t_i} = X^n_{t_{i-1}} + v_{t_i} \quad \text{and} \quad Y^n_{t_i} = \mathbb{M}^n X^n_{t_i} + u_{t_i} \quad \text{for}\quad {t_1,\ldots, t_n}\in[0,t], \] for some $k\times m$ matrix $\mathbb{M}^n$ and suitable innovations $(u_{t_i})_{i=1}^n$ and $(v_{t_i})_{i=1}^n$. As in park_phillips, one can then show that the least square estimator $\hat{\mathbb{M}}^n$ of $\mathbb{M}^n$ is of the form
where $\mathbb{X} $, $\mathbb{Y}$, and $ \mathbb{U}$ are appropriate matrices collecting the two time series and the first innovation sequence (dependence on $t$ and $n$ has been suppressed). To formulate hypothesis tests, the idea is to look for an invariance principle for $\hat{\mathbb{M}}^n - \mathbb{M}^n $ as the sample size $n$ tends to infinity, meaning that we pass to observations in continuous time in $[0,t]$. Naturally, this will rely on first assuming some invariance principle for the innovations $(u_{t_i})_{i=1}^n$ and $(v_{t_i})_{i=1}^n$, and it will involve suitable scaling factors depending on the assumption on the laws of these innovations.
While park_phillips considers Gaussian invariance principles, the objective of paulaskas_rachev (and the related works caner2,caner1) is to allow for heavy-tailed innovations with infinite variance. Based on stable limit theorems, paulaskas_rachev thus consider the re-scaled differences
where \[ \mathbb{T} = \textrm{diag}(\tfrac{1}{n^{1/\alpha_1}},\ldots, \tfrac{1}{n^{1/\alpha_p}})\quad \text{and} \quad \tilde{\mathbb{T}} = \textrm{diag}(\tfrac{1}{n^{1/\tilde{\alpha}_1}},\ldots, \tfrac{1}{n^{1/\tilde{\alpha}_q}}), \] for suitable parameters $1 < \alpha_i,\tilde{\alpha}_j \leq 2$, $i=1,\ldots,p$, $j=1,\ldots,q$. Rewriting $\mathbb{T} \mathbb{U}^\intercal \mathbb{X} \tilde{\mathbb{T}}$ in (ref), it turns out that the main challenge towards an invariance principle for (ref) is the joint weak convergence of the stochastic integrals $\int_0^\bullet Z^{n,i}_{s-} \mathop{}\!\mathrm{d} \tilde{Z}^{n,j}_{s}$, for $i=1,\ldots p$ and $j=1,\ldots, q$, where
In paulaskas_rachev, the paired innovations $(u_{t_\ell},v_{t_\ell})_{\ell=1}^n$ are assumed to form an i.i.d. sequence of $(p+q)$-dimensional random variables in the normal domain of attraction of a stable law on $\mathbb{R}^{p+q}$ with parameters $(\alpha,\tilde{\alpha})=(\alpha_1,\ldots,\alpha_p,\tilde{\alpha}_1,\ldots, \tilde{\alpha}_q)$. In this setting, the processes (ref) converge weakly on $(\mathbf{D}_{\mathbb{R}^{p+q}}[0,t],\mathrm{d}_{\text{J1}})$ to a pair of $(\alpha,\tilde{\alpha})$-stable L{\'e}vy processes $(Z,\tilde{Z})$ and the desired stochastic integral convergence becomes a direct consequence of the general framework in andreasfabrice_theorypaper.
We note that the arguments in paulaskas_rachev only apply to $\alpha_i,\tilde{\alpha}_j \in (1,2]$. The main technical work is concerned with verifying the UCV condition of kurtzprotter in order to apply kurtzprotter (see Remark (ref) below). Whilst not explicit, a close inspection of the proof of paulaskas_rachev reveals that these arguments boil down to a bound on $\mathbb{E}^n[\operatorname{sup}_{s\in[0,t]}|\Delta\tilde{Z}^{n}_s|]$ uniformly in $n\geq 1$ (something that is specific to the range $\alpha_i,\tilde{\alpha}_j \in (1,2]$\,). As the $\tilde{Z}^{n}$ can be seen to be martingales in this range, they immediately satisfy the martingale condition for good decompositions in andreasfabrice_theorypaper, and hence we have convergence of the stochastic integrals without any additional work.
Results similar to (ref) are central to a wide-ranging literature on cointegration tests, see e.g. the recent book Wang_book. Directly related to the above, the more general aymptotic analysis of heavy-tailed near-integrated time series with autoregression in chan_zhang also relies on the arguments of paulaskas_rachev through the proof of the key chan_zhang. It should be noted that marginal convergence is often sufficient, but functional convergence can be of independent interest and we refer to davis, davis2 for cases where functional convergence plays an essential role.
Finally, we wish to highlight how the framework of andreasfabrice_theorypaper can also shed light on cases where the innovations are given by linear processes (violating the assumptions of paulaskas_rachev) and where a nonlinear function is involved. Consider, for example, stochastic integrals $\int_0^\bullet f(Z^{n}_{s-}) \mathop{}\!\mathrm{d} \tilde{Z}^{n}_{s}$, for a continuous function $f$, where $Z^{n}$ and $\tilde{Z}^{n}$ are as in (ref). If the innovations $u$ and $v$ have the form of $\zeta$ from (ref), then $Z^{n}$ and $\tilde{Z}^{n}$ become strictly M1 convergent moving averages. It is straightforward to show that $(f(Z^n),\tilde{Z}^n)$ satisfy the weaker convergence assumptions outlined in andreasfabrice_theorypaper. Depending on the application, it can be natural for the innovations $(u_{t_\ell})_{\ell \geq 1}$ and $(v_{t_\ell})_{\ell \geq 1}$ to model two independent sources of randomness: in such a case, we obtain the following functional convergence result. For simplicity of notation, we take $i=j=1$.
Famously, already in the 60's, Mandelbrot mandlebrot and Fama fama advocated for $\alpha$-stable distributions in finance. In particular, mandlebrot illustrated how $\alpha =1.7$, rather than $\alpha=2$, gave a much better correspondence with observed time series for cotton prices. Inspired by such observations and driven by a phenomenological approach to market microstructure with random inter-trade durations, there has been a growing interest in CTRW models of asset prices, first formalized by Scalas, Gorenflo and Mainardi in scalas_finance (building on the classical works clark,parkinson,press).
Following on from scalas_finance, we consider a sequence of asset prices $S^n_t$ that are determined, in a `tick-by-tick' fashion, by random price moves occurring only at unevenly-spaced random times $(t_i)_{i\geq 0}$. For each $n\geq 1$, we work with waiting times $t_i-t_{i-1}=n^{-1}J_i$ and log-price innovations $\ln(S^n_{t_i})-\ln(S^n_{t_{i-1}})=n^{-\beta/ \alpha}\zeta_i+n^{-1}J_i\,r$, where $r$ is the risk free rate which is reflected as an upward trend in the price innovations (investors should expect to be compensated for the risk free return that they could have earned by holding the risk free asset). By summing up these tick-by-tick price moves, we obtain the exponential CTRWs
with $N(t)$ as in (ref). Here, the components $\zeta_i$ of the log-price innovations are assumed to be i.i.d. in the normal domain of attraction of an $\alpha$-stable law, for $\alpha \in [1,2]$, and the waiting times $J_i$ are assumed to be i.i.d. in the normal domain of attraction of a $\beta$-stable law with $\beta \in (0,1)$, as in the standard literature on CTRW models for tick-by-tick prices. Note that, for all $t\ge 0$,
and $J_{N(nt)}/n \Rightarrow 0$, so we readily obtain that the increasing sums in (ref) are weakly M1 convergent to the continuous limit $t\mapsto rt$ (hence weakly uniformly convergent). Writing $\tilde{R}^n_t$ for the sum of the $n^{-\beta/ \alpha}\zeta_i$ terms in (ref), we know from Section (ref) that the CTRWs $\tilde{R}^n$ converge weakly in J1 to $\tilde{R}:=Z_{D^{-1}}$ or $\tilde{R}:=((Z^-)_{D^{-1}})^+$, for an $\alpha$-stable L\'evy process $Z$ and $\beta$-stable subordinator $D$, depending on whether they are coupled or uncoupled. As one limit is continuous, we conclude that $R^n \Rightarrow \tilde{R}_\bullet+r\bullet$ in J1 and then also $S^n \Rightarrow S:=e^{\tilde{R}_\bullet+r\bullet}$ in J1.
For the limiting price $S$, the $\beta$-stable subordinator $D$ with $\beta\in(0,1)$ allows one to capture burst of activity in between periods of relative dormancy, and it yields price paths that are locally constant at almost all times in line with the intended tick-by-tick interpretation of price moves. For option pricing, one needs $\tilde{R}_t$ to have an exponential moment so that the discounted asset price $e^{-rt}X_t=e^{\tilde{R}_t}$ can be a martingale. This can be realised by `tempering' the probability of large innovations, leading to the notion of tempered stable L\'evy processes. To this end, Chakrabarty_Meerschaert has shown that one can take an i.i.d. sequence of innovations in the domain of attraction of a stable law, as in (ref), and truncate these jumps in a certain random way (mimicking the tempering function for the desired limit) such that one obtains a continuous-time random walk for which a suitable (deterministic) scaling yields convergence to the desired tempered stable L\'evy process. Combining the triangular scheme of Chakrabarty_Meerschaert with the results in henry_straka,jurlewicz, this also works with random waiting times resulting in a subordinated limit (see in particular jurlewicz).
Jacquier & Torricelli jack have developed a remarkably tractable option pricing theory for two such limiting models, motivated by the CTRW formalism (with appropriate tempering) and capitalising on analytical results of Becker-Kern, jurlewicz,leonenko,meerschaert_straka,meerschaert. The first is the subdiffusive L\'evy model, corresponding to $\tilde{R}=Z_{D^{-1}}$, where $Z$ and $D$ are independent (many earlier works are special cases of this). The second is the dependent returns and trade duration model, corresponding to $\tilde{R}=((Z^-)_{D^{-1}})^+$, where $Z$ and $D$ are dependent but enjoy a particular structure $Z=\tilde{Z}_D$ for a L\'evy process $\tilde{Z}$ independent of $D$. The point of the latter is that one then has $\tilde{R}=\tilde{Z}_G$ up to indistinguishability with $G:=g^+$, for the last-passage time $g(t):=\text{sup} \{ s < t : s\in \{D_r:r\geq 0\} \}$, where the new (right-continuous, increasing) time-change $G$ is now independent of the parent $\tilde{Z}$ and one can exploit powerful analytical results for the law of $g$. In jack, these models are shown to produce volatility surfaces that capture important phenomena observed in the market much better than classical L\'evy or stochastic volatility models. As in jurlewicz, one can obtain $\tilde{R}=\tilde{Z}_G$ as the limit directly, by using a suitable triangular array scheme.
Similarly to the analysis of Duffie & Protter duffie_protter for the Black--Scholes model, it is important to understand the stability properties of financial gain processes $\int_0^\bullet H^n_{t-}\mathop{}\!\mathrm{d} S^n_t$, for trading strategies $H^n_{-}$, as the asset prices $S^n$ converge to their scaling limit. With $S^n$ as in (ref), so that $S^n\Rightarrow S =e^{\tilde{R}_\bullet+r\bullet}$, we obtain the following stability result for the financial gains.
We note that, in Proposition (ref), one could also consider weakly M1 convergent trading strategies or, more generally, trading strategies that converge in the sense of Proposition andreasfabrice_theorypaper. However, then one must check that AVCI or its alternatives from andreasfabrice_theorypaper are satisfied, as this is no longer automatic. Since it is natural to allow the limiting (left continuous) trading strategies to jump at the same time as the limiting (right continuous) price processes, the simple sufficient criterion in andreasfabrice_theorypaper is unlikely to be helpful here. However, it can then be useful to seek to verify the conditions of andreasfabrice_theorypaper for the interplay of the $H^n$ and $S^n$ (as an alternative to AVCI) in order to deduce the J1 stability of the financial gains.
In order to make the above precise, for any given process $Y$, define $\operatorname{Disc}_{\mathbb{P}}(Y):=\{s>0: \mathbb{P}(\Delta Y_s\neq 0)>0\}$ and the number of its $\delta$-increments, $$N_{\delta}^T(Y):= \operatorname{sup}\{m\ge 1\, : \, \exists (t_i)_{i=1}^{2m}\subseteq [0,T], \, t_{i}< t_{i+1}, \text{ such that } |Y_{t_{2i+1}}-Y_{t_{2i}}|>\delta\}.$$ Further, we shall write $(H^n, X^n) \Rightarrow_{f.d.d.\times\text{J1}} (H,X)$ on $D\subseteq [0,\infty)$ if $$ ( H^n_{t_1},...,H^n_{t_\ell}, X^n) \; \Rightarrow \; (H_{t_1},...,H_{t_\ell},X) \qquad \text{ on } \quad (\mathbb{R}^{\ell d}, |\cdot|\, )\times (\mathbf{D}_{\mathbb{R}^{d}}[0,\infty),\, \mathop{}\!\mathrm{d}_{\operatorname{J1}})$$ for all $\ell\ge 1$ and $t_1,...,t_\ell \in D$.
For specific trading strategies, such as the type displayed in andreasfabrice_theorypaper with $J=0$, it turns out that one can also readily verify AVCI, if for example the $g_{n,i}$ are equicontinuous. Indeed, using this property and the J1 tightness criteria given in billingsley, we see that AVCI follows directly from the J1 tightness of the integrators. Furthermore, this then also holds for uncorrelated coupled CTRWs due to (ref).
Whilst not considered in jack, it is worth briefly examining what happens to the stability of the financial gains if the price processes exhibit a correlation structure with respect to the innovations and therefore are weakly M1 convergent (but not J1 convergent). Indeed, the picture changes rather drastically in this case. Albeit in a different setting from the CTRW framework considered here, we note that A{\"i}t-Sahalia & Jacod ait-sahalia_jacod define limiting semimartingale asset prices to be compatible with tick-by-tick models precisely if there is weak M1 convergence.
Suppose the log-prices $R^n$ in (ref) have innovations $\zeta_i$ of the form (ref) with $c_j>0$ for at least two indices so that we have the strict M1 convergence (ref). Up to the appropriate rescaling by a constant, this results in the same form of the limiting asset price $S$ and hence gives rise to the same option pricing theory as in jack. However, the stability of the financial gains is now lost in general. Indeed, for $1<\alpha<2$, andreasfabrice_theorypaper shows that there exist admissible trading strategies, almost surely uniformly vanishing in the limit, for which there is nevertheless a blow-up of the associated financial gain. Intuitively, this lack of stability is coherent with the autoregressive correlation structure of the price innovations, as this in effect opens up the floodgates to anticipating---and hence exploiting---the nature of future price moves (noting also that the total variation of the price processes is not tight).
In the spirit of the Bachelier model, `arithmetic' L\'evy models (as opposed to `geometric'/exponential Lévy models) are also considered in the financial literature. For this final part of the section, we switch to consider `arithmetic' CTRW price processes $X^n$, where the $X^n$ are correlated CTRWs as in (ref) with strict M1 convergence to a (sub-diffusive) subordinated $\alpha$-stable L\'evy process. As above, stability of the financial gains is lost (owing to a lack of good decompositions for the $X^n$), but the particular structure of having the $X^n$ as integrators nevertheless allows us to derive simple and intuitive sufficient criteria for recovering stability. For example, we can restrict the class of admissible trading strategies $H^n$ in such a way that, at any time, they are independent of the lagged price innovations that are part of determining the future price innovations: that is, if the lagged dependence of $X^n$ stretches over $\mathcal J+1$ price moves, then the portfolio at time $t$ is based solely on decisions made at least $\mathcal J+1$ prices moves ago. As $n\rightarrow \infty$, this window converges to zero, so, asymptotically, it is a mild restriction on how fast an investor can act relative to how the market moves.
To be specific, a class of trading strategies $H^n$ that are admissible in the sense of the above discussion are processes of the form\\[-2ex] \[\label{eq:Hn_fin_application} H^n_t \; := \; \sum_{i=1}^{\infty} \, g_{n}(t^n_i,X^n_{ \Xi^n(t^n_i)}) \, \operatorname{\mathbf{1}}_{[t^n_i, \, t^n_{i+1})}(t) \] with $\Xi^n(t^n_i):= \operatorname{max}\{\sigma^n_{k-\mathcal J-1} \, : \, k\ge 1 \text{ with } k-\mathcal J-1\ge 1, \, \sigma^n_k\le t^n_i\}$, where $\sigma^n_k=L_k/n$ is the $k$-th jump time of $X^n$, and where $t^n_i\uparrow \infty$ as $i \to \infty$, $\operatorname{sup}_{i}|t^n_{i+1}-t^n_i| \to 0$ as $n\to \infty$ (we set $X^n_{\operatorname{max}\{\emptyset\}}:=0$). If the $g_{n}$ are sufficiently well-behaved, e.g. continuous and uniformly converging to some continuous, non-decreasing map $g$, then it is not hard to deduce the weak convergence of the $H^n$ on the M1 space from that same convergence of the $X^n$. In addition, we take the i.i.d. innovations of the CTRWs $X^n$ to be centered for $1<\alpha<2$ and their common law to be symmetric if $\alpha=1$. The particular form of the $H^n$ directly implies andreasfabrice and since (ref) is trivially satisfied, on behalf of andreasfabrice we obtain the convergence $(X^n, \int_0^\bullet H^n_{s-} \mathop{}\!\mathrm{d} X^n_s) \Rightarrow (X, \int_0^\bullet H_{s-} \mathop{}\!\mathrm{d} X_s)$ in M1 with $X=(\sum_{j=0}^\mathcal{J} c_j)Z_{D^{-1}}$ and $H_t=g(t,X_{t-})$.
Alternatively, there is another intuitive approach to recovering the stability of the financial gains in the framework of finitely lagged CTRWs (i.e., $c_j=0$ for all $j> K$, for some $K$). Indeed, one can require the trading strategies $H^n$ to exhibit sufficient inertia in a continuous fashion, thus making them `too gradual' in their reaction to critical new price moves. More details on this particular approach can be found in andreasfabrice.
Consider an insurance company whose total cash flow of premiums net of claims is described by a stochastic process $P_t$. Assume further that this company is investing the entirety of its current reserves, denoted by $U_t$, in a risky asset (or index fund) with price $S_t$ at time $t$. Then, the reserves will evolve according to the SDE
Following Paulsen Paulsen_AAP and Kalashnikov & Norberg Kalashnikov_Norberg, consider two independent L\'evy processes $R$ and $P$, where $R=\log (S)$, and note that the solution to (ref) is then given by
As in the previous section, motivated by a CTRW model, one could also consider modelling the log-return $R$ by a time-changed L\'evy process. Similarly, we refer to Constantinescu, Ramirez & Zhu constantinescu for examples of CTRWs with non-exponential waiting times in the modelling of claims, which could lead to $P$ being modelled by a time-changed L\`evy process (possibly with an additional non-time-changed component for the premiums, which we leave out here for simplicity of notation). Maintaining the independence of $P$ and $R$ (due to independence of the financial and insurance risk), one still has $[P,R]=0$, so the representation (ref) continues to hold.
Now suppose there is an underlying CTRW model, where the log-returns $R^{n}$ and the net cash flows $P^{n}$ are given by moving averages or uncoupled CTRWs (ref), either uncorrelated or correlated, e.g. to capture large losses followed by further large losses. As before, we take $R^n$ and $P^n$ to be independent. Allowing for heavy-tailed waiting times and heavy-tailed innovations, the usual assumptions (see Section (ref)) give that $R^n$ and $P^n$ will be weakly convergent, in the M1 or J1 topology, to a pair of independent, possibly subordinated stable Lévy processes. We denote these limits by $P$ and $R$, in line with the above. Furthermore, for each $n\geq 0$, we write $S^n:=\operatorname{exp}(R^n)$ for the price process and note that the corresponding reserves are given by \[ U^n_t= e^{R^n_t}\Bigl(U_0 + \int_0^\bullet e^{-R^n_{s-}} \mathop{}\!\mathrm{d} P^n_s\Bigr),\quad t\geq0. \]
The following result makes precise the realisation of (ref)-(ref) as a limit of the CTRW models just described. We note that a similar result for limiting processes $R$ and $P$ given by Brownian motion with drift was established in Paulsen_Gjessing_Advances.
\noindentAcknowledgments. We thank Lorenzo Torricelli for insightful discussions concerning the contents of Section (ref). The research of FW was funded by the EPSRC grant EP/S023925/1.
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