We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by the asset value reaching a lower boundary. We prove that if our volatility models are picked from a class of mean-reverting diffusions, the system converges as the portfolio becomes large and, when the vol-of-vol function satisfies certain regularity and boundedness conditions, the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space. The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. In contrast to the CIR volatility setting covered by the existing literature, our results hold even when the systemic Brownian motions are taken to be correlated.

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). The goal of this work is twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver $A$. Second, we study penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a constrained optimal stopping problem or a constrained Dynkin game in which the set of minimizer’s admissible exercise times is constrained to the right support of the measure generated by $A$.

In the context of risk measures, the capital allocation problem is widely studied in the literature where different approaches have been developed, also in connection with cooperative game theory and systemic risk. Although static capital allocation rules have been extensively studied in the recent years, only few works deal with dynamic capital allocations and its relation with BSDEs. Moreover, all those works only examine the case of an underneath risk measure satisfying cash-additivity and, moreover, a large part of them focuses on the specific case of the gradient allocation where Gateaux differentiability is assumed.

The main goal of this paper is, instead, to study general dynamic capital allocations associated to cash-subadditive risk measures, generalizing the approaches already existing in the literature and motivated by the presence of (ambiguity on) interest rates. Starting from an axiomatic approach, we then focus on the case where the underlying risk measures are induced by BSDEs whose drivers depend also on the y-variable. In this setting, we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations (BSVIEs).

Via a forward SDE solution $\left(k_{t}, t \geq 0\right)$ that captures money supply dynamics, a macroeconomic model known as the monetary model generates a backward exchange rate process $\left(y_{t}, t \geq 0\right)$. For any $t \geq 0$, $y_{t}=k_{t}+\alpha^{-1} \mu_{t}$ where $\left(\mu_{t}, t \geq 0\right)$ is a backward process and $\alpha>0$ is a constant. Thus, $\left(y_{t}, t \geq 0\right)$ does not satisfy a conventional BSDE. Our paper proves $\left(y_{t}, t \geq 0\right)$ is a continuous semimartingale when restrictions on the SDE for $\left(k_{t}, t \geq 0\right)$ capture anti-inflationary initiatives. This new result in economic dynamics does not require the filtration to be the Brownian filtration.

In this paper, we focus on mean-field linear-quadratic games for stochastic large-population systems with time delays. The $\epsilon$-Nash equilibrium for decentralized strategies in linear-quadratic games is derived via the consistency condition. By means of variational analysis, the system of consistency conditions can be expressed by forward-backward stochastic differential equations. Numerical examples illustrate the sensitivity of solutions of advanced backward stochastic differential equations to time delays, the effect of the the population’s collective behaviors, and the consistency of mean-field estimates.

We extend the information-based asset-pricing framework by Brody, Hughston & Macrina to incorporate a stochastic bankruptcy time for the writer of the asset. Our model introduces a non-defaultable cash flow ${Z}_{T}$ to be made at time $T$, alongside the time $\tau $ of a possible bankruptcy of the writer of the asset are in line with the filtration generated by a Brownian random bridge with length $\nu =\tau \wedge T$ and pinning point $\sigma {Z}_{T}$, where $\sigma $ is a constant. Quantities ${Z}_{T}$ and $\tau $ are not necessarily independent. The model does not depend crucially on the interpretation of $\tau $ as a bankruptcy time. We derived the price process of the asset and compute the prices of associated options. The dynamics of the price process satisfy a diffusion equation. Employing the approach of P.-A. Meyer, we provide the explicit computation of the compensator of $\nu $. Leveraging special properties of the bridge process, we also provide the explicit expression of the compensator of ${Z}_{T}error\; "Character\; not\; currently\; supported:\; (FullDesc)"{\mathbb{I}}_{[\nu,+\infty )}$. The resulting conclusion highlights the totally inaccessible property of the stopping time $\nu $. This characteristic is particularly suitable for financial markets where the time of default of a writer cannot be predictable from any other signal in the system until default happens.