Mon travail de thèse est composé de deux parties bien distinctes, la première partie est consacrée à l’analysestochastique en temps discret des marches aléatoires obtuses quant à la deuxième partie, elle est liée aux probabili-tés libres. Dans la première partie, on donne une construction des intégrales stochastiques itérées par rapport à unefamille de martingales normales d-dimentionelles. Celle-ci permet d’étudier la propriété de représentation chaotiqueen temps discret et mène à une construction des opérateurs gradient et divergence sur les chaos de Wiener correspon-dant. [...] d’une EDP non linéaire alors que la deuxième est de nature combinatoire.Dans un second temps, on a revisité la description de la mesure spectrale de la partie radiale du mouvement Browniensur Gl(d,C) quand d ! +¥. Biane a démontré que cette mesure est absolument continue par rapport à la mesurede Lebesgue et que son support est compact dans R+. Notre contribution consiste à redémontrer le résultat de Bianeen partant d’une représentation intégrale de la suite des moments sur une courbe de Jordon autour de l’origine etmoyennant des outils simples de l’analyse réelle et complexe. / My PhD work is composed of two parts, the first part is dedicated to the discrete-time stochastic analysis for obtuse random walks as to the second part, it is linked to free probability. In the first part, we present a construction of the stochastic integral of predictable square-integrable processes and the associated multiple stochastic integrals ofsymmetric functions on Nn (n_1), with respect to a normal martingale.[...] In a second step, we revisited thedescription of the marginal distribution of the Brownian motion on the large-size complex linear group. Precisely, let (Z(d)t )t_0 be a Brownian motion on GL(d,C) and consider nt the limit as d !¥ of the distribution of (Z(d)t/d)⋆Z(d)t/d with respect to E×tr.

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在WTO成立，各國國際化程度日益提高的同時，企業與個人進行跨國投資的情形也愈來愈普遍，跨國投資除了要考慮標的資產之報酬與波動性之外，尚須考量匯率變動所產生之風險與不確定性。當某一國外資產具有正向預期報酬率的同時，實現後的報酬率卻又不一定為正，正是因為匯率波動所產生的影響。又，傳統財務理論告訴我們，藉由增加投資組合中所有非完全正相關的資產個數可以有效的降低投資組合的非系統風險，因此投資人在進行投資時往往採用建構投資組合的方式取代持有少數資產的型態。然而，在建構跨通貨避險投資組合時，若是對於投資組合中的各項資產與外幣分別進行避險（分別利用衍生性商品避險），往往是費時、費力又不具有效率。因此，對於整個投資組合進行避險反而是一個比較好的方法，當投資組合價值發生變動時，可以即時對於各項資產部位與外幣分別做調整，遠較於對個別資產進行避險來的方便、快速且有效。 / In most cases, investment is made of building a portfolio rather than single asset. Therefore, it is necessary to develop techniques of valuing portfolio derivatives. Moreover, we consider a cross-currency portfolio that account for currency and interest rate risk. As interest rate is stochastic, we use Heath-Jarrow Morton (HJM) Approach to describe its dynamics. Applying Vorst (1992); Geman, Karoui and Rochet(1995), we derive the approximated close-form of the cross-currency portfolio option. In HJM Approach, it is difficult to acquire hedge ratios of options. We apply another method to build a hedging portfolio. Then, we perform numerical simulations to test its hedging efficiency and sensitivity with respect to different variables.

有鑑於附有最低保證給付投資型保險期末現金流量與選擇權如出一轍，是以應用平賭訂價理論(The Martingale Pricing Method)嵌入HJM利率模型，對隨機利率下附有最低保證給付投資型保險進行評價。並對繳費方式與利率型態兩議題所構成四種類型附有最低保證給付投資型保險作實地數據模擬與評價，以及敏感度分析。 研究結果可以歸納為四點結論。 (1) 單就附有最低保證給付投資型保險簡化版（忽略期中死亡理賠與期滿生存機率）而言： 可視為是最低保證給付折現與以之為履約價的買權組合。因此，當影響因子僅與買權有相關性時，附有最低保證給付投資型保險與理論買權的敏感度分析結果，如出一轍。連動標的期初價格與波動度變動於附有最低保證給付投資型保險影響便是實證。 (2) 延續上點論述衍生： 當影響因子同時對買權與附有最低保證給付折現具有相關性時，由於買權佔整個保險價值比重過低，是以主要影響力皆來自附有最低保證給付的變動。附有最低保證給付與固定利率折現因子變動對於保險價值影響，即反應此結果。 (3) 分別就繳費方式不同下，投保年齡與投保期限變動對於附有最低保證給付投資保險的影響而言： 躉繳型繳費方式下，由第二點結論可得，投保期限越長保費越低，是以當投保年齡越大，期中死亡率提高，且期間短的保費較高的情況下，投保年齡變動對於附有最低保證給付投資型保險影響為正向；分期繳型繳費方式下，由於條款設定不同，無法與躉繳型一概而論，反映在投保期間越長保單價值與保費皆增加，但若是比較其增加的幅度（二階條件小於零）逐漸減少，倒是與躉繳型投資保險投保期間與保費關係意思相同，只是呈現方式不同。分期繳型投資型保險保單價值與投保年齡關係，從投保期限與保費關係以及高年齡層死亡率較高，可以得知，隨著投保年齡的增加，分期繳型投資保險中因為死亡理賠的現金流量產生機會提高，而此部分期間短保單價值較低，是以投保年齡與保單價值呈現反比關係，但是保單價值平準化後的保費，源於平準因子每期存活率因投保年齡增加而減少，造成投保年齡越高，保費也越高。 (4) 就性別而言： 躉繳型附有最低保證給付投資保險，由於女性相較於男性死亡率較低，容易取得期間較長的期滿保證金，而此部分價值較低，是以女生保費較男生便宜；分期繳型附有最低保證給付投資保險，則是相反的表現，由於此部分價值較高，是以女性的保險價值高於男性，同時因女性平準因子中的存活率也比男性高，是以每期所要繳交的保費也比男性低廉。 (5) 就利率型態而言： 隨機利率下躉繳型投資型保險與固定利率下躉繳型投資保險相較，便宜許多，主要是因為利率型態為隨機，且期初利率期間結構打破水平狀態的假設，真實反應正常期初利率期間結構(Normal Interest Rate Term Structure)，是以評價出的保費較固定利率型態下的保費低廉，甚至於分期繳型附有最低保證給付投資保險，在隨機利率下，隨著投保期限增加，保費反而下降。

The purpose of this thesis is to use the martingale approach to solve dynamic international portfolio problems. This thesis consists of three essays in dynamic international portfolio allocation. In demonstrating that foreign consumption plays an important role in international portfolio allocations, in Chapter 2, we present the first essay where we provide the optimal consumption plan and portfolio allocation for a representative investor with continuoustime and complete market assumptions in a simple two-country setting. Due to the demand for foreign consumption, the optimal portfolio allocation requires suitable foreign bonds to hedge against the changes in the foreign investment opportunity set and the exchange rate. The empirical results not only show that the optimal portfolio allocation with domestic and foreign consumption is different from that without consumption or with domestic consumption only, but also demonstrate the need for the foreign bonds to hedge against the change in the exchange rate risk. We present the second essay in which we extend the research of the investor's portfolio allocation problem into a continuous dynamical international market where the investment barrier of international portfolio exists. With deterministic market prices of risks, CRRA utility function and the existence of a simple investment barrier, the investor optimally hedges against the investment opportunity by allocating funds into three portfolios which are constructed by unconstrained bank accounts, equities and bonds. The first portfolio is the so called mean-variance portfolio, the second is the hedge portfolio, and the third is the synthetic portfolio which mimics the expected excess return of the constrained security in foreign country. This issue displays in Chapter 3. The third essay is presented in Chapter 4. Here we develop a continuous-time intertemporal portfolio allocation model in an international mildly segmented market. With deterministic market prices of risks and CRRA utility function, the domestic investor in the segmented market optimally hedges against the stochastic interest rates by allocating funds into two portfolios. The restricted mean-variance portfolio is derived from the traditional mean-variance portfolio without foreign constrained securities. The hedge portfolio is comprised of domestic bonds with a specific horizon for hedging against the change in the domestic interest rate. The numerical results indicate that when the volatility of the stochastic discount factor increases due to the less diversification caused by market segmentation, the less risk-averse investor benefits accordingly. Chapter 5 summarizes the main findings of the three studies and concludes the thesis by suggesting some future research venues related the current subject. / The purpose of this thesis is to use the martingale approach to solve dynamic international portfolio problems. This thesis consists of three essays in dynamic international portfolio allocation. In demonstrating that foreign consumption plays an important role in international portfolio allocations, in Chapter 2, we present the first essay where we provide the optimal consumption plan and portfolio allocation for a representative investor with continuoustime and complete market assumptions in a simple two-country setting. Due to the demand for foreign consumption, the optimal portfolio allocation requires suitable foreign bonds to hedge against the changes in the foreign investment opportunity set and the exchange rate. The empirical results not only show that the optimal portfolio allocation with domestic and foreign consumption is different from that without consumption or with domestic consumption only, but also demonstrate the need for the foreign bonds to hedge against the change in the exchange rate risk. We present the second essay in which we extend the research of the investor's portfolio allocation problem into a continuous dynamical international market where the investment barrier of international portfolio exists. With deterministic market prices of risks, CRRA utility function and the existence of a simple investment barrier, the investor optimally hedges against the investment opportunity by allocating funds into three portfolios which are constructed by unconstrained bank accounts, equities and bonds. The first portfolio is the so called mean-variance portfolio, the second is the hedge portfolio, and the third is the synthetic portfolio which mimics the expected excess return of the constrained security in foreign country. This issue displays in Chapter 3. The third essay is presented in Chapter 4. Here we develop a continuous-time intertemporal portfolio allocation model in an international mildly segmented market. With deterministic market prices of risks and CRRA utility function, the domestic investor in the segmented market optimally hedges against the stochastic interest rates by allocating funds into two portfolios. The restricted mean-variance portfolio is derived from the traditional mean-variance portfolio without foreign constrained securities. The hedge portfolio is comprised of domestic bonds with a specific horizon for hedging against the change in the domestic interest rate. The numerical results indicate that when the volatility of the stochastic discount factor increases due to the less diversification caused by market segmentation, the less risk-averse investor benefits accordingly. Chapter 5 summarizes the main findings of the three studies and concludes the thesis by suggesting some future research venues related the current subject.

本文研究保險人於匯率風險下之最適跨期投資組合。隨著資本市場全球化發展，從事國外投資受到匯率風險之影響加劇。本研究提出動態投資組合模型，針對壽險業之利變型商品，分析保險人於匯率風險之下之最適跨期投資組合。考慮資產集合包含本國現金、本國指數型股票基金、外國現金和外國指數型股票基金四項標的。本文研究方法主要以Cox & Huang(1989, 1991)平賭理論處理最適投資議題，將多期問題變為單期，求得保險人之最適投資組合。最後本文針對不同的匯率走勢與匯率風險波動度，利用電腦模擬，觀察不同風險趨避程度保險人之投資組合變化。 本文結果歸納如下： 1. 於風險市場價值、波動度和國內外無風險短期利率為定值下，保險人最適組合分別是擁有固定比例本國股票部位，外國股票部位則與匯率走勢呈負相關，而本國現金部位與外國現金部位呈現相反趨勢。發現匯率增量趨勢與外國現金帳戶、外國指數型股票基金和本國現金部位趨勢相同。 2. 匯率風險將影響保險人持有外國資產意願。若匯率風險波動度由0.1提高至0.3，則外國現金部位之最大值會從6.23下降到0.66。而外國股票持有部位於短期會增加，但隨著投資期限增加而逐漸遞減。同時短期增加之幅度小於外國現金減少之部分。整體而言，持有外國資產比例隨匯率風險波動度變大而遞減。 關鍵字：匯率風險、跨期投資組合、平賭理論、風險波動度、電腦模擬

Cette thèse présente quelques résultats de la théorie des probabilités quantiques et de l'analyse harmonique non commutative. Elle est constituée de trois parties. La première partie démontre l'analogue non commutatif de l'inégalité de John-Nirenberg et la décomposition atomique pour les martingales non commutatives. Ces résultats étendent et améliorent ceux qui existent déjà, et correspondent exactement à ceux que l'on connaît dans le cas classique. La deuxième partie est consacrée à l'étude des espaces de Hardy à valeurs opérateurs via la méthode d'ondelettes. Il est montré que les espaces de Hardy définis par ondelettes coïncident avec ceux définis par les fonctions carrées de Littlewood-Paley et Lusin. Cette approche est similaire à celle du cas des martingales non commutatives, mais l'utilisation des outils de martingales en analyse harmonique permet une démonstration plus rapide. Dans la troisième partie, nous nous tournons vers des applications de la théorie bien établie des espaces de Hardy, c'est-à-dire des opérateurs de Calderón-Zygmund (OCZ pour abréviation) associés à des noyaux à valeurs matricielles. On obtient des estimations de type faible (1, 1) pour des OCZ dyadiques parfaites et des shifts de Haar annulateurs associés à des noyaux non commutatifs, ainsi que des estimations de type H1 → L1 pour des OCZ arbitaires d'après une décomposition d'une fonction en ligne/colonne. En conjonction avec L∞ → BMO, nous établissons certaines estimations de type Lp. Cette approche s'applique aussi à des paraproduits et des transformées de martingales avec des symboles et coefficients non commutatifs respectivement.

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Wir studieren das Verhalten des Maximums, des Minimums und des Endwerts zeithomogener eindimensionaler Diffusionen auf endlichen Zeitintervallen. Zuerst beweisen wir mit Hilfe des Malliavin-Kalküls ein Existenzresultat für die gemeinsamen Dichten. Außerdem leiten wir Entwicklungen der gemeinsamen Momente des Tripels (H,L,X) zur Zeit Delta bzgl. Delta her. Dabei steht X für die zugrundeliegende Diffusion, und H und L bezeichnen ihr fortlaufendes Maximum bzw. Minimum. Ein erster Ansatz, der vollständig auf den elementaren Abschätzungen der Doob’schen und der Cauchy-Schwarz’schen Ungleichung beruht, liefert eine Entwicklung bis zur Ordnung 2 bzgl. der Wurzel der Zeitvariablen Delta. Ein komplexerer Ansatz benutzt Partielle-Differentialgleichungstechniken, um eine Entwicklung der einseitigen Austrittswahrscheinlichkeit für gepinnte Diffusionen zu bestimmen. Da eine Entwicklung der Übergangsdichten von Diffusionen bekannt ist, erhält man eine vollständige Entwicklung der gemeinsamen Wahrscheinlichkeit von (H,X) bzgl. Delta. Die entwickelten Verteilungseigenschaften erlauben es uns, eine Theorie für Martingalschätzfunktionen, die aus wertebereich-basierten Daten konstruiert werden, in einem parameterisierten Diffusionsmodell, herzuleiten. Ein Small-Delta-Optimalitätsansatz, der die approximierten Momente benutzt, liefert eine Vereinfachung der vergleichsweise komplizierten Schätzprozedur und wir erhalten asymptotische Optimalitätsresultate für gegen 0 gehende Sampling-Frequenz. Beim Schätzen des Drift-Koeffizienten ist der wertebereich-basierte Ansatz der Methode, die auf equidistanten Beobachtungen der Diffusion beruht, nicht überlegen. Der Effizienzgewinn im Fall des Schätzens des Diffusionskoeffizienten ist hingegen enorm. Die Maxima und Minima in die Analyse miteinzubeziehen senkt die Varianz des Schätzers für den Parameter in diesem Szenario erheblich. / We study the behavior of the maximum, the minimum and the terminal value of time-homogeneous one-dimensional diffusions on finite time intervals. To begin with, we prove an existence result for the joint density by means of Malliavin calculus. Moreover, we derive expansions for the joint moments of the triplet (H,L,X) at time Delta w.r.t. Delta. Here, X stands for the underlying diffusion whereas H and L denote its running maximum and its running minimum, respectively. In a first approach that entirely relies on elementary estimates, such as Doob’s inequality and Cauchy-Schwarz’ inequality, we derive an expansion w.r.t. the square root of the time parameter Delta including powers of 2. A more sophisticated ansatz uses partial differential equation techniques to determine an expansion of the one-barrier hitting time probability for pinned diffusions. For an expansion of the transition density of diffusions is known, one obtains an overall expansion of the joint probability of (H,X) w.r.t. Delta. The developed distributional properties enable us to establish a theory for martingale estimating functions constructed from range-based data in a parameterized diffusion model. A small-Delta-optimality approach, that uses the approximated moments, yields a simplification of the relatively complicated estimating procedure and we obtain asymptotic optimality results when the sampling frequency Delta tends to 0. When it comes to estimating the drift coefficient the range-based method is not superior to the method relying on equidistant observations of the underlying diffusion alone. However, there is an enormous gain in efficiency at the estimation for the diffusion coefficient. Incorporating the maximum and the minimum into the analysis significantly lowers the asymptotic variance of the estimators for the parameter in this scenario.

運用傳統再保險契約移轉風險受限於承保能量的逐年波動，尤其自90年代起，全球巨災頻繁，保險人損失巨幅增加，承保能量急遽萎縮，基於巨災市場之資金需求，再保險轉向資本市場，預期將巨災風險移轉至投資人，促成保險衍生性金融商品之創新，本研究針對佔有顯著交易量的巨災風險債券進行分析，基於Cummins和Geman (1995)所建構巨災累積損失模型，引用Duffie 與Singleton (1999)於違約債券的計價模式，將折現利率表示為短期利率加上事故發生率及預期損失比例之乘積，並將債券期間延長至多年期，以符合市場承保的需求，應用市場無套利假設及平賭測度計價的方法計算合理的市場價值，巨災損失過程將分成損失發展期與損失確定期，以卜瓦松過程表示巨災發生頻率，並利用台灣巨災經驗資料建立合適之損失幅度模型，最後以蒙地卡羅方法針對三種不同型態的巨災風險債券試算合理價值，並具體結論所得的數值結果與後續之研究建議。 / Using traditional reinsurance treaties to transfer insurance risks are restrained due to the volatility of the underwriting capacity annually. Catastrophe risks have substantially increased since the early 1990s and have directly resulted significant claim losses for the insurers. Hence the insurers are pursuing the financial capacities from the capital market. Transferring the catastrophe risks to the investor have stimulated the financial innovation for the insurance industry. In this study, pricing issues for the heavily traded catastrophe risk bonds (CAT-bond) are investigated. The aggregated catastrophe loss model in Cummins and Geman (1995) are adopted. While the financial techniques in valuing the defaultable bonds in Duffie and Singleton (1999) are employed to determine the fair prices incorporating the claim hazard rates and the loss severity. The duration of the CAT-bonds is extended from single year to multiple years in order to meet the demand from the reinsurance market. Non- arbitrage theory and martingale measures are employed to determine their fair market values. The contract term of the CAT-bonds is divided into the loss period and the development period. The frequency of the catastrophe risk is modeled through the Poisson process. Taiwan catastrophe loss experiences are examined to build the plausible loss severity model. Three distant types of CAT-bonds are analyzed through Monte Carlo method for illustrations. This paper concludes with remarks regarding some pricing issues of CAT-bonds.